Paper Title: Freiman's $3k-4$ Theorem for Function Fields.


§ ABSTRACT 
Freiman's 3k-4 Theorem states that if a subset A of k integers has a Minkowski sum A+A of size at most 3k-4, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a generalisation: it states that if K is a perfect field and if S⊃ K is a vector space of dimension k inside an extension F/K in which K is algebraically closed, and if the K-vector space generated by all products of pairs of elements of S has dimension at most 3k-4, then K(S) is a function field of small genus, and S is of small codimension inside a Riemann-Roch space of K(S).


A method for verifying the generalized Riemann hypothesis
    [
    
=========================================================





§ INTRODUCTION


We are interested in analogues of addition theorems that occur
in field extensions F/ of a base field . If S and T are
finite-dimensional -vector subspaces of F, we denote by ST the
-linear span of the set of all products st, s∈ S, t∈ T.
The general purpose of this area of research is to characterise
subspaces S and T whose product ST has unusually small
dimension: it is naturally inspired by one of the goals of additive
combinatorics which is to characterise subsets A,B of elements of a
group that have sumsets A+B of small cardinality, where A+B
denotes the set of elements a+b, a∈ A, b∈ B.

This line of research was arguably triggered by Hou, Leung and Xiang
when they proved <cit.> an extension field analogue of Kneser's
classical addition Theorem for Abelian groups <cit.>. The
extension field version essentially states that, given a field extension F/, if S and T are
-linear spaces of finite dimension contained in F, then whenever ST< S + T -1, it must be the case that the product space
ST is stabilised by a non-trivial (different from ) subfield of
F. This theorem was initially proved for separable extensions in
<cit.>, and was shown to be true for all extensions in
<cit.>.  Versions of Kneser's Theorem for products of vector
spaces were also shown to hold in other algebras than field extensions
<cit.>.  Many applications of the theory of products of
spaces in the algebra _q^n with componentwise multiplication are
discussed in <cit.> : the latter theory is concerned in particular
with the existence and the construction of products of linear codes
of small dimension, and since many of these codes are constructed by
evaluating functions drawn from a function field over _q, this
makes the study of the dimension of products of spaces in function
fields relevant to this topic as well.

If F/ does not contain a finite extension of , and if
S,T are -subspaces of F such that ST≠ F, then Kneser's
Theorem for extension fields implies that ST≥ S +
T -1. A study of the limiting case when ST= S +  T
-1 was undertaken in <cit.>. A particularly simple case occurs
when  is algebraically closed. One can then easily uncover the
structure of the spaces S and T by considering sets A and B of
discrete valuations of the field elements in S and T and arguing
that their Minkowski sum A+B must satisfy |A+B|=|A|+|B|-1, from
which addition in  tells us that A and B must be arithmetic
progressions with the same difference: It can then be inferred that the spaces S and T
must have bases in geometric progression.  
Switching to the symmetric case S=T for the sake of
simplicity, it becomes then tempting to ask for the structure of
spaces S such that

    S^2 = 2 S -1 +γ
 for increasing values of γ. It was suggested in
<cit.> that one should draw inspiration from what is often known
as Freiman's “3k-4” Theorem <cit.>, (called Freiman's 3k-3
Theorem in <cit.>). Recall that it states that, if A
is a set of integers,
  
    |A+A|=2|A|-1+γ

  implies, when γ≤ |A|-3, that A is included in an
  arithmetic progression of length |A|+γ. Natural analogues of
  such sets A are spaces S that are of codimension at most
  γ inside a space with a basis in geometric
  progression. However, the point was made in <cit.> that other
  examples of spaces satisfying (<ref>) are Riemann-Roch
  spaces Ł(D) of an algebraic curve of genus γ.  Note that a
  space with a basis in geometric progression is a particular instance of a
  Riemann-Roch space in a genus 0 function field.  In the present article
  we show that Riemann-Roch spaces play the role of arithmetic progressions in
  the extension field setting and that the following analogue of
  Freiman's 3k-4 Theorem holds:
  
   
    Let  be a perfect field and F
    be an extension field of  in which  is algebraically
    closed (i.e. any element of F that is algebraic over
     is in ). Let S be a K-subspace of finite dimension in
    F such that ⊂ S and F = K(S). If
    
    S^2 = 2 S -1 +γ  with  
        0≤γ≤ S -3,

    then F has transcendence degree 1 over K and its genus g
    satisfies g≤γ. Furthermore, there exists a Riemann-Roch
    space Ł(D) of F that contains S and such that
    Ł(D) - _ S ≤γ -g.


Theorem <ref> was first conjectured in <cit.> where the
statement was proved for γ≤ 1. Moreover, it was also proved that, under the conditions of the theorem, F must always be of transcendence degree 1 over K. Later on, the conjecture was proved
for γ = 2 under the additional assumption that F =  (x) <cit.>.

The integer γ in (<ref>)
was called the combinatorial genus of the space S because
of the way it relates to the geometric genus g of the associated
function field.  We note that Theorem <ref> is not only a
linear analogue of Freiman's 3k-4 Theorem, but can also be seen a
generalisation since it easily recaptures the original additive
statement. Indeed, if A is a set of integers satisfying
(<ref>) with γ≤ |A|-3, then, supposing without loss
of generality that A⊂ [0,n=max A] and gcd(A)=1, we
may form the vector space S generated by x^a, a∈ A, inside the
rational function field extension (x). We have that S^2 is
generated by x^a, a∈ A+A, and Theorem <ref> therefore
applies. The relevant Riemann-Roch space Ł(D) in the conclusion of
the Theorem can only be Ł(nP_∞), where P_∞ is the place
at infinity and n = max A, and that S is of codimension at most
γ inside Ł(nP_∞) means exactly that A equals the arithmetic
progression [0,n] with at most γ missing elements. 

The way we prove Theorem <ref> borrows from Lev and
Smeliansky's elegant proof of Freiman's additive 3k-4 Theorem
<cit.> (see <cit.> for a concise description). The
idea was to project the set of integers under study onto a finite
quotient /n and apply Kneser's Theorem in this finite group. Our
starting point will be to view F = K(S) as a finite extension
of a transcendental extension K(x) of K, for a suitably chosen element
x∈ S. The goal will then be to apply the extension field version
of Kneser's Theorem to K(S)/K(x) and extract non-trivial information
from it: there will be significant differences with the additive situation
however. In the next section we give an overview of the paper and the
structure of the proof of Theorem <ref>, which will draw upon
a number of algebraic and combinatorial techniques.

 

§ OVERVIEW



The additive case.
To describe the main steps of the proof of Theorem <ref>, it
is useful to have in mind the template for Lev and Smeliansky's proof
of the original 3k-4 Theorem. One starts with a set of integers A
satisfying (<ref>) 

    |A+A|=2|A|-1+γ

with min A = 0, max A=n and
gcd(A)=1: the goal is to prove that |A|≥
n+1-γ. To this end one reduces the set A to the set
A of integers modulo n, so that we have
|A|=|A|-1. Since the sets A and A+n have only n as a
common element, and since both reduce to A modulo n, we
have that

    |A+A|≤ |A|+γ≤ 2|A|-2.

Now Kneser's Theorem tells us that there exists a subgroup H of /n
such that A+A=A+A+H and |A+A|≥ 2|A|-|H|
(with (<ref>) therefore implying that |H|>1). It could happen that
A + A = /n, in which case we are done: indeed, the left-hand
inequality of (<ref>) entails precisely that
|A| ≥ n - γ and hence |A| ≥ n+1 -
γ. 
The rest of the proof consists of showing that H≠/n cannot happen.
This is achieved by partitioning A into sets that represent cosets modulo H:
by studying the behaviour of sums of these sets, one finds elements of A+A that are equal modulo H and counting them contradicts the hypothesis γ≤ |A|-3 in (<ref>). The argument is detailed concisely in <cit.>.

The function field setting.
It will be shown that under the hypothesis of the Theorem,
 the field F must have transcendence degree 1 so that it is a function field. It is then natural enough to consider the smallest Riemann-Roch space Ł(D) containing S as the analogue of the interval [0,n] in the additive case. The divisor D is therefore the least sum of all the places P, counted with suitable multiplicities, for which there exist functions in S with poles at P. We then identify a function x in S whose pole divisor is exactly D. Next, we augment the space S by considering its linear span S=K(x)S over K(x). The choice of x will ensure that S∩ xS=Kx so that _K(S+xS)=2 S -1. This guarantees that

    _K(x)S^2 ≤_K(x)S +γ≤ 2_K(x)S -2

similarly to (<ref>). Viewing now F as an algebraic extension of K(x) (rather than K), 
we can apply Kneser's Theorem for field extensions <cit.> which says that there exists a field L,
K(x)⊂ L⊂ F, such that S^2=LS^2 and _K(x)S^2≥ 2_K(x)S-_K(x)L. Since 2_K(x)S-_K(x)L≤ 2_K(x)S -2 we have that the field L must be strictly larger than K(x).
It will not be difficult to obtain that if L=F, then, somewhat similarly to the additive case, the conclusion of Theorem <ref> holds. This is a consequence of Clifford's Theorem and is detailed in Section <ref>.

It will remain therefore to show that the situation L≠ F cannot occur. At this point, similarities with the additive case break down rather seriously and we are confronted with new challenges. 
Following the additive template would be easier if we could prove the existence of an L-adapted basis of S, i.e. the existence of elements s_i∈ S such that LS decomposes as the direct sum of the Ls_i, and S decomposes as the direct sum of S_i=S∩ Ls_i. We could then study the dimensions of the products S_iS_j without fearing the appearance of unexpected linear relations between these subspaces.
We do not know how to argue for the existence of such a decomposition of S however, but we did manage to obtain a partial result in this direction. We showed that the dimension over K of L∩ S is a non-trivial quantity: this is the object of Section <ref>. The way this is achieved is by studying the evaluations of functions of S on suitably chosen places, and showing that some of these functions must have block-wise constant evaluations, exactly as elements of L do. Galois Theory arguments then enable us to show that these evaluations must come from functions of S that also fall into L. A shorter proof is provided in the case when K has characteristic 0.

Section <ref> switches from the algebraic techniques developed in Section <ref> to a much more combinatorial approach. It uses the result of Section <ref> on S∩ L to decompose S into a direct sum of spaces that is much cruder than what would be provided by a hypothetical L-adapted basis, but which turns out to be sufficient to give an evaluation of _KS^2 that only just contradicts the hypothesis (<ref>) of Theorem <ref>.

A geometric comment.
Using K(x)S as we did may not seem like the most natural approach. 
Since the additive case resorted to a quotient of the ring of integers, a tempting analogue, 
that may seem closer in spirit to the additive proof, would be to consider a subring of F containing S and reduce it modulo some ideal.
In Section <ref> this is exactly what we do: we choose a place O of K(x) which splits totally into a sum Q of places in F and we study S modulo Q. However, we have not seen how to derive a proof of Theorem <ref> using only this approach, even though there exist Kneser-type theorems for these quotients <cit.>.

However, we believe that using (x)S definitely makes sense from a geometric point of view. 
Indeed, the extension F/(x) corresponds to some curve covering
𝒴→𝒳, where 𝒳, 𝒴
have respective function fields (x) and F. The space (x)S can be
interpreted as the restriction of S to the generic fibre of the
covering 𝒴→𝒳, while reducing
S modulo Q corresponds to a restriction to some special fibre.
In the end, proving the result required us to restrict S both to the generic fibre and to some special fibre.

Organisation of the article
In Section <ref> we provide the setup for the proof of Theorem <ref>, introduce the subextension fields of K, K(x)⊂ L⊂ F, and prove that the conclusion of the Theorem holds if L=F. In Section <ref> we study the dimension over K of S∩ L. In Section <ref> we conclude the proof by showing that L≠ F cannot happen.
Both Sections <ref> and <ref> rely on the results of Section <ref>:
Section <ref> relies upon the result of Section <ref> on the dimension of S∩ L, but it is possible to put off its proof and read Section <ref> before Section <ref>, if one so wishes. 
The proof of Theorem <ref> developed through Sections <ref> to <ref> uses the hypothesis that the ground field K is algebraically closed: in Section <ref> we show how to extend the result to perfect fields, yielding Theorem <ref> in full generality.

 


§ SETTING UP THE PROOF OF THEOREM <REF>: THE SUBFIELDS K(X) AND L



Background and Notation
The arithmetic of function fields
Number theoretic background is fairly basic and most of it can be found in Chapters 1 and 3 of <cit.>, from which we borrow notation.

Given a field K,
the function fields over K that we will encounter (see Section <ref> below) will be exclusively function fields in one variable, i.e. finite extensions of the field K(x) of rational functions.


In this setting, a valuation ring of the function field F is a proper subring 
K ⊊𝒪⊊ F that is maximal for inclusion. Such rings are
discrete valuation rings, i.e. they are local rings whose unique maximal
ideal is principal. Following <cit.> and other authors we call places of F the maximal ideals of the discrete valuation rings of F. The induced discrete valuation of a place P is denoted
v_P : ^×→ℤ.
In most of the article, with the notable exception of Section <ref>, the field 
 will be supposed to be algebraically closed, so that we have Ø / P ≃, meaning places have degree 1.
The image of a function f by the evaluation map at P: 𝒪→𝒪 /P, will be denoted by f(P).



A divisor of F is a formal sum n_1 P_1 + ⋯ + n_s P_s
where the n_i's are integers and the P_i's are places of F.
The degree of D = n_1 P_1 + ⋯ n_s P_s is defined as ∑_i n_i when the places P_i are of degree 1.
A divisor D = n_1 P_1 + ⋯ n_s P_s is said to be positive if n_i ≥ 0 for all i. 
We write D ≥ D' to mean that D-D' is positive. Since 
a function f ∈ F^× has non-zero valuation at only a finite number of places,
we can associate to f its principal divisor

    (f)_F ∑_P v_P(f) P,

where the sum is taken over all the places of F. The positive and negative part 
of the above divisor are denoted (f)_F,0 and (f)_F,∞ which means that
(f)_F = (f)_F,0 - (f)_F,∞. To lighten notation, when there is no ambiguity on the ambient field, the subscript F is omitted and we write (f), (f)_0, (f)_∞.
Principal divisors are known
to have degree 0.
Finally, given a divisor D, the Riemann–Roch space associated to D
is defined as

    Ł_F (D) {f ∈ F^× |  (f)+D ≥ 0}∪{0}.

This is a finite dimensional -vector space. We will again omit the subscript
F and write Ł (D) when the ambient field is clear from context.
Finally, to any function field is associated
a non-negative invariant g called its genus and the Riemann–Roch Theorem asserts that

    _Ł(D) ≥ D + 1 - g

with equality when D ≥ 2g-2.




Products of spaces
As mentioned in the introduction, when S,T are K-vector subspaces of an
extension field F of K, we denote by ST the K-vector space generated by
products st of elements of S with elements of T. Note that this is simply the
set of sums of products of elements of S and T. In the particular case when
T=M is itself a subfield of F, MS is simply the M-subspace generated by
S. Also, if S and T are not only K-subspaces of F but are also
M-subspaces, for some intermediate field K⊂ M⊂ F, then we have
M(SF)=(MS)(MT) and we denote this space simply by MST. More generally, the product ST of two K–vector spaces can be defined in any –algebra. In particular, in Section <ref>, products of spaces in the product ring ^n are involved.

We assume the setting of Theorem <ref>. Precisely, let K be a fixed
perfect field and F/K be an extension field such that no element of
F∖ K is algebraic over K. Let S be a finite-dimensional
K-subspace of F, such that K⊂ S and F=K(S), i.e. S generates F
as an extension of K. We denote henceforth the dimension _KS  of S by
k. Finally, we suppose that S has combinatorial genus 0≤γ≤ k-3,
where the combinatorial genus γ is defined to be the integer such that

    S^2=2k-1+γ.



The fields F and K in Theorem <ref>

The transcendence degree of F/K.

As already mentioned in the introduction, given the above assumptions on
K,S,F, it is already proved in <cit.> that the extension F/K
has transcendence degree 1.

Hence, one can assume from now on that F is a one variable function field over K.

The base field K can be assumed to be algebraically closed.
We will prove Theorem <ref> in the special case when K is an
algebraically closed field. 
We will address the issue of the generalisation from an algebraically closed field K
to all perfect fields K in Section <ref>.

From now and till the end of Section <ref>, K is assumed to
be algebraically closed.

Choosing the function x∈ S

Translating S
One easily checks that, since K⊂ S, we have F=K(S)=K(u^-1S) for any
u ∈ S∖{0}, and that proving Theorem <ref> for S is
equivalent to proving it for u^-1S. It will be convenient for us to thus
switch to some multiplicative translate of S.

Specifically, start by fixing an arbitrary place of F that we shall denote by Q_∞. 
The reason for the subscript ∞ will become apparent later
(see Remark <ref>).
Consider a filtered basis f_0, …, f_k-1 of S, 
i.e. a basis of S such that v_Q_∞(f_0) > v_Q_∞(f_1) > ⋯ >
v_Q_∞(f_k-1), where v_Q_∞ is the discrete valuation at Q_∞ (such a basis is easily constructed). 
Replacing if need be S by f_0^-1S
and setting e_i  f_0^-1f_i for any 0 ≤ i <k, we get a filtered basis
e_0, …, e_k-1 with e_0 = 1 and

    0 = v_Q_∞(e_0) > v_Q_∞(e_1) > ⋯ > v_Q_∞(e_k-1).

We hereafter suppose that S admits a basis e_0=1,…,e_k-1 satisfying
(<ref>).

The poles of x and the divisor D.
Note that for almost any place Q of F, we have v_Q(e_i) ≥ 0
for all 0 ≤ i < k. Consequently, the pole locus of S is finite,
i.e.  there is only a finite set of places at which some element
of S∖{0} has a negative valuation. Therefore, there exists
a divisor D such that S ⊂Ł(D) and
S ⊄Ł (D_0) for any D_0 < D. Since 1 ∈ S, the divisor D is positive.

We can think of Ł(D) as the smallest Riemann-Roch space which contains S.


  There exists x ∈ S such that its pole divisor (x)_∞ satisfies
  
    (x)_∞= D.




  If we do not have (x)_∞= D, then there is a function
  y∈ S that has v_Q(y)<v_Q(x) for some place Q in the support
  of D. Since  is infinite, we may add a -multiple
  λ y of y to x to obtain a function x+λ y such
  that v_Q(x+λ y)=v_Q(y) and v_Q'(x+λ y)≤ v_Q'(x) for
  every place Q'≠ Q and supporting D. We then replace x by x+λ y: a finite number of
  iterations of this process leads to an x ∈ S satisfying the
  required property (<ref>).



Note that the previously introduced place Q_∞ of F
is a pole of x and hence this place lies above the place at infinity of K(x).


Choosing x to be separating
In Section <ref> we will need the extension F/ (x) to be separable. 
For this reason we potentially modify the choice of x
while keeping Property (<ref>).


  If F/ (x) is inseparable, there exists x' ∈ S such that F/(x')
  is separable and (x')_∞ = (x)_∞.



  Denote by p the characteristic of . Suppose p> 0 (otherwise
  any extension is separable).  From <cit.>,
  F/ (x) is separable if and only if x ∉ F^p. So if x ∉ F^p ti suffices to choose x'=x. Suppose now that x ∈ F^p. By assumption F = K(S), and hence S cannot be
  contained in F^p. Consequently, there exists y∈ S
  such that y ∉ F^p and for any λ∈,
  x + λ y ∉ F^p.  Since y ∈Ł(D), we have
  (y)_∞≤ (x)_∞ and one observes that for
  almost any λ∈, 
  (x + λ y)_∞ = (x)_∞.  Setting x' = x+λ y
  for such a λ yields the result.


Replacing x by x' thus constructed if need be, we may suppose that 
(x)_∞ = D and that F/K(x) is separable. Furthermore, since
x has been constructed so as to minimise, among functions of S, its valuation at any place,
we have that the Q_∞–filtered basis e_0,… ,e_k-1 satisfies v_Q_∞(e_k-1)=v_Q_∞(x). We may therefore choose to have e_k-1=x without
losing the Q_∞–filtered structure of the e_i.

The zeros of x
    In the sequel we will need one last property of x, namely that all its
    zeros in F be simple. In other words, we want the divisor (x)_0 to be a sum of places all occurring with multiplicity 1. Note that this divisor is nothing but the decomposition in F of the
    single place O of (x) at which x vanishes (the
place O will play a significant role in Section <ref>). Now,
non-simple zeros of x appear in F if and only if the
place O of K(x) ramifies in F. Note that the places of K(x) are the place at infinity and
    the unique zero of (x-a) when a ranges over K.
    Since in a given extension there are only finitely many ramified places, replacing if need be
    x by x-a for some a, one can suppose that the zero of x in (x) is unramified
    in F. Finally, note that this last change of variable has no influence on the pole locus of x
    and does not change the field (x). Hence it can be applied without harming the previous properties of x and its pole locus.



Summarising: from now on, we suppose that

    e_0 = 1,   e_k-1 = x,
         (x)_∞ = D, 
       F/(x) is separable,
       All the zeros of x in F are simple.





Finally, we recall that:

    (x)_∞ =  D= [F:(x)].


Applying Kneser's Theorem for extension fields
Let us recall Kneser's Theorem for extension fields <cit.>.


Let E⊂ M be fields and let A,B be finite-dimensional E-subspaces of
M such that A≠{0},B≠{0}. Then

    _EAB ≥_EA+_EB-_ESt(AB)

where St(AB)={x∈ M, xAB⊂ AB} is the stabiliser of AB in M.

Note that since AB is finite-dimensional over E, so is St(AB), which
therefore must be a field.

We shall apply Theorem <ref> to E=K(x), M=F, and A=B=K(x)S.
Let L=St(K(x)S^2) be the stabiliser of K(x)S^2. We first show that L is
non-trivial, i.e. L≠ K(x).

L is a non-trivial extension of K(x)

Consider the subspace S+xS of S^2. We have that S ∩ xS=Kx. Indeed, an element
f ∈ S ∩ xS is an element of xS and hence satisfies
(f)_∞≥ (x)_∞. On the other hand, as an element of
S it should satisfy (f)_∞≤ (x)_∞.  Therefore,
(f)_∞ = (x)_∞ and then (x^-1f) ≥ 0, i.e. x^-1f has no
poles, meaning  x^-1f ∈ and hence f ∈ Kx.
Consequently,
  
    _(S+xS) = 2k-1.

  Since _KS^2=2k-1+γ, there exists some –subspace T of F satisfying
  
    S^2=(S+xS)+T  and  _ T = γ.

  Now, consider the spaces (x)S and
  (x)S^2. Since all Q_∞-adic valuations of functions in (x)
  are multiples of v_Q_∞(x), we have that
  
    1,e_1,… ,e_k-2

  must be linearly independent over (x), hence
  
    _(x) (x)S=k-1.

  We also have (x)(S+xS)= (x)S, which, together with
  (<ref>) entails
  
    _(x) (x)S^2≤_(x) (x) S+_(x) (x) T≤ k-1+γ.

From (<ref>) and (<ref>) we therefore have

    _ (x) (x)S^2 ≤ k-1+γ   = _(x)(x)S + γ
        < 2 _ (x) (x)S - 1.

since γ≤ k-3. Theorem <ref> therefore implies that
L K(x).





The hypothesis L=F
It could happen that L=F, i.e. that K(x)S^2 is stabilised by F, which
simply means that K(x)S^2=F. We now proceed to show that if this is the case, then the conclusion of Theorem <ref> holds. 
This is captured by the following proposition.

If K(x)S^2=F then the genus g of F satisfies g≤γ and 
 _Ł (D) - _KS ≤γ - g, i.e.  the space S is of
codimension at most
γ-g inside a Riemann-Roch space.


From (<ref>) and (<ref>),
the hypothesis L=F gives 

    D ≤ k-1+γ.

Let us now prove that the genus g of F
satisfies g≤γ. By Clifford's Theorem <cit.>, if D≤ 2g-2, then
we have Ł(D)≤ 1+1/2 D. Since S⊂Ł(D) we have
k≤Ł(D) so that

    2k-2≤ D,

which together with (<ref>) gives k≤γ +1 in
contradiction with the hypothesis k≥γ +3. Therefore,
D≤ 2g-2 cannot hold, and by the Riemann-Roch Theorem D≥ 2g-1 implies Ł(D)= D+1-g. Hence,

    k   ≤Ł(D) =  D+1-g
       ≤ k-1+γ +1-g,

where the last inequality is a consequence of (<ref>).
From this, we obtain

    g≤γ.

Furthermore, Ł(D)= D+1-g and (<ref>) also gives

    _Ł (D) - _KS ≤γ - g.



We have therefore shown that if L=F all conclusions of Theorem <ref> hold. 
It
remains to study the situation when (x)  L  F.
In the following sections we proceed to show that such a situation cannot
happen.

Choosing a basis of LS
We conclude this section by fixing a basis of LS that will play a role in the two subsequent sections.
Let us set ℓ [L: (x)] and κ = _LLS, so that we have
_K(x)LS=κℓ. 



We have (2κ-1)ℓ≤ 2k-4.


Since L is the
stabiliser of LS^2=K(x)S^2, Kneser's Theorem <ref> gives us 
that _K(x)LS^2≥ 2_K(x)LS - _K(x)L=2κℓ-ℓ.
Together with (<ref>) and γ≤ k-3 this gives us the lemma.



 Let us focus on the
place Q_∞ relative to which we have chosen a filtered basis of S. We have v_Q_∞(x)=-N for a certain positive integer N,
and the set of Q_∞–adic valuations of the functions of K(x) is therefore N. The set
of Q_∞–adic valuations of the functions of L is therefore a subgroup of  which
contains N, so it is the subgroup d for some d|N. Since elements of
L of valuations 0,-d,-2d,… ,-(N/d-1)d must be linearly independent over
K(x), we have N/d≤ℓ.
Now in the interval [-N,0] we have N/d+1 integers that are multiples of d,
and exactly N/d representatives of every non-zero class of integers modulo
d. Therefore, the number of integers of [-N,0] that fall into the union of
some κ-1 classes modulo d is bounded from above by (κ-1)ℓ +1.
But Lemma <ref> implies that (κ-1)ℓ +1<k.
Since the set of valuations of the functions of S must have cardinality equal
to k=_KS, it has to include at least κ distinct classes modulo d. Let us choose
κ elements of S, s_1=1,s_2,… ,s_κ that have distinct
valuations v_Q_∞(s_i) modulo d: these elements are necessarily linearly
independent over L. Summarising for future reference:



There exist κ elements s_1=1,s_2,…,s_κ of S that have
valuations v_Q_∞(s_i) at Q_∞ that are distinct modulo d, where
v_Q_∞(L)=d, and make up therefore a basis of LS.





 

§ THE DIMENSION OF S ∩ L


Let τ be the integer defined as:

    τ_(x) LS - _(x)(x)S.

Our objective in this section is to prove the following lower bound on the dimension of S∩ L:


    We have
    
    _K(S∩ L)≥ℓ + 1-τ  or equivalently  _ S∩ L ≥_ S - (κ - 1)ℓ.


Whereas the previous section was mainly focused on the poles of x and the space Ł(D) for D=(x)_∞, in the present section the zeros of x will play a dominant role. Since we have taken care to ensure that every pole of every function of S is also a pole of x, it must be that every zero of x is not a pole of any function of S. It makes sense, therefore, to evaluate the functions of S on the zeros of x and study the K-vector space generated by these evaluations.
In the first part of this section we will prove that the subspace S_0 of those functions of S that evaluate to blockwise constant vectors (to be defined formally below) satisfies the lower bound of Proposition <ref>: this will be given by Proposition <ref>. The second part of the section will then be devoted to proving that this space S_0 must coincide with the space S∩ L, which will yield Proposition <ref>.

The dimension of the space S_0

Decomposing places in extensions
Given a separable extension M' of a function field M over K, and given a place P' of M', there is a unique place P of M defined as P  P' ∩ M, called the
restriction of P' to M. 
According to the representation on Figure <ref>, we follow standard practice and
say that P' is above P and that the latter is below
the former.

In the other direction, starting from a place P of M, there exist finitely many places P_1', …, P_s' of M whose restriction to M equals P. Given an element
t ∈ M of valuation 1 at P, then for any P' above P the ramification index of P' above P is defined as

    e(P'|P)  v_P'(t)

and is independent of the choice of t. We recall the fundamental identity (for K algebraically closed)

    [M' : M] = ∑_P'  above P e(P'|P).



The place O and its decomposition in L and F
According to Section <ref>, the function x has only simple zeros in F or equivalently, the place O  (x)_K(x),0 of (x) does not ramify in F.
Consequently, O is also
unramified in L and decomposes into places
_1, …, _ℓ of L. Next, these places of L all split totally into places Q_11, …, Q_1m, ⋯, Q_ℓ 1, …, Q_ℓ m of F. 
  Denote by  the local ring at O of (x),
  that is to say:
  
    {f(x)/g(x)∈(x)  |  g(0) ≠ 0
        }.

  Then, denote by  the integral closure of  in L and by  the
  integral closure of  in F. The rings  and  are
  semi-local (i.e., have a finite number of maximal
  ideals) and, from <cit.>, they satisfy
  
    = ⋂_1 ≤ i ≤ℓØ__i  and   = ⋂_1 ≤ i ≤ℓ
    1 ≤ j ≤ mØ__ij.

 Moreover, we have
  
    / x ≃,      / x ≃^ℓ  and   / x ≃^n

  where n [F:K(x)] which is equal, from (<ref>), to (x)_F,∞=(x)_F,0.
  Finally,  is a discrete valuation ring, and, from
  <cit.> the semi–local rings  and  are
  principal ideal domains.
  
The situation is summarised in 
Figure <ref>.



The space of evaluations
Note that when regarded as belonging to a subspace of / x≃^n, the
elements of /x are constant vectors and those of /x are
blockwise constant vectors. That is to say, vectors indexed by the
places _ij whose entries are equal on places _i_0 j for
j ∈{1, …, m} and a fixed i_0. They are the vectors of evaluations
of functions of  at places _ij and hence take the same value
at places lying above a given place of L.

We now focus on the functions of S that have blockwise constant vectors in the sense just introduced.
Recall that the integer τ is defined by (<ref>).



  Let
    S_0 {f ∈ S  | ∀ i ∈{1, …, ℓ},    f(_i1) = ⋯ = f(_im)}.
  In words, let S_0 be the subspace of functions of S whose evaluation vectors
are blockwise constant. The dimension of S_0 then satisfies:
  
    _ S_0 ≥ℓ + 1 - τ.




We will prove Proposition <ref> in several steps. Observe first that S_0 contains both 1 and x
and hence has dimension at least 2. Next, we will focus on
the evaluation map itself

    π : ^nf(f(_11), …, f(_ℓ m)),

which, by the Chinese remainder theorem, is equivalent to the canonical map

    ⟶/x.

In the sequel, for an element a or a subspace A of , we
denote respectively by a̅ and A their images by the map π of
(<ref>), or equivalently by the map (<ref>).

Our objective is to evaluate the dimension of  from which the
dimension of S_0 can be easily be deduced as explained by the two
following statements. Notice that Ł(D)⊂, so that
π is well-defined on Ł(D).


  The kernel of the map π_|Ł(D) : Ł(D) ⟶^n is the
  space Ł (D - ∑_i,j Q_ij), which equals the
  one–dimensional space spanned by x.



  Let f ∈Ł(D)∖{0} such that π(f) = 0. By (<ref>)
  f vanishes at all the places Q_ij and hence (f) ≥∑_i,j Q_ij -D.
  Recall that D was defined as D=(x)_∞ and since the zeros of x are the Q_ij, we have

    (x) = (x)_0-(x)_∞ = ∑_i,jQ_ij - D.

  Thus, (f) ≥ (x) and, since principal divisors have
  degree 0, we deduce that (f) = (x) or equivalently (f/x) = 0 and hence that f and x
  are –collinear.



  The spaces  and  have respective –dimensions
  _ S_0 -1 and _ S - 1.



  Both spaces are contained in Ł(D) and contain x.


From now on, our objective will be to derive a lower bound for the dimension of
. For this purpose, we will study the dimensions of  and .


  _ = ℓ



  First observe that  is isomorphic to
  / (∩ x ) =  / x. From <cit.>,
   is an –free module of rank ℓ. Then, by Nakayama's
  Lemma <cit.>, / x is a vector space over / x≃
  of dimension ℓ.

The previous lemma can be made more explicit: indeed, since / x, when regarded as a
subspace of ^n, is made up of blockwise constant vectors, it follows from _K = ℓ that 
is exactly the subspace of blockwise constant vectors of ^n, meaning that it
is spanned by the vectors
  
    v̅_1     =     (1, …, 1, 0, …, 0, ⋯, 0, …, 0);
    v̅_2     =     (0, …, 0, 1, …, 1, ⋯, 0, …, 0);
       ⋮   
    v̅_ℓ    =     (0, …, 0, 0, …, 0, ⋯, 1, …, 1).



  Now that we have the dimension of , observing that  contains , we
  will therefore obtain the dimension of  if we compute the codimension of  in .
  For this, we proceed as follows. We observe that since both S and
   contain 1, then both  and  are contained in
  =, where the product  is taken   in the algebra /x or equivalently in ^n equipped with the component-wise product.


  
    The module S is a free –module whose rank satisfies
    
    Rk_ S ≤_(x)LS = k-1+τ,

    where τ is defined by (<ref>). 

  
    Since S is finite dimensional and  is free and finitely
    generated over  (<cit.>), we deduce that
    S is a finitely generated module over  which is a
    principal ideal domain. Hence, by the structure theorem of
    finitely generated modules over a principal ideal domain, S
    is the direct sum of a free -module and a torsion module.
    However, we have that S ⊆ F and that F, being a field, has no zero
    divisors and is therefore torsion–free as an –module. Therefore, S is torsion–free also and hence is a free finitely generated –module.

    Next, let r be the –rank of the module S and let
    t_1, …, t_r be an –basis of S, we claim that
    t_1, …, t_r are (x)–independent. Indeed, suppose that
    
    f_1 t_1 + ⋯ + f_r t_r = 0

    for some f_1, …, f_r ∈ (x). Then, since  is a
    discrete valuation ring of (x) with local parameter x, for
    some integer j, the functions x^j f_1, …, x^j f_r all lie
    in . Therefore, multiplying (<ref>) by a suitable x^j we get an -linear relation between the t_i's which
    entails that the x^j f_i's are all zero and hence so are the
    f_i's. Thus, the t_i's are (x)–independent in LS and hence
    r ≤_ (x) LS.
  

  
    Actually, pushing further the arguments of the above proof, one can
    easily obtain that S has –rank exactly k-1+τ
    but we do not require this in the sequel.
  

  We now have all we need to prove Proposition <ref>.

  
    Since both  and  contain 1̅, we deduce that
     contains both  and . Since  is exactly the subspace of blockwise vectors of K^n, we have from the definition of S_0 that
    
    = ∩.

    Let
    us estimate the dimension of =.
    The evaluation map
    
    S ⟶ / x

    has x  S in its kernel, inducing a map
    
    S/ x S ⟶ / x

    whose image is nothing but .
    Thus, by the rank nullity theorem,
    
    _≤_ S / x  S

    and, from Nakayama's Lemma <cit.> the dimension of
    the right–hand side is the –rank of S. Hence, from
    Proposition <ref>, we deduce that
    
    _≤ k-1+ τ.

    From Corollary <ref>, _ = k-1.
    Thus, (<ref>) entails that  has codimension
    at most τ in .
    From Lemma <ref>, _ = ℓ
    and hence = ∩ has dimension
    at least ℓ - τ, which, again from Corollary <ref>, yields
    
    _ S_0 ≥ℓ + 1 - τ.

  

The space S_0 equals S ∩ L 
Since the evaluation of elements of S on the places Q_ij is well-defined,
we have that S∩ L⊂, and since we have seen that functions of  have
blockwise constant evaluations, we have that S∩ L⊂ S_0.
Our present objective is now to prove that the converse statement holds:

  S_0  = S ∩ L.

Together with Proposition <ref>, Proposition <ref>
will therefore yield Proposition <ref>.










Let us briefly recall the setting described in Section <ref>. There is a place  above the place
at infinity of (x) for which S has a filtered basis with
decreasing valuations from 0 to -N for some N>0.  Denote by
 the restriction of  to L. The elements of L have a
-adic valuation in d for some integer d or
equivalently, the ramification index e( | ) equals d.

To prove Proposition <ref>, we suppose the existence of u ∈ S_0 ∖ S∩ L and work towards a contradiction.
We will use Galois theory and since F/L has no reason
to be a Galois extension, we introduce  to be the Galois closure of
F/L, that is to say the smallest Galois extension of L containing F.
Since F/(x) is separable, then so is F/L and hence such a
closure exists and is a finite extension of F. We denote by G
the Galois group of the extension F'/L.

Let us observe some first properties of the Galois action of  on
the objects we are dealing with. Recall first a well–known fact
 (see for instance <cit.>). Given f ∈,  a
place of  and σ∈, then

    v_σ()(σ(f)) = v_(f).

Finally, we denote by  the conorm of D in , i.e.,
the pole locus of x when regarded as an element of . In
addition, let  be a place of  above , which is a place
in the support of .


  The divisor  is –invariant and Ł() is
  a linear representation of .



  Since  is the pole locus of x and x ∈ L,
  it is –invariant and hence, so is its pole locus .
  Now, if for some
  f ∈ we have (f) ≥ -, then,
  from (<ref>), we deduce that for any
  σ∈, we have
  (σ (f)) ≥ -σ () = -. Thus, σ leaves
  Ł () globally invariant and the action is linear since σ
  is a field automorphism of .


From now on let u be a fixed, hypothetical, element u ∈ S_0 ∖ S ∩ L.


  There
  exists an additive character χ_u : →,
  satisfying
  
    ∀σ∈,  σ(u) = u + χ_u(σ)x.




  Let σ∈. By definition of S_0, for any
  i ∈{1, …, ℓ}, u has constant evaluation at the places Q_ij that lie above P_i in F.
  Since u∈ F, we also have that u, viewed as a function of F', has constant evaluation at all the places of F' that lie above any given Q_ij. Therefore, u has constant evaluation at all the places of F' that lie above a given place P_i of L.
Since σ acts by permutation
  on the places of F' above _i, the evaluations of σ(u) at the
  places above _i are exactly the same as the evaluations of u.
  Equivalently, σ(u) - u vanishes at any place of F' above the
  _i's. In other words, σ(u)- u vanishes at any zero of x.

  Moreover, since u ∈Ł_F(D) and hence u ∈Ł_F'() when
  regarded as an element of , then from
  Lemma <ref>, σ (u) ∈Ł() and hence
  we have σ(u) - u∈Ł() also. Since  is nothing but the pole locus of x
  when regarded as an element of , we deduce that
  
    (σ (u) - u)_≥ (x)_

  and since principal divisors have degree zero, we deduce that the
  above inequality is an equality. Thus σ (u) - u = λ x
  for some λ∈ which we now denote by χ_u(σ).

  There remains to prove that the map
  σχ_u(σ) is an additive character.
  Let μ∈,
  
    μ∘σ (x)     = μ (u + χ_u(σ)x) 
       = μ(u) + μ(χ_u(σ)x) 
        = u + χ_u(μ)x + μ (χ_u(σ) x)

  Now, since χ_u(σ)x ∈ L, it is fixed by any element of 
  and hence
  
    μ∘σ (x) = u + (χ_u(μ)+χ_u(σ)) x

  and hence χ_u(μ∘σ) = χ_u(μ) + χ_u(σ)
  which means that χ_u is a group morphism.



  Note that if K is of characteristic zero, the proof of Proposition <ref> stops here since
  the additive character χ_u would be a morphism from a finite group
  into a torsion-free group, hence must be trivial, implying u∈ L.



Actually in characteristic zero, Proposition <ref> can be proved without involving Galois action as follows. Given f ∈ S_0, consider t = 1/m_F/L(f). One can prove using
<cit.> that t, when regarded as an element of F is in Ł(D) and that
for any i,j, f(Q_ij) = t(Q_ij). Therefore, from Lemma <ref>,
f = t + λ x for some λ∈ K and since both t and x are in L we deduce
that f ∈ L. Details are left to the reader.



  The extension L(u)/L is Galois, with Galois group / χ_u.



For σ∈ G, by definition of χ_u, we have that σ(u)=u if and only if σ∈χ_u.
Therefore, L(u) is exactly the subfield of  fixed by χ_u.
Moreover,  χ_u is a normal subgroup of , and by Galois
  correspondence its invariant subfield L(u) is Galois over
  L with Galois group / χ_u.


Now, let us consider the places at infinity. Recall that we chose a
pole  of x in F at which x has valuation -N. We also chose a
place  above . Next,  is the restriction of 
(or of ) to L. Finally, we denote by  the restriction of  
to L(u). The situation is summarised in Figure <ref>.


  Recall that elements of L have –adic valuation in d
  which means that the ramification index e( | ) equals d
  as shown on the figure.

  Recall also that u ∈Ł(D), which entails that
  v_(u) ≥ v_(x). When regarded as elements of
  L(u), we deduce that v_(u) ≥ v_(x).
  Note that, 
if the inequality is not strict, then for some choice of δ∈, we have v_(u+δ x) > v_(x).
  Moreover, when replacing u by u' = u + δ x, the character χ_u
  is unchanged. Indeed, for σ∈,
  
    σ (u')     = σ (u + δ x) = σ (u) + σ (δ x_∈ L) = σ (u) + δ x 
        = u + δ x + χ_u(σ)x
        = u' + χ_u(σ) x.

  In short, χ_u = χ_u'. Thus, replacing u by u' if need be, from now on one can assume that
  
    v_(u) > v_(x).


  
    The place  is unramified over .
  

  
    From <cit.>, the decomposition group
    
    D_|{σ∈Gal(L(u)/L)  | σ() = }
 is non-trivial if and only if the place is ramified.
    Let σ be an element of the decomposition group.
    Then, from (<ref>),
    
    v_ (u)     = v_σ ()(σ (u)) 
        = v_(σ (u)) 
       v_ (u)      = v_ (u + χ_u(σ)x),

    the second equality being due to the definition of the decomposition group.
    If χ_u(σ) ≠ 0, then, from (<ref>),
    the right–hand side of (<ref>) is v_(χ_u(σ) x), but then equality with the left-hand side
    immediately contradicts (<ref>) itself.

    Thus, χ_u (σ) = 0.  But from
    Corollary <ref>, the Galois group of L(u)/L is
    / χ_u. Hence, for any 
    σ∈Gal(L(u)/L) ∖{id}, χ_u(σ) ≠ 0.  Thus,
    σ = id and the decomposition group is trivial. Hence
    the result.
  

  Consequently, since
  
    e( | ) = e( | ) e( | ) = d,

  Lemma <ref> yields
  
    e( | ) = d.

  
  
    The –adic valuations of elements of L(u) are in d.
  

  Now we have all we need to finally prove Proposition <ref>.

  
    Let u ∈ S_0 ∖ S ∩ L. Using the L–basis of LS guaranteed by Lemma <ref>,
    and since u ∈ S, there exist ℓ_1, …, ℓ_κ∈ L such that
    
    u     = ℓ_1 + ℓ_2 s_2 + ⋯ + ℓ_κ s_κ
     
          u - ℓ_1     =  ℓ_2 s_2 + ⋯ + ℓ_κ s_κ.

    The left–hand side is in L(u) and, from
    Corollary <ref>, has its –adic valuation in
    d.  The terms ℓ_i s_i on the right–hand side have distinct valuations since, from Lemma <ref>, they must be distinct
    modulo d. Thus, the
    –adic valuation of the right–hand side is that of one of
    its terms, but all of them have a valuation non divisible by d.
    Thus, the only possibility would be that both sides are zero, yielding u = ℓ_1 ∈ L. A contradiction.
  


 

§ FROM PROPOSITION <REF> TO A PROOF OF THEOREM <REF>:
COMBINATORIAL ARGUMENTS


Laying the groundwork

Notice that the hypothesis of Theorem <ref>, namely _KS^2≤
2k-1+γ and γ≤ k-3, where k=_KS, implies

    _KS^2≤ 3k-4

which we will refer to as the Freiman hypothesis (hence the name
“3k-4 Theorem”). 

Recall that κ denotes κ=_LLS so that we have
_K(x)LS=κℓ, and recall from definition (<ref>)

    τ=_K(x)LS-_K(x)K(x)S=κℓ -k+1.


From Theorem <ref> (Kneser for extension fields) we have

    _LLS^2≥ 2κ -1

since the stabiliser of LS^2 is L, by definition of L.
If κ=1 we have LS^2=LS=L and, since S generates F over , we have L=F.
That situation was dealt with in
Section <ref>, so we are now in the setting κ >1. Our goal is to
derive a contradiction with the Freiman hypothesis (<ref>), thus
proving that we can only have κ=1 which will prove
Theorem <ref>. 

The following lemma will be useful:


Let W be a K-subspace of S^2 such that _LLW=w≤ 2κ -1. Then

    _K S^2≥_KW+(2κ-1-w)ℓ.



Let W' be a complementary subspace of W in S^2. Since (x)S^2=LS^2, from (<ref>)
we have _K(x)K(x)S^2≥ (2κ-1)ℓ. Since _LLW=w, we have _K(x)LW=wℓ and _K(x)K(x)W≤ wℓ. Hence, K(x)S^2=K(x)W+K(x)W' implies _K(x)K(x)W'≥ (2κ-1-w)ℓ. Finally,
S^2=W⊕ W' and _KW'≥_K(x)K(x)W' imply therefore the result.

Applying Lemma <ref> to W=S^2∩ LS gives therefore

    _KS^2≥_K(S^2∩ LS) + (κ-1)ℓ.

We have (S^2∩ LS)⊃ (S+xS), and recall from
Section <ref> that _K(S+xS)=2k-1. Inequality
(<ref>) therefore gives us

    _KS^2≥ 2k-1 + (k+τ-1) - ℓ =3k-2 -(ℓ -τ).

This does not contradict the Freiman hypothesis (<ref>), but it is
not too far from it: to obtain a contradiction, what we need to do is to improve
the lower bound (<ref>) by ℓ-τ-1.

We will improve upon (<ref>) 
in two ways. We will use valuation arguments to obtain that (S^2∩ LS) is
potentially strictly larger than S+xS, yielding a larger term than 2k-1 in
(<ref>). We will also need to improve upon the (κ-1)ℓ
estimate in (<ref>). For this we will use the Cauchy-Davenport inequality:


If U and V are K-vector subspaces of F of finite dimension, then 

    _KUV≥_KU+_KV-1.



This is a direct consequence of Theorem <ref>, since F does not
contain finite-dimensional extensions of K.

The strategy will be to look for K-subspaces U,V of S for which
Lemma <ref> will yield a subspace UV of S^2 such that  
_KUV > ℓ_LLUV. Care will be needed to ensure that these “extra
K-dimensions” add up to a sufficiently large improvement of the lower bound
(<ref>).

Proposition <ref> allows us to forget how S_0 was originally defined in Section <ref>, and we may now
use the notation S_0 simply to refer to S∩ L.


Finally, recall from Lemma <ref> that there exist κ elements s_1=1,s_2,…,s_κ of S
whose valuations at Q_∞ lie in distinct classes modulo d. We may suppose
additionally that each v_Q_∞(s_i) is maximal within
its class modulo d. Let these elements s_i of S remain fixed
throughout Section <ref>. 

The decomposition of LS and the induced subspaces of S

The following technical lemma gives us the existence of a special decomposition of the
space LS that will play an important role in bounding from below the dimension
of S^2.
We use bold uppercase letters to denote
vector spaces over L, while K-subspaces of S are denoted with
regular uppercase letters.




There exists an element,
s ∈{s_2, s_3, …, s_κ},
and three L-subspaces 𝐀, 𝐁, 𝐂
of respective L-dimensions a≥ 1,b≥ 1 and c, such that L ⊂𝐀 
and

    LS = 𝐀⊕𝐁⊕𝐂,

that satisfy the following properties:


  *  𝐀s ⊂ LS;

  *  LS ∩𝐁s = {0};

  *  The subspace A = S ∩𝐀
has a dimension _K A satisfying 
    _K A ≤_K S_0 + (a-1)ℓ,
 

  *  The subspace A^+ = S ∩ (𝐀 + 𝐁)
has a dimension satisfying 
    _K A^+≥_K S - cℓ.
 




  We will iteratively construct a sequence of triples
  𝐀_i, 𝐁_i, 𝐂_i satisfying all conditions
  of the lemma except possibly condition <ref>, and show that we must
  eventually arrive at a triple satisfying all conditions, including condition <ref>. 
  The associated element s changes at each iteration,
  though we will keep denoting it by s to lighten notation: this should not generate undue confusion. 
  The L–dimensions of the
  spaces 𝐀_i, 𝐁_i, 𝐂_i will be denoted as
  a_i, b_i, c_i respectively, and the associated subspaces of S
  will be denoted as A_i and A_i^+.

We start by choosing any s ∈{s_2,… s_κ}. 
We then define the L-subspace 𝐀_0 of LS consisting of those 𝐲∈ LS such that
𝐲s ∈ LS. Clearly L⊂𝐀_0; 
note also that LSs ≠ LS, otherwise
LS would be stabilised by s, and L would not be the
stabiliser of LS^2, since s∉L. Therefore, 𝐀_0≠ LS.
For 𝐁_0 we take any subspace of LS such that 
LS = 𝐀_0 ⊕𝐁_0, and we set 𝐂_0 = {0}, so
that we have LS=𝐀_0 ⊕𝐁_0 ⊕𝐂_0, with
a_0≥ 1 and b_0≥ 1.
Note that conditions <ref> and <ref> of the lemma are fulfilled by
construction. Condition <ref> is trivially satisfied since A_0^+ = S.


There remains the issue of condition <ref>: it is either satisfied, and we are done: or it is not, in which case we will look for another triple (𝐀_1,𝐁_1,𝐂_1)
(together with an element s)
satisfying all conditions of the Lemma except possibly the third one. More generally, given 
a triple (𝐀_i,𝐁_i,𝐂_i) satisfying all the requirements except condition <ref>,
we now construct another triple (𝐀_i+1,𝐁_i+1,𝐂_i+1) satisfying all requirements except possibly condition <ref>, and such that 𝐀_i+1⊊𝐀_i.

First observe that we have 

    a_i = _L 𝐀_i≥ 2,

otherwise we would have 𝐀_i = L and
A_i = S_0, satisfying condition <ref>.
Next, we change the current value of s to a value s ∈{s_2, … , s_κ} such
that 𝐀_i s ⊄LS. Such an s exists because
otherwise LS 𝐀_i ⊂ LS and LS would be stabilised by
𝐀_i, implying that the stabiliser of LS^2 would be not L but a
field containing 𝐀_i. We now set
𝐀_i+1 = {𝐚∈𝐀_i, 𝐚s ∈ LS}⊊𝐀_i and take 𝐁_i+1 to be any
complementary subspace of 𝐀_i+1 in 𝐀_i. Finally we set
𝐂_i+1 = 𝐁_i ⊕𝐂_i, so that 
LS=𝐀_i+1⊕𝐁_i+1⊕𝐂_i+1.

Note that L⊂𝐀_i+1⊊𝐀_i, and that b_i+1≥ 1.
By definition of A_i+1^+, we have A_i+1^+ = A_i: since A_i does not
satisfy condition <ref>, we have

    _K A_i+1^+ > _K S_0 + (a_i - 1)ℓ.

Applying Proposition <ref>, we therefore get

    _K A_i+1^+ > _K S - (κ - a_i)ℓ.

But κ - a_i = b_i + c_i = c_i+1: thus, the new triple
(𝐀_i+1, 𝐁_i+1, 𝐂_i+1) satisfies
condition <ref>. Conditions <ref> and <ref> are naturally satisfied by construction,
therefore we again have a triple satisfying all conditions of the Lemma except possibly for condition <ref>.

As already observed, condition <ref> is
trivially satisfied when 𝐀_i = L. We have therefore constructed a sequence of triples
(𝐀_i,𝐁_i,𝐂_i) that, since the 𝐀_i are strictly decreasing,
must eventually lead to L-spaces 𝐀, 𝐁, 𝐂 and an
element s satisfying all conditions of the lemma.


From now on we let 𝐀, 𝐁, 𝐂 and s ∈ S be fixed
as guaranteed by Lemma <ref>. 

Studying S_0S and S_0s∩ S
Let us write S_0 = S_0' ⊕ S_0”
where S_0' is the subspace of S_0 consisting of the elements y such
that ys ∈ (S + xS), i.e. S_0' = S_0 ∩ s^-1(S + xS), and
S_0” is any complementary subspace of S_0' in S_0. 
Since 1,x∈ S_0 we have S_0S⊃ (S+xS), and we also have S_0S⊃ S_0”s. By definition of S_0” we have S_0”s∩(S+xS)={0}, therefore

    _K S_0S ≥_K(S + xS) + _K S_0”.

Note that 1, x ∈ S_0'. Let T_0 be the subspace of elements of S_0' whose valuation at Q_∞ is larger than -N=v_∞(x). We clearly have S_0'=T_0⊕ Kx, so that _K T_0=_K S_0'-1.



We have T_0s⊂ S.



We have T_0⊂ S_0' and by definition of S_0' we have have therefore T_0s⊂ S+xS.
Suppose there exists λ∈ T_0 such that λ s∉S. 
Then, since λ s∈ S+xS, there must be some y∈ S such that v_Q_∞(λ s)=v_Q_∞(yx), i.e.

    v_Q_∞(λ)+ v_Q_∞(s)=v_Q_∞(y)+v_Q_∞(x).

Since x,λ∈ L, their valuations at Q_∞ are in d, so we have
v_Q_∞(y) = v_Q_∞(s)  d. Now recall from Lemma <ref>
that s is chosen to be one of the elements s_1,… ,s_κ and therefore 
(from the end of Section <ref>)
v_Q_∞(s) is maximal within its class modulo d. In other words we have v_Q_∞(y) ≤ v_Q_∞(s).
But the definition of T_0 implies v_Q_∞(x) < v_Q_∞(λ), so we have a contradiction with (<ref>).


Since A^+⊂ S, Lemma <ref> implies T_0sA^+⊂ S^2, and we have
S^2 ⊃ S_0S + T_0sA^+. 
We will study the sum S_0S + T_0sA^+ and its dimension, aiming to derive a contradiction
with the Freiman hypothesis (<ref>).  More precisely, since A^+⊂𝐀+𝐁, by condition 1 of Lemma <ref> we have A^+s⊂ LS+𝐁s: and since T_0⊂ L and LS+𝐁s is L-linear,
we have

    T_0sA^+ ⊂ LS + 𝐁s,

and finally, since S_0⊂ L, we also have S_0S + T_0sA^+ ⊂ LS + 𝐁s.
Therefore,

    _L L(S_0S + T_0sA^+) ≤κ + b

(where b = _L 𝐁). 
Applying Lemma <ref>, we therefore get:

    _K S^2 ≥_K(S_0S + T_0sA^+) + (κ - 1 - b)ℓ.

We will find a contradiction with the Freiman hypothesis as soon as we
show that

    _K (S_0S + T_0sA^+) ≥ 3_K S - 3 - (κ - 1 - b)ℓ,

which is now our objective. The dimension of S_0S + T_0sA^+ will be bounded from below by the sum of two terms: 
the dimension of S_0S given by (<ref>) and the contribution of elements of T_0sA^+ that
“fall outside of LS”, in other words, those that have a nonzero
component in 𝐁s.

The lower bound on S_0S+T_0sA^+ that contradicts the Freiman hypothesis

Consider the projection map

    Φ : LS⊕𝐁s𝐁sσ+𝐛s𝐛s

where σ and 𝐛 are arbitrary elements of LS
and 𝐁 respectively. The following lemma is a simple remark, but is a key to the argument to follow.


For any finite-dimensional K-subspace V of LS+𝐁s and any finite-dimensional K-subspace U of
L, we have Φ(UV)=UΦ(V).


This is a simple application of the L-linearity of the map Φ.





All dimensions below are K-dimensions. Let us write

    (S_0S+T_0sA^+)    ≥ S_0S+  T_0sA^+ -(S_0S∩ T_0sA^+)
       ≥ S_0S+  T_0sA^+ - (LS∩ T_0sA^+)

since S_0S⊂ LS. From (<ref>) and the rank-nullity theorem we have 

    T_0sA^+=Φ(T_0sA^+) + (Φ∩ T_0sA^+).

Therefore, since LS=Φ,

    (S_0S+T_0sA^+) ≥ S_0S + Φ(T_0sA^+).

Applying Lemma <ref> we therefore have, since T_0⊂ L,

    (S_0S+T_0sA^+) ≥ S_0S +  T_0Φ(sA^+).

Since A^+s⊂ LS+𝐁s, applying again the rank-nullity theorem,

    Φ(A^+s)   = A^+s - (Φ∩ A^+s)
       = A^+ -  (LS∩ A^+s).

But from the definition of A^+=S∩(𝐀+𝐁) and conditions 1 and 2 of Lemma <ref>,
we have LS∩ A^+s=As. Therefore,

    Φ(A^+s) =  A^+ -  A.

Lemma <ref> now gives us the “Cauchy-Davenport inequality”:

    T_0Φ(sA^+)    ≥ T_0 + Φ(A^+s) - 1
       ≥ T_0 +  A^+- A -1

by the above computation of Φ(A^+s).
Substituting into inequality (<ref>) we therefore have:

    (S_0S + T_0sA^+) ≥ S_0S +  T_0 +  A^+ -  A - 1.

Applying (<ref>) we now obtain:

    (S_0S + T_0sA^+)    ≥(S + xS) +  S_0” +  T_0 +  A^+ -  A - 1 
       = (S + xS) +  S_0” +  S_0' +  A^+ -  A - 2 
       = (S + xS) +  S_0 +  A^+ -  A - 2

where we have used S_0=S_0'⊕ S_0” and T_0= S_0'-1.
Applying conditions <ref> and <ref> from Lemma <ref>, we obtain:

    (S_0S + T_0sA^+)    ≥(S + xS) - 2 +  S - cℓ - (a - 1)ℓ
       ≥ 2 S - 1 - 2 +  S - (κ - 1 - b)ℓ
       = 3 S - 3 - (κ - 1 - b)ℓ

where we have used _LLS=κ=a+b+c.
Finally, substituting into (<ref>) gives the desired contradiction
with the Freiman hypothesis (<ref>). This concludes the proof of Theorem <ref> for algebraically closed fields K.

 

§ WHEN K IS NOT ALGEBRAICALLY CLOSED


Once Theorem <ref> is proved in the special case of an algebraically closed ground field K, there remains to extend it to the more general case a perfect ground field K. Suppose that K is perfect and algebraically closed
in F. The proof to follow is in the same spirit as that of <cit.>.
Denote by  the algebraic closure of K. Since K is algebraically closed in F,
the compositum F'  F is a function field over . Moreover, from <cit.>,
_ T = _K T for any K-subspace T, so that we have

    _ S = _K S

and also 

    _ S^2 = 2 _ S - 1 + γ.

Theorem <ref> for algebraically closed fields therefore applies, so that
F' has genus g ≤γ and the least divisor D' of F' such that 
S ⊂Ł_F' (D') satisfies

    _Ł_F' (D') - _ S ≤γ - g.







From <cit.>, F' has the same genus as F which yields the upper
bound on the genus. Let D be the least divisor of F such that S ⊂Ł_F (D), there remains to evaluate the codimension of S in Ł_F (D). 
Following the lines of <cit.>,
one proves easily that D' is nothing but the conorm (also called the pullback)
of D in F' (see <cit.>) and, from <cit.>,

    _Ł_F'(D') = _KŁ_F (D).

Therefore, (<ref>) and (<ref>) together with (<ref>)
permits to conclude.





Note that the assumption “K is perfect” can be relaxed to “any finite extension of K is primitive”. This is indeed the only requirement for the above argument to hold. Therefore, Theorem <ref> remains true, for example, when K is an algebraic extension of the rational function field 𝔽_q(T).


 


§ ACKNOWLEDGEMENTS

The authors express their deep gratitude to Christine Bachoc and Hugues Randriambololona for inspiring discussions. This work was supported by
the French Agence Nationale de la Recherche project ANR-21-CE39-0009 Barracuda and by Horizon-Europe MSCA-DN project Encode.
  
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 Paper Title: Polyhedral geometry of refined $q,t$-Catalan numbers.








Department of Mathematics, San Francisco State University

<https://matthbeck.github.io/>
mattbeck@sfsu.edu


Department of Mathematics, University of California, Berkeley

<https://math.berkeley.edu/ mhanada>
mhanada@berkeley.edu


Department of Mathematics & Statistics, Pomona College

<https://math.berkeley.edu/ mhlava>
magda.hlavacek@pomona.edu


Department of Mathematics, University of California, Berkeley

<https://math.berkeley.edu/ jlentfer>
jlentfer@berkeley.edu


Department of Mathematics, Harvey Mudd College, <https://math.hmc.edu/arvm>
arvm@hmc.edu


Department of Mathematics, University of Michigan

<https://sites.google.com/view/katie-waddle>
waddle@umich.edu



05A15 (Primary), 52B20 (Secondary)

§ ABSTRACT 
We study a refinement of the q,t-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. 
These refined q,t-Catalan numbers depend on a vector of parameters k⃗ and the classical q,t-Catalan numbers are recovered when k⃗ = (1,…,1).
We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on k⃗-Dyck paths and their associated area and bounce statistics.
Through this polyhedral approach, we recover Xin and Zhang's theorem on
q,t-symmetry of the refined q,t-Catalan numbers in the cases where k⃗
= (k_1,k_2,k_3) and (k,k,k,k), give some extensions, including the case k⃗ = (k,k+m,k+m,k+m), and discuss
relationships to other generalizations of the q,t-Catalan numbers.


Polyhedral geometry of refined q,t-Catalan numbers
    Matthias Beck, Mitsuki Hanada, Max Hlavacek, John Lentfer, 
 Andrés R. Vindas-Meléndez, Katie Waddle 
    30 July 2024
=========================================================================================================







§ INTRODUCTION


The q,t-Catalan numbers are a two-parameter deformation of the well-known Catalan numbers 1/n+12nn. 
They were first introduced by Haiman in <cit.> as the bigraded Hilbert series of the sign representation of a certain S_n-module called the diagonal coinvariants.
An elegant way to write down the q,t-Catalan numbers, due to Garsia and Haglund <cit.>, is

    C_n(q,t) = ∑_D ∈𝒟_n q^(D)  t^(D),

where 𝒟_n is the set of Dyck paths of length n and area and
bounce are certain statistics on Dyck paths; we give details below in Section <ref>.
We can immediately observe the symmetry of q and t from Haiman's original definition of C_n(q,t).
A famous open problem asks for a bijection on Dyck paths that interchanges area and bounce, which would provide a combinatorial proof of q,t-symmetry.
For more on the history and equivalent definitions of q,t-Catalan numbers, see <cit.> or <cit.>.

We study a refinement C_k⃗(q,t) of the q,t-Catalan numbers introduced by Xin and Zhang <cit.>.
Here k⃗ is a vector with positive integer entries that sum to n, and we take the sum in (<ref>) only over those Dyck paths whose north steps are precisely the entries of k⃗.
We recover C_n(q,t) as the special case where k⃗ has dimension n and all of its entries are 1.
It is a natural question to ask in which cases C_k⃗(q,t) is symmetric in q and t. Xin and Zhang proved symmetry when k is two- or three-dimensional, while giving a counterexample in dimension four <cit.>.
They also proved symmetry for the case that k⃗ has four equal entries.
Their proofs are constructive and involved: they compute, via MacMahon's Omega operator, the rational generating function of C_k⃗(q,t), from which the symmetry is plainly visible; for k⃗∈_ >0 ^3, they also give an intricate bijective proof.
One of our goals is to interpret (and re-derive) Xin–Zhang's generating functions

    (x_1, x_2, x_3, q, t) := ∑_k⃗∈_≥ 0 ^3  C_k⃗(q,t)   x_1^ k_1  x_2^ k_2  x_3^ k_3

and

    (x, q, t) := ∑_ k ∈_≥ 0  C_(k,k,k,k)(q,t)   x^k

through the use of polyhedral geometry: we develop polyhedral cones and certain subdivisions which appear organically from the (linear) constraints defining Dyck paths and their area and bounce statistics. 
It turns out these cones are arithmetically nice: their integer lattice-point structure are either unimodular or close to unimodular, and so their generating functions are manageable to compute.

Xin and Zhang were primarily interested in the q,t-symmetry of 

    C_λ(q,t)=∑_μ(k⃗) = λC_k⃗(q,t),

where μ(k⃗) = λ means that rearranging the entries of k⃗ in decreasing order gives the partition λ.
In general, if all C_k⃗(q,t) are q,t-symmetric for a fixed μ(k⃗) = λ, then C_λ(q,t) is q,t-symmetric.
It is also possible for C_λ(q,t) to be q,t-symmetric even if some of the C_k⃗(q,t) summands used in its definition are not; for example, 

    C_λ = (1,1,1,2)(q,t) = C_k⃗ = (1,1,1,2)(q,t) + C_k⃗ = (1,1,2,1)(q,t) + C_k⃗ = (1,2,1,1)(q,t) + C_k⃗ = (2,1,1,1)(q,t)
 is q,t-symmetric, but individually, C_k⃗ = (1,1,2,1)(q,t) and C_k⃗ = (1,2,1,1)(q,t) are not. 
It is unclear how common this symmetry phenomenon is; starting with partitions
λ of length 4 many C_λ are not symmetric in q,t.

After giving background on both q,t-Catalan numbers and the integer-point structure of polyhedral cones in Section <ref>, we compute (x_1, x_2, x_3, q, t) in Section <ref> and (x, q, t) in Section <ref>. 
Naturally, this geometric ansatz can be used in other situations, as we exhibit
this in Section <ref> by computing

    (x, y, q, t) := ∑_ k, m ∈_≥ 0  C_(k,k+m,k+m,k+m)(q,t)   x^k   y^m ,

realizing symmetry again along the way. We conjecture that C_(k,a,…,a)(q,t) is
q,t-symmetric for any positive k and a.
We conclude in Section <ref> with various extensions of our work,
including a proof that 
    C_k⃗ = (k_1,…,k_j,m)(q,t) = C_k⃗ =
    (k_1,…,k_j,l)(q,t)   .

Finally, we discuss the relationship between the refined q,t-Catalan numbers and other generalizations of the q,t-Catalan numbers. 





§ BACKGROUND




 §.§ q,t-Catalan numbers


The Catalan numbers C_n = 1/n+12nn are well known to have numerous combinatorial interpretations (see, for example, <cit.>) and  satisfy the recurrence relation 

    C_n = ∑_k=1^n C_k-1 C_n-k  ,

for n≥ 1, with the initial condition C_0=1.
There is not always a canonical way to create a q-analogue of a combinatorial
sequence, and indeed,
the Catalan numbers have two well-known q-analogues. 
One of them is the Carlitz–Riordan q-Catalan numbers C_n(q), which are defined by the recurrence relation 

    C_n(q) = ∑_k=1^n q^k-1  C_k-1(q)   C_n-k(q)   ,

for n≥ 1, with the initial condition C_0(q) = 1. 
The second is the MacMahon q-Catalan numbers D_n(q), which are
defined via the usual q-integers and q-binomial coefficients by 

    D_n(q) = 1/[n+1]_q2nn_q   .


Based on computational data, Haiman first observed that the bigraded Hilbert series of the sign representation of the diagonal coinvariants was a q,t-deformation of the Catalan numbers <cit.>. 
Haiman first defined the q,t-Catalan numbers by C_n(q,t) := ⟨ e_n, ∇ e_n ⟩, where ∇ is a certain Macdonald eigenoperator for the Macdonald symmetric functions.
These q,t-Catalan numbers C_n(q,t) were further studied in <cit.>, where the first combinatorial formula was presented.
Several conjectured properties were then proven in <cit.>.

Garsia and Haiman <cit.> showed that their combinatorial construction of the q,t-Catalan numbers C_n(q,t) included both the Carlitz-Riordan and MacMahon q-Catalan numbers as specializations:

    C_n(q) = C_n(q,1) = C_n(1,q)      and      D_n(q) = C_n(q,  1
    q)   q^n2  .

Garsia and Haiman's combinatorial formula for C_n(q,t) was not obviously a polynomial in [q,t], but a rational function of polynomials. 
Later, an elegant characterization of the q,t-Catalan numbers, which made polynomiality clear, was conjectured by Haglund <cit.> and proven by Garsia and Haglund <cit.> in terms of two statistics on Dyck paths, namely, area and bounce.


A Dyck path is a lattice path on the lattice ^2 that starts at (0,0), uses only (0,1) and (1,0) as steps (called north and east, respectively), ends at (n,n), and never goes below the line y=x. 
Denote the set of all Dyck paths from (0,0) to (n,n) by 𝒟_n. 
For a Dyck path D∈𝒟_n we define (D) to be the number of complete cells between the Dyck path and the line y=x.  




For D∈𝒟_n, the statistic (D) is due to Haglund <cit.> and is defined by the following algorithm. 
Begin at (0,0) and travel north until an east step of D is encountered. 
Turn at a right angle to the east and travel east until the line y=x is reached. 
Then turn north and travel until an east step of D is encountered and repeat until (n,n) is reached. 
This defines the bounce path of D. 
The bounce path “bounces” off the line y=x at positions (0,0), (j_1, j_1), (j_2, j_2), …, (j_b, j_b) = (n,n). 
Then define 

    (D) := ∑_i=1^b-1 n - j_i   .



For the purposes of this paper, we take the following as the definition of the q,t-Catalan numbers <cit.>:

The q,t-Catalan numbers are given by
    
    C_n(q,t) = ∑_D ∈𝒟_n q^(D)  t^(D).



The symmetry of C_n(q,t) follows immediately from Haiman's original
definition of the q,t-Catalan numbers. 
Thus, we obtain the following as a consequence.

 For any nonnegative integer n,
    
    ∑_D ∈𝒟_n q^(D)  t^(D) = ∑_D ∈𝒟_n q^(D)  t^(D).



A notable open problem <cit.> is to find a bijection on Dyck paths that interchanges area and bounce, which would prove the q,t-symmetry in a combinatorial way.
Figure <ref> provides an example of the calculation of C_3(q,t), highlighting the aforementioned terminology and q,t-symmetry.  





 §.§ k-Dyck paths


Xin and Zhang refined the q,t-Catalan numbers by modifying Haglund's area-bounce formula <cit.>. 
They provided three equivalent models which can be used to define a k⃗-Dyck path. 
Here, we recall their first model.


Let k⃗=(k_1,…,k_m) be a vector of positive integers which sum to n. 
A k⃗-Dyck path is a lattice path in ^2 from (0,0) to (n,n), which never goes below the diagonal y=x, such that the north steps are of length k_i, for 1≤ i ≤ m, from bottom to top. 
The east steps are of length 1 without additional restriction. 
Denote the set of all k⃗-Dyck paths by D_k⃗.
Define the rank of a point (x,y) of a k⃗-Dyck path as y-x. 



In general, a k⃗-Dyck path encodes more information than a standard Dyck path because it also includes information about how long the north steps are. 
For example, consider a k⃗-Dyck path that contains a north step of length k_i immediately followed by a north step of length k_i+1: if one considers this as a standard Dyck path by sending north steps of length k_i to k_i north steps of length 1, we cannot then recover the lengths k_i, k_i+1. 


We recover the k-Dyck paths studied by Loehr <cit.> as a special case by using the vector k⃗ = (k,k,…,k) (and shearing the k-Dyck path as necessary so that the slope of the main diagonal becomes 1).
We can also recover the standard Dyck paths as a special case by using the vector k⃗ = (1,1,…,1).
Xin and Zhang also extended the definition of the area and bounce of a Dyck path to k⃗-Dyck paths (such that they are compatible with the definitions of area and bounce for k-Dyck paths). 
To describe area for a k⃗-Dyck path, we need an auxiliary definition <cit.>.


Let D∈ D_k⃗ be a k⃗-Dyck path. 
Define the red ranks of a k⃗-Dyck path to be the sequence (r_1,…,r_n), where r_i is the rank of the starting point of the north step corresponding to k_i.


The red ranks can also be computed by counting complete lattice cells between the path and the line y=x. 
Specifically, r_i is the number of such cells in the row whose bottom left corner is the south end of the start of the k_i set of north steps.
A k⃗-Dyck path is uniquely determined by its red ranks since they determine the location of all north steps. 
Note that the first red rank r_1 is always equal to 0.


For D∈ D_k⃗, define the statistic area to be the sum of its red ranks, that is,

    (D) := r_1+⋯+r_n   .



We recall from <cit.> the algorithm to compute bounce for a k⃗-Dyck path. 
For most of this paper, with the exception of Section <ref>, it suffices use a few special cases.



Xin and Zhang call the final tableau created during the algorithm the rank tableau and they note that (D) is equal to the sum of the entries in its first row.
For an example showing the construction of the rank tableau, see <cit.>. 
The  bounce path is a k⃗-Dyck path, distinct from the original Dyck path D, where each Q_i is the start of an east step of D. 
Note that the bounce path for D does not necessarily touch the diagonal line y=x (or “bounce” off the line) after each sequence of consecutive east steps like it does in the classical k⃗ = (1,…,1) case.

For small cases, one can give a piecewise closed-form formula for bounce.
In general, this becomes more difficult as the number of entries in k⃗ increases. 
For example, we extensively use the following characterization of bounce when k⃗ has three parts <cit.>. 


    For a (k_1,k_2,k_3)-Dyck path D, 
    
    (D)=
    
       2(k_1-r_2)+r_2+k_2-r_3-min(r_2,k_2)      if  r_2+k_2-r_3≥ 2min(r_2,k_2), 
    
       2(k_1-r_2)+⌈r_2+k_2-r_3/2⌉    otherwise.



In Section <ref>, we also state and use an involved formula for bounce when k⃗ = (k,k,k,k) due to Niu <cit.>.



Let 𝒟_k⃗ be the set of all k⃗-Dyck paths and define the refined q,t-Catalan number by

    C_k⃗(q,t)=∑_D∈𝒟_k⃗q^(D)  t^(D) .

Arranging the entries of k⃗ in decreasing order gives a partition μ(k⃗) and we further define

    C_λ(q,t)=∑_μ(k⃗)=λ∑_D∈𝒟_k⃗q^(D)  t^(D).



Xin and Zhang proved the q,t-symmetry of C_(k_1,k_2)(q,t), C_(k_1,k_2,k_3)(q,t), and C_(k,k,k,k)(q,t) for any positive integers k_1,k_2,k_3, and k <cit.>. 
For an example, see Figure <ref>.
They also investigated the symmetry of C_λ(q,t), observing that symmetry holds for partitions of length 2.
They infer symmetry in the case that λ is a partition of length 3 as a corollary to the q,t-symmetry of C_(k_1,k_2,k_3)(q,t) and exhibit examples showing that symmetry does not hold in general for partitions of length 4. 
Furthermore, Xin and Zhang conjectured that symmetry of C_λ(q,t) holds given λ=((a+1)^s,a^n-s) with 0≤ s≤ n.
Niu <cit.> proves this conjecture in the case that n=4 using similar partition analysis methods as those employed by Xin and Zhang <cit.>.


    When k⃗ or λ are given specific values in C_k⃗(q,t) or C_λ(q,t), respectively, it can be ambiguous which definition is being referred to. 
    In this paper, we adopt the convention that if otherwise unlabelled by k⃗ or λ, we are referring to C_k⃗(q,t) when we use an expression like C_(1,2,3)(q,t).









 §.§ Integer-point generating functions of polyhedral cones


As mentioned in the introduction, our computations employ the arithmetic of integer points in polyhedral cones, and we sketch how to compute their generating functions here, following <cit.>. 
A polyhedral cone is a nonnegative linear combination of a finite set of vectors in ^d. 
The cone is simplicial if its generators are linearly independent, and rational if we can choose its generators to be in ^d. 
To a rational cone C we associate its integer-point transform

    σ_C (z⃗) := ∑_m⃗∈ C ∩^d  z_1^ m_1  z_2^ m_2 ⋯ z_d^ m_d  .

We abbreviate the monomial in the sum as z⃗^m⃗. 
We compute integer-point transforms of simplicial rational cones, with a twist, namely, some of its facets (faces of codimension 1) removed.
To be precise, fix linearly independent vectors v⃗_1, v⃗_2, …, v⃗_k ∈^d and let 

    C := _≥ 0 v⃗_1 + … + _≥ 0 v⃗_m-1 + _ >0 v⃗_m + …
    + _ >0 v⃗_k   ,

which we call a half-open cone; here 1 ≤ m ≤ k.
Because C is simplicial, each facet of C is opposite one of its generators, and so we may think of C as having the facets opposite v⃗_m, v⃗_ m+1 , …, v⃗_k removed.
We call the vectors v⃗_1,…, v⃗_k  generators of this cone.
A standard tiling argument and geometric series yield (see for example <cit.>),

    σ_C (z⃗) = σ_Π(C) (z⃗) /∏_ j=1 ^k (1 - z⃗^v⃗_j ) ,

where

    Π(C) := [0,1) v⃗_1 + … + [0,1) v⃗_m-1 + (0,1] v⃗_m + … + (0,1] v⃗_k

denotes the fundamental parallelepiped of C.
In particular, σ_C (z⃗) is a rational function and so (because we can triangulate any cone into simplicial cones) is the integer-point transform of any rational cone.
Now, we are ready to apply these methods to our first application.





§ THE CASE K = K1,K2,K3


Throughout this section, let k⃗=(k_1,k_2,k_3) be a vector of three positive integers that sum to n. 
For a k⃗-Dyck path, denote its red ranks (as defined in Definition <ref>) r_1,r_2,r_3, recalling that r_1 is 0. 
As discussed in Section <ref>, the bounce of a k⃗-Dyck path D is given by equation (<ref>), and 

    area(D)=r_2+r_3   .

Our goal in this section is to replicate the following formula due to Xin and
Zhang <cit.> for the generating function (<ref>),
i.e., 

    (x_1,x_2,x_3,q,t)=∑_k_1,k_2,k_3≥
    0x_1^k_1x_2^k_2x_3^k_3∑_D∈𝒟_(k_1,k_2,k_3)
    q^area(D)  t^bounce(D).



The generating function (x_1,x_2,x_3,q,t) equals
 
    (1-x_1x_2qt^2)(1-x_1x_2q^2t)/(1-x_2q)(1-x_2t)(1-x_1qt)(1-x_1t^2)(1-x_1q^2)(1-x_1x_2qt)(1-x_3)  .



Note that this formula is q,t-symmetric, and so the q,t-symmetry of
C_(k_1,k_2,k_3)(q,t) follows immediately.
We represent the 
(k_1,k_2,k_3)-Dyck paths as integer points of cones and then compute
(x_1,x_2,x_3,q,t) via integer-point transforms.

Xin and Zhang <cit.> split their computation of this generating function into two parts, each divided into two cases based on numerical relationships between the quantities k_2, r_2, and r_3 that are relevant in the computation of bounce. 
We consider both the conditions in the bounce formula and the geometry of our cones to form a coarser decomposition than that of Xin and Zhang.

Each (k_1,k_2,k_3)-Dyck path corresponds to a vector of the form (k_1, k_2, k_3, r_2, r_3) where we omit r_1, as it always equals 0. 
By definition the following conditions hold for k_1,k_2,k_3-Dyck paths:

    k_1, k_2, k_3 ≥ 0 ,
       k_1 ≥ r_2 ≥ 0,
       r_2 + k_2 ≥ r_3 ≥ 0 .


These inequalities define a 5-dimensional cone with six generators: four of
them depicted in Figure <ref> plus the unit vectors in direction k_1 and k_3.
This cone naturally decomposes into three simplicial cones, also shown in
Figure <ref>.
For each of these subcones, we write its integer-point transform

    σ_C_i (z_1, z_2, z_3, w_2, w_3) = ∑_(k⃗, r⃗) ∈ C_i ∩^5 z_1^k_1 z_2^k_2 z_3^k_3 w_2^r_2 w_3^r_3.

We now compute these.



 §.§ Cone C1
 
Let C_1 be the cone defined by the inequalities (<ref>)-(<ref>) along with

    r_2+k_2-r_3 ≥ 2k_2    and    r_2 ≥ k_2   .

Thus, C_1 has the generators

    [ 1; 0; 0; 0; 0 ], [ 0; 0; 1; 0; 0 ], [ 1; 0; 0; 1; 0 ], [ 1; 0; 0; 1; 1 ],[ 1; 1; 0; 1; 0 ]

and is unimodular: these generators are a lattice basis for ^5.
The corresponding integer-point transform is

    σ_C_1(z_1,z_2,z_3,w_2,w_3)    =
    1/(1-z_1)(1-z_3)(1-z_1w_2)(1-z_1w_2w_3)(1-z_1z_2w_2)  .


Any k⃗-Dyck path corresponding to an integer point in C_1 satisfies both 
    min(r_2,k_2)=k_2   and   r_2+k_2-r_3≥ 2min(r_2,k_2),
 so its bounce is computed by

    bounce(D)=2k_1-r_2-r_3

and thus, the contribution of C_1 to (<ref>) is the generating function

    _1(x_1,x_2,x_3,q,t)    = ∑_(k⃗, r⃗) ∈ C_1 ∩^5
    x_1^k_1 x_2^k_2x_3^k_3 q^r_2 + r_3 t^2k_1-r_2-r_3 .

Writing down the generating function _1 is equivalent to replacing each monomial term z_1^k_1z_2^k_2z_3^k_3w_2^r_2w_3^r_3 of σ_C_1 with the corresponding term x_1^k_1 x_2^k_2x_3^k_3 q^r_2 + r_3 t^2k_1-r_2-r_3, and so

    _1(x_1,x_2,x_3,q,t)   = ∑_(k⃗, r⃗) ∈ C_1 ∩^5
    (x_1t^2)^k_1(x_2)^k_2(x_3)^k_3(qt^-1)^r_2 (qt^-1)^r_3
       =
    σ_C_1(x_1t^2,x_2,x_3,qt^-1,qt^-1)
       =1/(1-x_1t^2)(1-x_3)(1-x_1qt)(1-x_1q^2)(1-x_1x_2qt)  .

 



 §.§ Cone C2

Let C_2 be the cone defined by the inequalities (<ref>)-(<ref>) along with

    r_2+k_2-r_3 ≥ 2k_2    and     r_2 ≤ k_2   .

The cone C_2 is also unimodular, generated by 

    [ 1; 0; 0; 0; 0 ], [ 0; 1; 0; 0; 0 ], [ 0; 0; 1; 0; 0 ], [ 1; 1; 0; 1; 0 ], [ 0; 1; 0; 0; 1 ],

and the corresponding integer-point transform is

    σ_C_2(z_1,z_2,z_3,w_2,w_3)    =
    1/(1-z_1)(1-z_2)(1-z_3)(1-z_1z_2w_2)(1-z_2w_3)  .


For any k⃗-Dyck path D corresponding to an integer point in this cone, 

    bounce(D)=2(k_1-r_2)+k_2-r_3   ,

and so the contribution of C_2 to (<ref>) is the generating function

    _2(x_1,x_2,x_3,q,t)    = ∑_(k⃗, r⃗) ∈ C_2 ∩^5
    x_1^k_1 x_2^k_2x_3^k_3 q^r_2 + r_3 t^2(k_1-r_2)+k_2-r_3
       =∑_(k⃗, r⃗) ∈ C_2 ∩^5 (x_1t^2)^k_1(x_2t)^k_2(x_3)^k_3(qt^-2)^r_2
    (qt^-1)^r_3
       = σ_C_2(x_1t^2,x_2t,x_3,qt^-2,qt^-1) 
       =1/(1-x_1t^2)(1-x_2t)(1-x_3)(1-x_1x_2qt)(1-x_2q)  .




 §.§ Cone C3

Let C_3 be the cone defined by the inequalities (<ref>)-(<ref>) along with

    r_2+k_2-r_3 < 2k_2    and    r_2+k_2-r_3 < 2r_2   .

The cone C_3 is generated by

    [ 1; 0; 0; 0; 0 ],  [ 0; 0; 1; 0; 0 ], [ 1; 1; 0; 1; 0 ], [ 1; 0; 0; 1; 1 ],[ 0; 1; 0; 0; 1 ]

and thus, not unimodular. 
The fundamental parallelepiped of C_3 is 
 
    Π(C_3) = {λ_1 [ 1; 0; 0; 0; 0 ] + λ_2 [ 0; 0; 1; 0; 0 ] + λ_3[ 1; 1; 0; 1; 0 ] + λ_4 [ 1; 0; 0; 1; 1 ]+λ_5[ 0; 1; 0; 0; 1 ] |   0 ≤λ_1,λ_2,λ_3 < 1  and  0< λ_4,λ_5 ≤ 1},

with Π(C_3) ∩^5 = {(1,1,0,1,2), (1,1,0,1,1)}. 
Thus,

    σ_C_3(z_1,z_2,z_3,w_2,w_3)=     z_1 z_2 w_2 w^2_3 + z_1 z_2
    w_2w_3/(1-z_1)(1-z_3)(1-z_1z_2w_2)(1-z_1w_2w_3)(1-z_2w_3)  .


For the cones C_1 and C_2, we used the linearity of the bounce formulas to write _1,_2 using a change of variables in σ_C_1,σ_C_2. 
However, for a k⃗-Dyck path D corresponding to an integer point in C_3, 

    bounce(D)=  2(k_1-r_2)+⌈r_2+k_2-r_3/2⌉ .

In order to obtain a rational formula for _3, we can partition the integer points in C_3 into two subsets, such that bounce is a linear function on each subset.
This partition corresponds to the two summands in the numerator
of (<ref>).
Let

    A = {α_1[ 1; 0; 0; 0; 0 ]  +
    α_2[ 0; 0; 1; 0; 0 ]+
    α_3[ 1; 1; 0; 1; 0 ]+
    α_4[ 1; 0; 0; 1; 1 ]+
    α_5[ 0; 1; 0; 0; 1 ]
     |  α_1,α_2,α_3∈_≥ 0 and α_4,α_5∈_>0}.

For the points in A, area is computed by α_3+2α_4+α_5, and (since A falls under the second bounce condition) bounce is computed by
 
    2(k_1-r_2)+⌈r_2+k_2-r_3/2⌉   =
         2(α_1+α_3+α_4-α_3-α_4)+⌈α_3+α_4+α_3+α_5-α_4-α_5/2⌉
       =2α_1+α_3
      .

Hence, A contributes the generating function

    _3A(x_1,x_2,x_3,q,t)
         = ∑_α_i ∈ℤ_≥ 0,  α_4,α_5≠
    0(x_1t^2)^α_1 (x_3)^α_2 (x_1x_2qt)^α_3
    (x_1q^2)^α_4 (x_2q)^α_5
       =   x_1x_2q^3/(1-x_1t^2)(1-x_3)(1-x_1x_2qt)(1-x_2q)(1-x_1q^2)  .


The second set of the partition of integer points in C_3 is

    B = {[ 1; 1; 0; 1; 1 ] + 
    α_1[ 1; 0; 0; 0; 0 ]  +
    α_2[ 0; 0; 1; 0; 0 ]+
    α_3[ 1; 1; 0; 1; 0 ]+
    α_4[ 1; 0; 0; 1; 1 ]+
    α_5[ 0; 1; 0; 0; 1 ] |  α_i∈_≥ 0}.

Area is computed by α_3+2α_4+α_5 + 2, by summing the fourth and fifth coordinates corresponding to r_2 and r_3, and the constant 2 represents the contribution from the special point (1,1,0,1,1).
Since B falls under the second bounce condition, bounce is computed by
 
    2(k_1-r_2)+⌈r_2+k_2-r_3/2⌉=  2α_1 +
    α_3 + 1   ,
 
and so the generating function enumerating points in set B is

    _3B(x_1,x_2,x_3,q,t) =
    x_1x_2q^2t/(1-x_1t^2)(1-x_3)(1-x_1x_2qt)(1-x_2q)(1-x_1q^2)  .


Lastly, we combine the generating functions for sets A and B to obtain a generating function for (k_1,k_2,k_3)-Dyck paths in cone C_3:

    _3(x_1,x_2,x_3,q,t) = x_1x_2q^3 + x_1x_2 q^2
    t/(1-x_1t^2)(1-x_3)(1-x_1x_2qt)(1-x_2q)(1-x_1q^2)  .




 §.§ Summing over the full cone

By construction, any (k_1,k_2,k_3)-Dyck path corresponds to a point in either C_1, C_2 or C_3. 
The only points that appear in more than one cone are points in the cone generated by  [[ 1; 0; 0; 0; 0 ]],[[ 0; 0; 1; 0; 0 ]],[[ 1; 1; 0; 1; 0 ]]
(represented by the point labeled by [[ 1/2; 1/2;   0; 1/2;   0 ]]
in Figure <ref>), which appear in both cones C_1 and C_2 and are enumerated by the generating function
 
    ∑_(k⃗, r⃗) ∈ (C_1∩ C_2)∩^5 x_1^k_1
    x_2^k_2x_3^k_3 q^r_2 + r_3 t^2k_1-r_2-r_3 =
    1/(1-x_1t^2)(1-x_3)(1-x_1x_2qt)  .

Hence,
 
    (x_1,x_2,x_3,q,t)    = _1 + _2 + _3
    -1/(1-x_1t)(1-x_3)(1-x_1x_2qt)
       =
    (1-x_1x_2qt^2)(1-x_1x_2q^2t)/(1-x_2q)(1-x_2t)(1-x_1qt)(1-x_1t^2)(1-x_1q^2)(1-x_1x_2qt)(1-x_3)  ,
 
and this finishes our proof of Theorem <ref>.





§ THE CASE K4


We turn our attention to our second application, where we revisit ideas
from the previous section, as well as introduce some new ones.
Our goal in this section is to compute (<ref>), i.e.,

    (x, q, t) := ∑_ k ∈_≥ 0  C_(k,k,k,k)(q,t)   x^k,

re-establishing the following formula of Xin and Zhang <cit.>. 


The generating function (x,q,t) equals

    N/(1-q^3tx)(1-qt^3x)(1-q^2t^2x)(1-q^6x)(1-t^6x)  ,

where 

    N=1+   (q^5t+qt^5+q^4t^2+q^2t^4+q^4t+qt^4+q^3t^2+q^2t^3+q^3t^3)x
        + (-q^7t^3-q^3t^7+q^6t^5+q^5t^6-q^6t^4-q^4t^6-q^5t^5-q^5t^4-q^4t^5)x^2
        - (q^8t^8+q^9t^6+q^6t^9+q^8t^7+q^7t^8)x^3 .



Note again the q,t-symmetry of .
Throughout this section, let k⃗=(k,k,k,k), where k is a positive integer. 
Recall that a (k,k,k,k)-Dyck path D can be uniquely identified with the
vector (k,a,b,c) based on the red ranks (0, r_2, r_3,r_4), where

    r_2 = k-a,    r_3 = 2k-a-b,    and  r_4 = 3k-a-b-c.

We have from  <cit.> that area(D) = 6k-3a-2b-c and

    bounce(D)    = 
            6a + 3b + c - 4k     if  b ≥ 2k-2a  and  c≥ 4k-2a-2b, 
    
            5a + 2b + ⌈c/2⌉ - 2k      if  b ≥ 2k-2a  and  c< 4k-2a-2b,
    
            4a + 2b + c -2k      if  b < 2k-2,  b  is even, and c≥ 3k-a-3b/2,
     
            2a + b/2 + k + ⌈3a +3b/2+c - 3k/2⌉    if  b < 2k-2,  b  is even, and 
              3k-3a-3b/2≤ c<3k-a-3b/2,
     
            3a + b + c/3    if  b < 2k-2a,  b  is even, and  c< 3k-3a-3b/2,
     
            4a + 2b + c-2k + 1     if  b < 2k-2a,  b  is odd, and 
              c≥ 3k-a-3(b+1)/2 +1, 
     
            2a + b+1/2 + k +      if  b < 2k-2a,  b  
    is odd, and 
        3a + 3(b+1)/2 + c -3k-1 /2    3k-3a - 3(b+1)/2+1≤ c<3k-a-3(b+1)/2 +1 , 
    
            3a + b + 1 + ⌈c-1/3⌉    if  b < 2k-2,   b  is odd, and 
              c< 3k-3a - 3(b+1)/2+1.


We now use the techniques from Section <ref> to compute the generating function

    (x,y_1,y_2,y_3,q,t) = ∑_k≥ 0  x^k ∑_(k,a,b,c) ∈𝒟_(k,k,k,k) y_1^a   y_2^b  y_3^c  q^area(D) t^bounce(D),

as a sum of generating functions derived from cones. 
By definition, all vectors (k,a,b,c) corresponding to (k,k,k,k)-Dyck paths must satisfy the following inequalities:

    0    ≤ a ≤ k   ,
    
        0    ≤ b ≤ 2k-a  ,
    
        0    ≤ c ≤ 3k-a-b  .

This system of inequalities corresponds to the cone C generated by 

    [ 1; 0; 2; 0 ], [ 1; 0; 0; 0 ], [ 1; 1; 0; 0 ], [ 1; 1; 1; 0 ],[ 1; 1; 1; 1 ],  [ 1; 0; 2; 1 ], [ 1; 0; 0; 3 ],  [ 1; 1; 0; 2 ].

We denote these cone generators by v_1, v_2, …, v_8, respectively.
Figure <ref> depicts a projection of them to the hyperplane
k=1.

To compute the integer-point transform

    σ_C(y,z_1,z_2,z_3) = ∑_(k,a,b,c)∈ C y^k z_1^a z_2^b
    z_3^c   ,

we triangulate C depending on the bounce cases listed above.
We follow the notation in <cit.> and denote

    Part 1   :b ≥ 2k-2a,
    Part 2   : b < 2k-2a  and b even, 
    Part 3   : b < 2k-2a  and b odd.

Part 1 is split into two cases, 
Part 2 into three cases where b is always even,
and Part 3 into three cases where b is always odd.




 §.§ Part 1

We divide the points satisfying b ≥ 2k - 2a into two cases:

    Case 1:     c≥ 4k-2a-2b,
    Case 2:     c< 4k-2a-2b.

 

Let C_1.1 be the cone defined by the inequalities
 (<ref>)-(<ref>) and (<ref>); it has
generators v_1, v_4, v_5, v_6,v_8 and is
not simplicial, but can be triangulated into two simplicial cones:

    
  * Cone A generated by v_1, v_4, v_6,v_8, 
  * Cone B generated by v_4, v_5, v_6,v_8.

We remove the facet opposite v_5 in cone B; it equals the facet opposite v_1 in cone A which is included there (see Figure <ref>).



Both Cone A and Cone B are  unimodular, so we can directly compute the integer-point transforms 

    σ_C_1.1A(y,z_1,z_2,z_3)    =
    1/(1-yz_2^2)(1-yz_1z_2)(1-yz_2^2z_3)(1-yz_1z_3^2)  , 
    σ_C_1.1B(y,z_1,z_2,z_3)    =
    yz_1z_2z_3/(1-yz_1z_2)(1-yz_1z_2z_3)(1-yz_2^2z_3)(1-yz_1z_3^2)  ,

which in turn gives

    σ_C_1.1(y,z_1,z_2,z_3) = σ_C_1A + σ_C_1B= 1-y^2z_1z_2^3z_3/(1-yz_2^2)(1-yz_1z_2)(1-yz_2^2z_3)(1-yz_1z_3^2)(1-yz_1z_2z_3)  .

Using methods similar to the ones for first two cones in the Section
<ref>, we compute the generating function

    _1.1(x,y_1,y_2,y_3,q,t)  =
    1-x^2y_1y_2^3y_3q^2t^8/(1-xy_2^2q^2t^2)(1-xy_1y_2qt^5)(1-xy_2^2y_3qt^3)(1-xy_1y_3^2qt^4)(1-xy_1y_2y_3t^6)  .


Next, define C_1.2 to be the cone defined by inequalities  (<ref>)-(<ref>) and (<ref>); it has generators v_1,
v_3, v_4, v_8, and we remove the facet opposite v_3.
This cone is simplicial (though not unimodular), with integer-point transform

    σ_C_1.2(y,z_1,z_2,z_3)    = yz_1 +
    yz_1z_3/(1-yz_2^2)(1-yz_1)(1-yz_1z_2)(1-yz_1z_3^2)  ,

as Π(C_1.2) ∩^4  ={(1,1,0,0),(1,1,0,1) }. 
We apply the techniques of Section <ref> to compute

    _1.2(x,y_1,y_2,y_3,q,t)  = xy_1q^3t^3 +
    xy_1y_3q^2t^4/(1-xy_2^2q^2t^2)(1-xy_1q^3t^3)(1-xy_1y_2qt^5)(1-xy_1y_3^2qt^4)  .




 §.§ Part 2

We divide the points satisfying b < 2k-2a with b even into three cases:

    Case 1:     c≥ 3k-a-3b/2  ,
    Case 2:      3k-3a-3b/2≤
    c<3k-a-3b/2  ,
    Case 3:     c<
    3k-3a-3b/2  .




 §.§ Part 3

Similarly, we divide the points satisfying b < 2k-2a with b odd into three cases:

    Case 1:     c≥ 3k-a-3(b+1)/2 + 1   ,
    Case 2:      3k-3a-3(b+1)/2+1 ≤ c<3k-a-3(b+1)/2+1   ,
    Case 3:     c< 3k-3a-3(b+1)/2+1  .


We treat each case for Parts 2 and 3 in parallel. 
We start by disregarding the parity condition on b and consider all points that satisfy the given inequalities. 
Once we determine the relevant cone generating function, we extract the points
with the correct parity condition for b.



 §.§ Case 1

We start with Part 2, Case 1. 
Let C_2.1 be the cone defined by the inequalities  (<ref>)-(<ref>) and (<ref>); it has generators v_1,
v_6,v_7, v_8, and the facet opposite v_7 is missing.

Its fundamental parallelepiped contains the two integer points Π(C_2.1) ∩^4
={(1,0,0,3),(1,0,1,2) } and so the corresponding integer-point transform is

    σ_C_2.1(y,z_1,z_2,z_3)    = yz_3^3 + yz_2z_3^2/(1-yz_2^2)(1-yz_2^2z_3)(1-yz_3^3)(1-yz_1z_3^2)  .


We are interested in extracting all points (k,a,b,c) enumerated by this generating function such that b is even.
Since the coordinate b of a given point corresponds to the power of z_2 in the monomial, this is equivalent to considering monomial terms where z_2 is raised to an even power.
Since z_2 appears only with even powers in the denominator, 
the two numerator terms of σ_C_2.1 correspond to generating functions enumerating points (k,a,b,c) where b is even and odd, respectively. 
Since we are only interested in the case that b is even, we simply disregard the term y z_2 z_3^2 in the numerator.
It remains to apply the techniques of Sections <ref> and <ref> to compute the generating function

    _2.1( x,y_1,y_2,y_3,q,t)     =
    xy_3^3q^3t/(1-xy_2^2q^2t^2)(1-xy_2^2y_3qt^3)(1-xy_3^3q^3t)(1-xy_1y_3^2qt^4)  .


Next, we consider Part 3, Case 1, where we eventually require that b is odd.
The inequalities   (<ref>)-(<ref>) and (<ref>) reduce to the following system of inequalities:

    0  ≤ a,
       0 ≤ b ,
       0 ≤ c ,
       c ≤ 3k-a-b ,
       b < 2k-2a ,
       3k-a-3(b+1)/2 + 1  ≤ c.

The last inhomogeneous inequality makes computations more subtle.
For example,  (<ref>), (<ref>), and (<ref>) imply 

    -1/2≤ 2a -1/2≤ 3k -3k -a + 3a -1/2≤ 3k-a -3/2b -1/2≤ c,

but since c∈ the implication is the redundant assumption that 0≤ c.

Inequalities (<ref>) and (<ref>)  imply -1 ≤ b, which is weaker than (<ref>). 
To simplify our computations, we first assume -1≤ b and compute a cone generating function, and then impose the condition 0≤ b by eliminating the points where b = -1.
Hence, we consider the cone C_3.1 defined by the following minimal hyperplane description:

    0  ≤ a,
       c ≤ 3k-a-b,
       b < 2k-2a,
       3k-a-3(b+1)/2 + 1  ≤ c.

The cone C_3.1 has generators v_1, v_6, v_7, v_8, and the facet opposite v_7 is missing. 
Though these are the same generators as of C_2.1, the cone C_3.1 is a translation of C_2.1 such that its apex is at (-1/2, 0, -1, -1/2) rather than the origin.  
With  Π(C_3.1) ∩^4 = { (0, 0, -1 , 1), (1 , 0, 0, 2)}, we compute

    σ_C_3.1(y,z_1,z_2,z_3)     = z_2^-1z_3  + y
    z_3^2/(1-yz_2^2)(1-yz_2^2z_3)(1-yz_3^3)(1-yz_1z_3^2)  .

To impose the condition 0 ≤ b, we first subtract all terms where b = -1, i.e., z_2^ -1:

    σ_C_3.1(y,z_1,z_2,z_3) - z_2^-1 z_3/(1-yz_3^3)(1-yz_1z_3^2)   = yz_3^2 - y^2z_2^3z_3^2  + yz_2z_3 + yz_2z_3^2/(1-yz_2^2)(1-yz_2^2z_3)(1-yz_3^3)(1-yz_1z_3^2) .

Finally, we need to extract the terms for odd b, i.e., the monomials with odd powers of z_2.
Again, the denominator contributes only even powers of z_2, and so we simply need to disregard the first term in the numerator, giving rise to the resulting generating function

    - y^2z_2^3z_3^2 + yz_2z_3 + yz_2z_3^2
    /(1-yz_2^2)(1-yz_2^2z_3)(1-yz_3^3)(1-yz_1z_3^2)  .

From this, we obtain the contribution

    _3.1(x,y_1,y_2,y_3,q,t)     = xy_2y_3q^2 t^2 ( -xy_2^2y_3q^2t^3 + q + y_3t)
    /(1-xy_2^2q^2t^2)(1-xy_2^2y_3qt^3)(1-xy_3^3q^3t)(1-xy_1y_3^2qt^4)  .




 §.§ Case 2

We next consider Part 2, Case 2.
Let C_2.2 be the cone defined by the inequalities   (<ref>)-(<ref>) and (<ref>); it has generators v_1, v_3, v_7, v_8, and we remove the facets opposite to v_3, v_7.

Here, Π(C_2.2) contains four integer points, and the integer-point transform is 

    σ_C_2.2(y,z_1,z_2,z_3) = y^2z_1z_3^3 + y^2z_1z_3^4 +
    y^2z_1z_2z_3 + y^2z_1z_2z_3^2/(1-yz_2^2)(1-yz_1)(1-yz_3^3)(1-yz_1z_3^2)  .

Imposing the condition that b is even means disregarding the last two terms in the numerator, from which we obtain the generating function

    _2.2(x,y_1,y_2,y_3,q,t)     =x^2y_1y_3^3q^6t^4 +
    x^2y_1y_3^4q^5t^5/(1-xy_2^2q^2t^2)(1-xy_1q^3t^3)(1-xy_3^3q^3t)(1-xy_1y_3^2qt^4)  .


Part 3, Case 2 proceeds in a similar fashion.
We define the cone C_3.2 by the following system of inequalities reduced from inequalities (<ref>)-(<ref>) and (<ref>):


    0    ≤ b,
     
        b    < 2k-2a,
     
        c    ≥ 3k-3a-3(b+1)/2+1,
     
        c    < 3k-a-3(b+1)/2+1.

The cone C_3.2 is a translation of the cone C_2.2 such that its apex is at (0,0,0,-1/2).
Again, Π(C_3.2) contains four integer points, and the resulting integer-point transform is

    σ_C_3.2(y,z_1,z_2,z_3)
           = y^2 z_1 z_2 z_3  + y^2 z_1z_2z_3^2+  y^2 z_1 z_3^3+ y^2 z_1
    z_3^4/(1-yz_2^2)(1-yz_1)(1-yz_3^3)(1-yz_1z_3^2)  .

To impose the condition that b is odd, we disregard the last two terms in the numerator, from which we obtain the generating function

    _3.2(q,t, x,y_1,y_2,y_3)    = x^2 y_1 y_2 y_3q^5 t^5 (q   +
    y_3t)/(1-xy_2^2q^2t^2)(1-xy_1q^3t^3)(1-q^3txy_3^3)(1-xy_1y_3^2qt^4)  .





 §.§ Case 3

Let C_2.3 be the cone defined by inequalities  (<ref>)-(<ref>) and (<ref>); it has generators v_1, v_2, v_3, v_7, and we discard the facet opposite to v_2. 

Since 
    Π(C_2.3) ∩^4 =
    {(1,0,0,0),(1,0,1,0),(1,0,0,1),(1,0,1,1),(1,0,0,2),(2,0,1,2)}  ,
 the
integer-point transform is 

    σ_C_2.3(y,z_1,z_2,z_3)    =y + yz_2 + yz_3 + yz_2z_3 + yz_3^2
    + y^2z_2z_3^2/(1-yz_2^2)(1-y)(1-yz_1)(1-yz_3^3)  .

We are only interested in the points where b is even, and so we disregard the three terms in the numerator with z_2, arriving at the generating function

    _2.3(q,t, x,y_1,y_2,y_3)    =  xq^4(q^2 + y_3q t +
    y_3^2t)/(1-xy_2^2q^2t^2)(1- xq^6)(1-xy_1q^3t^3)(1-xy_3^3q^3t)  .


For Part 3, Case 3, let C_3.3 be the cone defined by inequalities  (<ref>)-(<ref>) and (<ref>); it has the same generators as C_2.3, but is translated to place its apex at (1/6, 0 ,0,0) rather than the origin.
Since 
    Π(C_3.3) ∩^4 = {(1,0,0,0),(1,0,1,0),(1,0,0,1),(2,0,1,1),(1,0,0,2),(2,0,1,2)}  ,
 
the integer-point transform is

    σ_C_3.3(y,z_1,z_2,z_3)      = y + yz_2 + yz_3 + y^2z_2z_3 +
    yz_3^2 + y^2 z_2z_3^2/(1-yz_2^2)(1-y)(1-yz_1)(1-yz_3^3)  .

We are now only interested in the points where b is odd and so we disregard the three terms in the numerator without z_2; the resulting generating function is 

    _3.3(q,t, x,y_1,y_2,y_3)    =  xy_2q^4t^2 + x^2y_2y_3q^9t^2 +
     x^2 y_2y_3^2q^8t^3/(1-xy_2^2q^2t^2)(1-xq^6)(1-xy_1q^3t^3)(1-xy_3^3q^3t)  .


Adding the eight generating functions  yields

    (q,t, x,y_1,y_2,y_3) =   1-q^2t^8x^2y_1y_2^3y_3/(1-q^2t^2xy_2^2)(1-qt^5xy_1y_2)(1-qt^3xy_2^2y_3)(1-qt^4xy_1y_3^2)(1-t^6xy_1y_2y_3)
        + xy_1q^2t^3(q + ty_3)/(1-q^2t^2xy_2^2)(1-q^3t^3xy_1)(1-qt^5xy_1y_2)(1-qt^4xy_1y_3^2)
        + 
    q^3txy_3^3/(1-q^2t^2xy_2^2)(1-q^3txy_3^3)(1-qt^3xy_2^2y_3)(1-qt^4xy_1y_3^2)
        + q^5t^4x^2y_1y_3^3(q+ty_3)/(1-q^2t^2xy_2^2)(1-q^3txy_3^3)(1-q^3t^3xy_1)(1-qt^4xy_1y_3^2)
        +  q^4x(q^2 + ty_3^2 + q ty_3)/(1-q^2t^2xy_2^2)(1- q^6x)(1-q^3txy_3^3)(1-q^3t^3xy_1)
        + q^2 t^2xy_2y_3 ( q + ty_3-q^2t^3xy_2^2y_3 ) /(1-q^2t^2xy_2^2)(1-q^3txy_3^3)(1-qt^3xy_2^2y_3)(1-qt^4xy_1y_3^2)
        +  x^2 y_1 y_2 y_3q^5 t^5 (q   + y_3t)/(1-q^2t^2xy_2^2)(1-q^3txy_3^3)(1-q^3t^3xy_1)(1-qt^4xy_1y_3^2)
        + q^4t^2xy_2(1+q^5xy_3+q^4txy_3^2)/(1-q^2t^2xy_2^2)(1-q^6x)(1-q^3txy_3^3)(1-q^3t^3xy_1)  .

Setting y_2=y_3=y_4=1 we obtain

    (x,1,1,1,q,t)=N/(1-q^3tx)(1-qt^3x)(1-q^2t^2x)(1-q^6x)(1-t^6x)  ,

where 

    N=1+   (q^5t+qt^5+q^4t^2+q^2t^4+q^4t+qt^4+q^3t^2+q^2t^3+q^3t^3)x
        + (-q^7t^3-q^3t^7+q^6t^5+q^5t^6-q^6t^4-q^4t^6-q^5t^5-q^5t^4-q^4t^5)x^2
        - (q^8t^8+q^9t^6+q^6t^9+q^8t^7+q^7t^8)x^3 ,

and this completes the proof of Theorem <ref>.  





§ THE CASE KA


Based on computational data, we make the following conjecture about q,t-symmetry of the refined q,t-Catalan numbers C_k⃗(q,t).


If k⃗ = (k,a,…,a), for positive integers k,a, then the refined q,t-Catalan numbers C_k⃗(q,t) are q,t-symmetric.


We settle this conjecture in the four-dimensional case when k ≤ a.  As in the case of k⃗=(k,k,k,k), our current understanding of the bounce formula makes further progress daunting. 
Our goal in this section is to compute (<ref>), i.e.,

    (x, y, q, t) := ∑_ k, m ∈_≥ 0  C_k =
    (k,k+m,k+m,k+m)(q,t)   x^k   y^m.



    If k,m≥ 0, then

    (x,y,q,t)=M/(1-xq^6)(1-xt^6)(1-xq^3t)(1-xqt^3)(1-xq^2t^2)(1-yq^3)(1-yt^3)(1-yqt)  ,

where

    M=
        (q^13t^7    + q^7t^13 + q^12t^8 + q^8t^12 + q^9t^12 + q^12t^9 + q^11t^11 + q^10t^11+ q^11t^10 + q^9t^11 + q^11t^9  + q^10t^10)x^3y^2 
       - (q^11t^5 + q^5t^11 + q^10t^6 + q^6t^10 + q^8t^9 + q^9t^8 + q^7t^9 + q^9t^7 + q^8t^8 - q^7t^8 - q^8t^7 + q^7t^7)x^2y^2 
       -(q^7t^4 + q^4t^7 + q^6t^5 + q^5t^6)xy^2
       -(q^12t^6 + q^6t^12 + q^8t^11 + q^11t^8 + q^10t^8 + q^8t^10 + q^7t^11 + q^11t^7  + q^7t^10 + q^10t^7 + 2q^9t^9 + q^8t^9 + q^9t^8)x^3y
        -(q^10t^3 + q^3t^10 + q^5t^7 + q^7t^5 - q^5t^9 -q^9t^5 + q^4t^9 + q^9t^4 + q^4t^8 + q^8t^4 + q^5t^6 + q^6t^5-q^8t^6 - q^6t^8)x^2y  
       +(qt^8+ q^8t + qt^7  + q^7t + q^6t^3 +q^3t^6 + q^4t^5 + q^5t^4  + q^2t^5 + q^5t^2 + 2q^4t^4 + q^5t^3+ q^3t^5 +2q^3t^4 +2q^4t^3 
               + q^7t^2 + q^2t^7 + q^6t^2 + q^2t^6)xy 
       -(q^2t+qt^2)y
       +(q^9t^6 + q^6t^9 + q^8t^7+ q^7t^8 + q^8t^8 )x^3 + (q^7t^3 + q^3t^7 - q^5t^6 - q^6t^5 + q^4t^6 + q^6t^4 + q^5t^5 + q^4t^5 + q^5t^4)x^2  
       -(qt^5 + q^5t + q^2t^4 + q^4t^2 + qt^4 + q^4t + q^3t^3 + q^2t^3 + q^3t^2)x  -  1  .




    M=   (q^9t^6 + q^6t^9 + q^8t^7+ q^7t^8 + q^8t^8 )x^3 + (q^7t^3 + q^3t^7 - q^5t^6 - q^6t^5 + q^4t^6 + q^6t^4 + q^5t^5 + q^4t^5 + q^5t^4)x^2  
       +[(q^13t^7 + q^7t^13 + q^12t^8 + q^8t^12 + q^9t^12 + q^12t^9 + q^11t^11 + q^10t^11+ q^11t^10 + q^9t^11 + q^11t^9  + q^10t^10)x^3  
       - (q^11t^5 + q^5t^11 + q^10t^6 + q^6t^10 + q^8t^9 + q^9t^8 + q^7t^9 + q^9t^7 + q^8t^8 - q^7t^8 - q^8t^7 + q^7t^7)x^2
       -(q^7t^4 + q^4t^7 + q^6t^5 + q^5t^6)x]y^2
        -(qt^5 + q^5t + q^2t^4 + q^4t^2 + qt^4 + q^4t + q^3t^3 + q^2t^3 + q^3t^2)x
       -[(q^12t^6 + q^6t^12 + q^8t^11 + q^11t^8 + q^10t^8 + q^8t^10 + q^7t^11 + q^11t^7  + q^7t^10 + q^10t^7 + 2q^9t^9 + q^8t^9 + q^9t^8)x^3
        +q^2t + qt^2+
        (q^10t^3 + q^3t^10 + q^5t^7 + q^7t^5 - q^5t^9 -q^9t^5 + q^4t^9 + q^9t^4 + q^4t^8 + q^8t^4 +
        q^5t^6 + q^6t^5-q^8t^6 - q^6t^8)x^2 - 
                (qt^8+ q^8t + qt^7  + q^7t + q^6t^3 +q^3t^6 + q^4t^5 + q^5t^4  + q^2t^5 + q^5t^2 + 2q^4t^4 + 
                   q^5t^3+ q^3t^5 + 2q^3t^4 +2q^4t^3 + q^7t^2 + q^2t^7 + q^6t^2 + q^2t^6)x]y - 1  .


Note once more the q,t-symmetry in this formula. 

A (k,k+m,k+m,k+m)-Dyck path D can be uniquely identified with the vector (k,m,a,b,c) based on the red ranks (0,r_2,r_3,r_4) where

    r_2=k-a,    r_3=2k+m-a-b,    r_4=3k+2m-a-b-c.

We compute area(D)=6k+3m-3a-2b-c and
    
    bounce(D)=
    {[                                 3a                             b,c=0,;                           3a+⌈c/3⌉                b=0,   0<c≤ 3(k-a),;                  2a+k+⌈c-3(k-a)/2⌉                   b=0,   c>3(k-a),;                          3a+2⌈b/2⌉                 0<b≤ 2(k-a),  c=0,;                    3a+2⌈b/2⌉+⌈c/3⌉       0<b≤ 2(k-a),  b  is even and;                                                   0< c≤ 3(k-a-⌈b/2⌉),;                  3a+2⌈b/2⌉+⌈c-1/3⌉        0<b≤ 2(k-a),  b  is odd and;                                                 0≤ c-1≤ 3(k-a-⌈b/2⌉),;      2a+⌈b/2⌉+k+⌈c-3(k-a-⌈b/2⌉)/2⌉       0<b≤ 2(k-a),  b  is even and;                                      3(k-a-⌈b/2⌉)<c≤ 3(k-⌈b/2⌉)-a+2m,;                        2a+⌈b/2⌉+k+        0<b≤ 2(k-a),  b  is odd and;               ⌈c-1-3(k-a-⌈b/2⌉)/2⌉ 3(k-a-⌈b/2⌉)<c-1≤ 3(k-⌈b/2⌉)-a+2m,;                   c-2k+4a+4⌈b/2⌉-m       0<b≤ 2(k-a),  b  is even and;                                                   c> 3(k-⌈b/2⌉)-a+2m,;                 c-1-2k+4a+4⌈b/2⌉-m        0<b≤ 2(k-a),  b  is odd and;                                                 c-1> 3(k-⌈b/2⌉)-a+2m,;                     5a+2b-2k+⌈c/2⌉     b>2(k-a),   0≤ c≤ 2(2k-a+m-b),;                       6a+3b-4k+c-m          b>2(k-a),  c>2(2k-a+m-b). ].

By definition, all vectors (k,m a, b, c) corresponding
to (k, k+m, k+m, k+m)-Dyck paths must satisfy the following inequalities:

    0   ≤ a≤ k ,
    
        0   ≤ m ,
    
        0   ≤ b≤ 2k+m-a ,
    
        0   ≤ c≤ 3k+2m-a-b.

This system of inequalities corresponds to the cone C generated by 

    [ 1; 0; 0; 2; 0 ],
        [ 1; 0; 1; 0; 0 ],
        [ 1; 0; 0; 0; 0 ],
        [ 0; 1; 0; 0; 0 ],
        [ 0; 1; 0; 1; 0 ],
        [ 1; 0; 1; 1; 0 ],
        [ 1; 0; 1; 1; 1 ],
        [ 1; 0; 0; 2; 1 ],
        [ 1; 0; 1; 0; 2 ],
        [ 1; 0; 0; 0; 3 ],
        [ 0; 1; 0; 0; 2 ],
        [ 0; 1; 0; 1; 1 ].

We denote these cone generators by v_1,…,v_12. 
Similarly to the (k,k,k,k) case we triangulate C depending on the bounce cases listed above.  We denote

    Part 1:     b=0 ,
    Part 2:     0<b≤ 2(k-a)  and  c=0 ,
    Part 3:     0<b≤ 2(k-a),   c 0, and b is even,
    Part 4:     0<b≤ 2(k-a),   c 0,  and b is odd,
    Part 5:     b>2(k-a).

Some parts are further split into subcases.
We do not exhibit the details of the ensuing computations, as the generating-function
methods are similar to those for the k⃗=(k_1,k_2,k_3) and k⃗=(k,k,k,k) cases above.
However, we do give the result for each subcase.



 §.§ Part 1

We divide the points with b=0 into three cases: 

    Case 1:     c=0,
    Case 2:     0<c≤ 3(k-a),
    Case 3:     c>3(k-a).

Case 1 gives rise to the unimodular cone C_1.1 defined by the inequalities <ref>-<ref>, <ref>, and <ref>,
with generating function 

    _1.1(x,y,z_1,q,t)=1/(1-xz_1q^3t^3)(1-yq^3)(1-xq^6)  .

Case 2 gives the simplicial cone C_1.2 defined by the inequalities <ref>-<ref>, <ref> and <ref>, 
with generating function

    _1.2(x,y,z_1,z_3,q,t)=xz_3q^3t(q^2+z_3q+z_3^2)/(1-xz_3^3q^3t)(1-xq^6)(1-xz_1q^3t^3)(1-yq^3)  .

Case 3 gives the cone C_1.3 defined by the inequalities <ref>-<ref>, <ref> and <ref>. It needs to
be triangulated, giving rise to the generating function

    _1.3(x,y,z_1,z_3,q,t)
       = yz_3q^2t+yz_3^2qt/(1-xz_3^3q^3t)(1-yq^3)(1-xz_1q^3t^3)(1-yz_3^2qt)
            + xz_1z_3q^2t^4+xz_1z_3^2qt^4/(1-xz_3^3q^3t)(1-xz_1q^3t^3)(1-xz_1z_3^2qt^4)(1-yz_3^2qt)  .




 §.§ Part 2

Let C_2 be the simplicial cone defined by the inequalities <ref>-<ref> and <ref>. 
It gives rise to the generating function 

    _2(x,y,z_1,z_2,z_3,q,t)=xz_2q^4t^2 + xz_2^2q^2t^2/(1-xz_1q^3t^3)(1-yq^3)(1-xz_2^2q^2t^2)(1-xq^6) .






 §.§ Part 3

We divide the points with b even, c 0, and 0<b≤ 2(k-a) into three cases:

    Case 1:     0<c≤ 3(k-a-b/2)  ,
    Case 2:     3(k-a-b/2)<c≤ 3(k-b/2)-a+2m  ,
    Case 3:     c>3(k-b/2)-a+2m  .

Case 1 gives the simplicial cone C_3.1 defined by the inequalities  <ref>-<ref>, <ref> and <ref>;
we need only consider the points with even b values.  
This gives the generating function 

    _3.1(x,y,z_1,z_2,z_3,q,t)=x^2z_2^2z_3q^7t^3+x^2z_2^2z_3^2q^6t^3+x^2z_2^2z_3^3q^5t^3/(1-yq^3)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)(1-xz_3^3q^3t)(1-xq^6)  .

Case 2 gives the cone C_3.2 defined by inequalities <ref>-<ref>, <ref> and <ref>; again we
need only consider the points with even b value.  
After a triangulation, we compute the generating function

    _3.2(x,y,z_1,z_2,z_3,q,t)
       =
    xyz_2^2z_3q^4t^3+xyz_2^2z_3^2q^3t^3/(1-yq^3)(1-yz_3^2qt)(1-xz_3^3q^3t)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)
            + x^2z_1z_2^2z_3q^4t^6+x^2z_1z_2^2z_3^2q^3t^6/(1-yz_3^2qt)(1-xz_3^3q^3t)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)(1-xz_1z_3^2qt^4)  .

Case 3 gives the simplicial cone C_3.3  defined by the inequalities <ref>-<ref> , <ref> and <ref>; once
more we need only consider the points with even b value.  
Its generating function is

    _3.3(x,y,z_1,z_2,z_3,q,t)=xz_2^2z_3qt^3/(1-xz_3^3q^3t)(1-yz_3^2qt)(1-xz_1z_3^2qt^4)(1-xz_2^2q^2t^2)(1-xz_2^2z_3qt^3)  .




 §.§ Part 4

We similarly divide the points with b odd and 0<b≤ 2(k-a) into three cases:

    Case 1:     0≤ c-1≤ 3(k-a-b+1/2)   ,
    Case 2:     3(k-a-b+1/2)<c-1≤ 3(k-b+1/2)-a+2m   ,
    Case 3:     3(k-b+1/2)-a+2m<c-1  .

For Case 1, we consider the points with odd b value of the cone C_4.1 defined by the inequalities <ref>-<ref>, <ref> and <ref>, the last of which in this context simplifies to  

    0<c<3k-3a-3b/2  .

It comes with the generating function 

    _4.1(x,y,z_1,z_2,z_3,q,t)=xz_2z_3q^3t^2+x^2z_2z_3^2q^8t^3+x^2z_2z_3^3q^7t^3/(1-yq^3)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)(1-xz_3^3q^3t)(1-xq^6)  .

Similarly, Case 2 is captured by the points with odd b value of the cone C_4.2 defined by the inequalities <ref>-<ref>, <ref>, and <ref>, the last of which in this context simplifies to

    3k-3a-3b/2≤ c<3k-3b/2-a+2m  .
  
After a triangulation, its generating function is

    _4.2(x,y,z_1,z_2,z_3,q,t)
         =
    xyz_2z_3^2q^5t^3+xyz_2z_3^3q^4t^3/(1-xz_3^3q^3t)(1-xz_2^2q^2t^2)(1-yz_3^2qt)(1-xz_1q^3t^3)(1-yq^3)
            + x^2z_1z_2z_3^2q^5t^6+x^2z_1z_2z_3^3q^4t^6/(1-xz_3^3q^3t)(1-xz_2^2q^2t^2)(1-yz_3^2qt)(1-xz_1q^3t^3)(1-xz_1z_3^2qt^4)  .


For Case 3, we consider the points with odd b value of the cone C_4.3 defined by the inequalities <ref>-<ref>, <ref>, and <ref>, the last of which simplifies to

    c≥ 3k-3b/2-a+2m  .

The corresponding generating function is 

    _4.3(x,y,z_1,z_2,z_3,q,t)=xz_2z_3^2q^2t^3/(1-xz_3^3q^3t)(1-yz_3^2qt)(1-xz_1z_3^2qt^4)(1-xz_2^2q^2t^2)(1-xz_2^2z_3qt^3)  .




 §.§ Part 5

We divide the points with b>2(k-a) into two cases: 

    Case 1:     0≤ c≤ 2(2k-a+m-b)  ,
    Case 2:     c>2(2k-a+m-b)  .

For Case 1, we consider the cone C_5.1 defined by the inequalities <ref>-<ref>, <ref>, and <ref>. 
After a triangulation, we compute its generating function as

    _5.1(x,y,z_1,z_2,z_3,q,t)
         =yz_2qt^2 +
    y^2z_2z_3q^3t^3/(1-yq^3)(1-yz_3^2qt)(1-yz_2qt^2)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)
            + xyz_1z_2z_3q^3t^6+xyz_1z_2z_3^2q^2t^6/(1-yz_3^2qt)(1-yz_2qt^2)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)(1-xz_1z_3^2qt^4)
            + xz_1z_2 qt^5 + x^2z_1^2z_2z_3q^3t^9/(1-yz_2qt^2)(1-xz_2^2q^2t^2)(1-xz_1q^3t^3)(1-xz_1z_3^2qt^4)(1-xz_1z_2qt^5)  .

Finally, for Case 2, let C_5.2 be the cone defined by the inequalities <ref>-<ref>, <ref> and <ref>. 
With another triangulation, we compute the generating function 

    _5.2(x,y,z_1,z_2,z_3,q,t)
           =yz_2z_3t^3/(1-yz_3^2qt)(1-yz_2qt^2)(1-yz_2z_3t^3)(1-xz_2^2q^2t^2)(1-xz_1z_3^2qt^4)
            + xyz_2^3z_3^2qt^6/(1-yz_3^2qt)(1-yz_2z_3t^3)(1-xz_2^2q^2t^2)(1-xz_2^2z_3qt^3)(1-xz_1z_3^2qt^4)
            + xyz_1z_2^2z_3qt^8/(1-yz_2qt^2)(1-yz_2z_3t^3)(1-xz_2^2q^2t^2)(1-xz_1z_3^2qt^4)(1-xz_1z_2qt^5)
            + x^2z_1z_2^3z_3q^2t^8/(1-yz_2z_3t^3)(1-xz_2^2q^2t^2)(1-xz_2^2z_3qt^3)(1-xz_1z_3^2qt^4)(1-xz_1z_2qt^5)
            + xz_1z_2z_3t^6/(1-yz_2z_3t^3)(1-xz_2^2z_3qt^3)(1-xz_1z_3^2qt^4)(1-xz_1z_2qt^5)(1-xz_1z_2z_3t^6)  .


Setting z_1,z_2,z_3=1 and adding all the generating functions yields Theorem <ref>.





§ FURTHER DIRECTIONS




 §.§ Revisiting Xin and Zhang's conjecture

As previously mentioned, Xin and Zhang <cit.> conjectured that if λ=((a+1)^s,a^n-s) with 0≤ s≤ n, then C_λ(q,t) is q,t-symmetric. 
Since C_λ(q,t) involves taking a sum over possibly multiple C_k⃗(q,t), it is interesting to further ask for which k⃗ we have q,t-Catalan symmetry. 
For example, Xin and Zhang note that C_λ = (1,1,1,3)(q,t) is not q,t-symmetric. 
Breaking this up into the constituent C_k⃗(q,t), we find that 

    C_k⃗ = (1,1,1,3)(q,t)      and      C_k⃗ =
    (3,1,1,1)(q,t)
 
are q,t-symmetric (see Figure <ref>), but 

    C_k⃗ = (1,1,3,1)(q,t)      and      C_k⃗ =
    (1,3,1,1)(q,t)
 
are not q,t-symmetric (see Figure <ref>).
Since the asymmetries present in C_k⃗ = (1,1,3,1)(q,t) and C_k⃗ = (1,3,1,1)(q,t) do not cancel out, C_λ = (1,1,1,3)(q,t) is not q,t-symmetric. 





This contrasts with examples which fall within Xin and Zhang's conjecture, such as C_λ = (1,1,1,2)(q,t) which is q,t-symmetric. 
There, 

    C_k⃗ = (1,1,1,2)(q,t)      and      C_k⃗ =
    (2,1,1,1)(q,t)
 
are q,t-symmetric (see Figure <ref>), but 

    C_k⃗ = (1,1,2,1)(q,t)      and      C_k⃗ =
    (1,2,1,1)(q,t)
 
are not q,t-symmetric (see Figure <ref>).
However, this time the asymmetries present in C_k⃗ = (1,1,2,1)(q,t) and C_k⃗ = (1,2,1,1)(q,t) do cancel out, so C_λ = (1,1,1,2)(q,t) is q,t-symmetric.







 §.§ Varying the last parameter

We start with a result which shows that the final entry in the vector k⃗ does not affect its area or bounce.


    Let 
    
    (k_1,…,k_j)∈_>0^j    and    (a_1,…,a_j)∈_≥ 0^j

    be such that the a_i are the number of sequential east steps of a valid (k_1,…,k_j)-Dyck path (i.e. take a_1 east steps after the size k_1 north steps, etc.). 
    Let l,m∈_>0. 
    The bounce and area formulas for the (k_1,…,k_j,l)-Dyck path and (k_1,…,k_j,m)-Dyck path with east steps (a_1,…,a_j) are identical.



In the course of applying the bounce algorithm (<ref>), bounce can be computed as soon as the top row of the rank tableau is complete.
We claim that all the numbers in the top row of the rank tableau are entered without knowledge of the length of the final north step. 

The initial numbers in the top row are entered while the bounce path lies below the final north steps. Eventually we arrive at a point P_i such that traveling north from P_i the next east steps are along the top boundary of the n× n
square containing the Dyck path.
We count v_i, the number of  north steps this last leg of the bounce path traverses, and enter this many copies of i in the rank tableau—completing the top row, and arriving at the moment we can compute bounce.  
Therefore, the values of l and m are immaterial to the bounce computation, only the fact that they are non-zero matters.

For either Dyck path we have
 
    r_1   =k_1-a_1  , 
    
         r_2   =k_1-a_1+k_2-a_2  . 
       ⋮
    
         r_j   =k_1-a_1+…+k_j-a_j  ,

and since area for either path is the sum of these r_i, area also does not depend on l,m.


By combining Proposition <ref> with the definition of
C_k⃗(q,t), we obtain the following result. 


For any positive integers k_1,…,k_j,m,l, 
    
    C_k⃗ = (k_1,…,k_j,m)(q,t) = C_k⃗ =
    (k_1,…,k_j,l)(q,t)   .



Without loss of generality, assume m<l. 
Given a (k_1,…,k_j,m)-Dyck path, we can add l-m units to the final north step, and l-m east steps after the final east step.  
The result is a valid (k_1,…,k_j,l)-Dyck path.  
    
On the other hand, given a (k_1,…,k_j,l)-Dyck path, we can subtract l-m units from the final north step. 
We need to delete l-m east steps from the end in order to create a valid (k_1,…,k_j,m)-Dyck path. 
The total number of east steps that can occur in the original (k_1,…,k_j,l)-Dyck path prior to the final north step is k_1+…+k_j, and the final total is k_1+…+k_j+l, so at least l east steps must occur after the final north step.  
Therefore, we can delete l-m east steps from the end and still have a valid Dyck path.

We have described a bijection on (k_1,…,k_j,m)- and (k_1,…,k_j,l)-Dyck paths, and by Proposition <ref> this bijection preserves area and bounce, proving the claim.



We can see Corollary <ref> in action in the following examples:

    
  * For m∈_>0 the bounce and area formulas for (k,k,k,m)-Dyck paths are the same as the bounce and area formulas for (k,k,k,k)-Dyck paths. 
    
  * The formula for all (k_1,k_2,m)-Dyck paths are also the same regardless of m∈_>0, the formula for bounce in the (k_1,k_2,k_3)-Dyck path case does not depend on k_3.
    
  * Similarly, Niu points out in <cit.> that the formula for (k,k,k,k-1)-Dyck paths is the same as that of (k,k,k,k)-Dyck paths.  
    
  * If Conjecture <ref> holds, Corollary <ref> implies that for positive integers k,a,l, that C_k⃗(q,t) with k⃗ = (k,a,…,a,l) is also q,t-symmetric. 

Note Corollary <ref> does not allow us to extend the bounce formula for a (k_1,…,k_m)-Dyck path to a bounce formula for a (k_1,…,k_m,k_m+1)-Dyck path; the fact that we have an additional north step of size k_m+1 significantly changes the computation. 





 §.§ Other generalized q,t-Catalan numbers


We briefly discuss the relationship between Xin-Zhang's refined q,t-Catalan numbers and other generalizations which have appeared in the literature. 
We already saw that with k⃗ = (1,…,1_n), we recover the classical q,t-Catalan numbers C_n(q,t). 
Taking k⃗ = (m,…,m_n), we recover the higher q,t-Catalan numbers C_n^(m)(q,t) <cit.>, which are a special case of the rational q,t-Catalan numbers. 
Combinatorial formulas for C_n^(m)(q,t) were conjectured by Haglund et al. <cit.> and Loehr <cit.>, and proven as a consequence of the (m,n)-shuffle theorem, conjectured by Haglund et al. <cit.>, refined by Bergeron et al. <cit.>, and proven by Mellit <cit.>.
Note that in both of these cases, there is no difference between C_k⃗(q,t) and C_λ(q,t) since there is only one unique rearrangement of the vector. 

In 2020, Gorsky et al. <cit.> introduced a
generalized q,t-Catalan number, which was further studied by Blasiak et al.
<cit.>, where it is defined as

    C_b := ⟨ s_(|b|), ω(D_b·
    1)⟩  ,

s_(|b|) is a single-row Schur function, ω is an involution on symmetric functions, and D_𝐛 is a certain operator in the Schiffmann algebra introduced by Negut. 
It is interesting to speculate on the relationship between C_b and the q,t-Catalan numbers of Xin-Zhang, since they depend on a vector of integers, or positive integers, respectively, and generalize the higher q,t-Catalan numbers.
For example, the generalized q,t-Catalan number 

    C_𝐛 = (1,2,1,0) = q^4 + q^3t + q^2t^2 + qt^3 + t^4 + q^2t + qt^2
 
equals the refined q,t-Catalan number 

    C_k⃗ = (1,2,1) = q^4 + q^3t + q^2t^2 + qt^3 + t^4 + q^2t + qt^2
 
(and neither is a higher q,t-Catalan number).
Furthermore, Gorsky et al. showed that the generalized q,t-Catalan number does not depend on b_1, the first entry of the vector 𝐛.
This is similar to our remark in Section <ref> that the refined q,t-Catalan number does not depend on the final entry of k⃗ (the way the diagrams are drawn is flipped between these settings).

This might suggest that C_𝐛(q,t) and C_k⃗(q,t) are equal when 𝐛  = (b_1, …, b_ℓ, 0) = (b_ℓ, …, b_1) = k⃗, however, the following example shows that, in general, this is not the case: 

    C_𝐛 = (2,1,2,2,0)(q,t) - C_k⃗ = (2,2,1,2)(q,t) = q^2t^7 +
    q^4t^4 + q^3t^5 - q^4t^3 - q^3t^4 - q^2t^5   .

Perhaps understanding when C_𝐛(q,t) = C_k⃗(q,t) would allow for some progress on a conjecture of Negut <cit.> (and extended by Blasiak et al. <cit.>).

Gorsky et al. <cit.> give an example where the coefficients of the generalized q,t-Catalan number C_b are negative.
Some of Xin-Zhang's q,t-Catalan numbers are not q,t-symmetric, while all of the generalized q,t-Catalan numbers are q,t-symmetric by construction from Negut's operator (see <cit.>). 
As a result, neither generalization is contained within the other. 





§ ACKNOWLEDGEMENTS & FUNDING

This work was initiated at the 2023 GEMS of Combinatorics Workshop held at the American Institute of Mathematics (AIM). 
The authors thank the workshop organizers and AIM for fostering a fruitful research atmosphere. 
We thank Carolina Benedetti, Laura Colmenarejo, and Sophie Rehberg for conversations at the start of the project.
We also thank Mark Haiman for helpful conversations.
Lentfer is supported by the National Science Foundation Graduate Research Fellowship DGE-2146752. 
Vindas-Meléndez was partially supported by the National Science Foundation under Award DMS-2102921. 
Waddle is partially supported by the National Science Foundation RTG grant DMS-1840234.

amsplain







 Paper Title: Efficiency of the singular vector of a reciprocal matrix and comparison to the Perron vector.





Efficiency of the singular vector of a reciprocal matrix and comparison to the
Perron vector
    Susana FurtadoEmail: sbf@fep.up.pt The work of this author was
supported by FCT- Fundação para a Ciência e Tecnologia, under
project UIDB/04721/2020. Corresponding author.
CEAFEL and Faculdade de Economia 
Universidade do Porto
Rua Dr. Roberto Frias
4200-464 Porto, Portugal
Charles R. Johnson Email: crjohn@wm.edu. 
Department of Mathematics
College of William and Mary
Williamsburg, VA 23187-8795
    
==================================================================================================================================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
In models using pair-wise (ratio) comparisons among alternatives (reciprocal
matrices A) to deduce a cardinal ranking vector, the right Perron
eigenvector was traditionally used, though several other options have emerged.
We propose and motivate another alternative, the left singular vector (Perron
eigenvector of AA^T). Theory is developed. We show that for reciprocal
matrices obtained from consistent matrices by modifying one column, and the
corresponding row, the cone generated by the columns is efficient, implying
that the Perron vector and the left singular vector of such matrices are
always efficient. In addition, the two vectors are compared empirically for
random matrices. With regard to efficiency, the left singular vector
consistently outperforms the right Perron eigenvector, though both perform
well for large numbers of alternatives.



Keywords: column perturbed consistent matrix, decision analysis,
efficient vector, Perron eigenvector, reciprocal matrix, singular vector.

MSC2020: 90B50, 91B06, 15A18, 15B48 





§ INTRODUCTION


An n-by-n positive matrix A=[a_ij] is called reciprocal if
a_ji=1/a_ij for every pair 1≤ i,j≤ n. The diagonal
entries are all 1. We denote the set of such matrices by 𝒫𝒞_n.
Reciprocal matrices A∈𝒫𝒞_n arise as an array of pair-wise
(ratio) comparisons among n alternatives. In this context, typically a
cardinal ranking vector is to be extracted from A for decision problems
about business alternatives
<cit.>; and other applications,
in voting theory and currency exchange rates, are emerging. Historically, the
(right) Perron eigenvector of Ahas been used for this vector of weights,
as suggested in <cit.>. However, many natural concerns about
this choice have emerged (see, for example, <cit.>). Here, we suggest an
entirely new, but natural alternative, the Perron eigenvector of AA^T,
sometimes called the left singular vector of A, as it is the left
singular vector of A associated with the largest singular value.

The comparisons embodied in a reciprocal A are consistent (a_ik
=a_ija_jk for all 1≤ i,j,k≤ n) if and only if, for some positive
n-vector w, A=ww^(-T) (in which w^(-T) denotes the entry-wise
inverse of the transpose of w). In this case w is the only natural weight
vector (up to a factor of scale), but consistency is unlikely in independent
pair-wise comparisons. One agreeable criterion to be a weight vector w is
that the consistent matrix built from w, W=ww^(-T), be a Pareto optimal
approximation to A. This means that, if | A-V|≤| A-W| , absolute value entry-wise, in which
V=vv^(-T) for a positive vector v, then v is proportional to w. In
this literature, such a positive vector w is called efficient (for
A). However, when A is inconsistent, there will be infinitely many,
projectively distinct efficient vectors. Denote by ℰ(A) the
collection of all vectors efficient for A. It is now known that every column
of A is efficient <cit.>, the entry-wise geometric convex hull of the
columns <cit.> and, in particular, the simple entry-wise geometric mean of
all the columns <cit.>, are efficient. However, the Perron vector is
not always efficient <cit.>. A graph characterization of
efficiency was given in <cit.> (see also <cit.>). Several other
developments on efficiency have appeared (see, for example,
<cit.>).

Since the definition of efficiency for A∈𝒫𝒞_n involves rank
1 approximation to A, and the best rank 1 approximation (in the
Frobenius norm) is the normalized outer product of the right and left singular
vectors of A (associated with the spectral norm) <cit.>, it seems
natural to test one of these for efficiency. Of course, the best rank 1
approximation is not usually reciprocal, let alone consistent.

Here, we primarily compare the left singular vector of A∈𝒫𝒞_n
(i.e. the Perron vector of AA^T) to the Perron eigenvector under the lens
of efficiency. In the next section we amass the technical background that we
need. Then, in Section <ref>, we develop some additional technology for
understanding efficiency. In particular, it is noted that, in the event that
ℰ(A) is convex, both the singular vector and the Perron vector are
efficient. In Section <ref> we give a unified proof that, among column
perturbed consistent matrices, both vectors are always efficient. Convexity of
ℰ(A) occurs when A lies in the class of simple perturbed
consistent matrices introduced in <cit.> (see also <cit.>), but
unfortunately it is not yet known in general for which A ℰ(A) is
convex. In Section <ref>, we accumulate considerable information comparing
the two vectors, via simulation. The singular vector performs better than the
Perron eigenvector in all dimensions n>3, both are more frequently efficient
the higher the dimension (approaching efficiency with probability 1), and
more detailed comparison is presented. We give some final remarks in Section
<ref>.



§ BACKGROUND


The set 𝒫𝒞_n is closed under monomial similarity, that is,
similarity via a product of a permutation matrix and a positive diagonal
matrix (monomial matrix) <cit.>. Fortunately, such transformation
interfaces with efficient vectors in a natural way.


<cit.> Suppose that A∈𝒫𝒞_n and w∈ℰ(A). If S is an n-by-n monomial matrix, then Sw∈ℰ(SAS^-1).




Given A=[a_ij]∈𝒫𝒞_n and a positive vector

    w=[
    [ w_1   ⋯ w_n ]]  ^T,

define G(A,w) as the directed graph (digraph) with vertex set {1,…
,n} and a directed edge i→ j if and only if w_i≥
a_ijw_j, i≠ j.

In <cit.> the authors proved that the efficiency of w can be
determined using G(A,w). In <cit.> a shorter matricial proof of
this result has been given.


<cit.> Let A∈𝒫𝒞_n. A positive
n-vector w is efficient for A if and only if G(A,w) is a strongly
connected digraph, that is, for all pairs of vertices i,j, with i≠ j,
there is a directed path from i to j in G(A,w).


We recall that the (right) Perron eigenvector of an n-by-n positive matrix
A is a (right) eigenvector of A associated with the spectral radius
ρ(A) of A (called the Perron eigenvalue of A) <cit.>. It is known
that, up to a constant factor, all its entries are positive and, with some
normalization, as having the sum of the entries equal to 1, it is unique.

If a positive matrix has rank 1, then any column is a Perron vector of the
matrix (all the other projectively distinct eigenvectors are associated with
the eigenvalue 0). When A is consistent, A=ww^(-T), any column of A,
and of AA^T, is a multiple of w. Thus, the Perron vectors of A and
AA^T (as any column) are projectively equal to w and, thus, are
efficient vectors for A. Here, we focus upon the efficiency of the Perron
eigenvector of AA^T for a reciprocal matrix A. Note that, since AA^T
is symmetric, its right and left Perron vectors coincide.



We denote by 𝒞(A) the (convex) cone generated by the columns of
A, that is, the set of nonzero, nonnegative linear combinations of the
columns of A. Note that, according to our definition, 0 is not in
𝒞(A). It is important to note that both the right Perron vector of
A and the Perron vector of AA^T lie in 𝒞(A).


For any positive n-by-n matrix A (in particular,
A∈𝒫𝒞_n), both the right Perron eigenvector of A and the
Perron eigenvector of AA^T lie in 𝒞(A).



If Av=ρ(A)v, v may be taken to be entry-wise positive. So Av lies in
𝒞(A), and since v is proportional to Av, via the positive
number ρ(A), v∈𝒞(A) also. If AA^Tu=ρ(AA^T)u, since
AA^T is positive, ρ(AA^T) is positive and u may be taken to be
positive. Now, A^Tu is positive so that A(A^Tu)∈𝒞(A), and,
as u is proportional, via ρ(AA^T), u∈𝒞(A) also.




Two simple lemmas will be helpful.


Let A and S be n-by-n matrices, with S monomial. If
𝒞(A) is the cone generated by the columns of A then
S𝒞(A) is the cone generated by the columns of SAS^-1.



It is enough to show that 𝒞(SAS^-1)⊆ S𝒞(A), as
then applying this result to A instead of SAS^-1 we get the other
inclusion. Let S=DP, in which D=*diag(λ_1
,…,λ_n) is a positive diagonal matrix and P is a permutation
matrix (the proof is similar if S=PD). Let a_i and a_i^'
denote the ith columns of A and A^'=SAS^-1, respectively,
i=1,…,n. Let x_1,…,x_n≥0 (not all 0). Then,

    x_1a_1^'+⋯+x_na_n^'=S(  x_1/λ_1a_σ(1)+⋯+x_n/λ_na_σ
    (n))  ∈ S𝒞(A),

where σ is the permutation of {1,…,n} associated with P^T.



If 𝒞 is a convex subset of ℝ^n, and
S is an n-by-n real matrix, then S𝒞 is also convex.



Let Sw_1,Sw_2 be arbitrary vectors in S𝒞. Let λ
_1,λ_2>0 summing to 1. Since w_1,w_2∈𝒞 and
𝒞 is convex, λ_1w_1+λ_2w_2∈𝒞.
Then, λ_1Sw_1+λ_2Sw_2=S(λ_1w_1+λ_2
w_2)∈ S𝒞.


Let A∈𝒫𝒞_n and S be an n-by-n monomial
matrix. Then, ℰ(A) is convex if and only if ℰ
(SAS^-1) is convex.



By Lemma <ref>, ℰ(SAS^-1)=Sℰ(A). Thus, by Lemma
<ref>, ℰ(A) convex implies ℰ(SAS^-1) convex.
The converse follows similarly.




§ EFFICIENCY AND THE CONE GENERATED BY THE COLUMNS OF A RECIPROCAL
MATRIX


The cone generated by the columns of A∈𝒫𝒞_n may or may not be
contained in ℰ(A). But it is quite helpful when it is.


Let A∈𝒫𝒞_n and suppose that 𝒞
(A)⊆ ℰ(A). Then, both the Perron vector of AA^T and the
right Perron vector of A lie in ℰ(A). Moreover, 𝒞
(AA^T)⊆ℰ(A).



By Lemma <ref>, both the right Perron eigenvector of A and the Perron
eigenvector of AA^T lie in 𝒞(A), so that the first claim
follows. The second claim is a consequence of the fact that 𝒞
(AA^T)⊆ 𝒞(A).


Recall that each column of A∈𝒫𝒞_n lies in
ℰ(A). So, if ℰ(A) is convex, 𝒞
(A)⊆ℰ(A).


Let A∈𝒫𝒞_n and suppose that ℰ(A) is convex. Then,
both the Perron vector of AA^T and the right Perron vector of A lie in
ℰ(A).




In Theorem <ref>, we have identified a situation (𝒞(A)
contained in ℰ(A)) in which both the left singular vector and the
Perron vector of a reciprocal matrix are efficient for the matrix. If n=3,
this situation always occurs since ℰ(A) is necessarily convex, as
we shall see. However, for any n>3, there exists A∈𝒫𝒞_n such
that 𝒞(A) is not contained in ℰ(A), in which case
ℰ(A) is not convex. Any Awith inefficient right Perron vector
establishes this (and this situation may occur for any n≥4
<cit.>). When 𝒞(A) is not contained in
ℰ(A), either or both the Perron vector of AA^T or the right
Perron vector of A (or none) may lie in ℰ(A).


Consider the reciprocal matrix

    A=[
    [      1 1.1742 0.5647 4.4912 0.3633; 0.8516      1 1.4198  0.734 0.8444; 1.7709 0.7043      1 1.3358 1.7356; 0.2227 1.3624 0.7486      1 5.4467; 2.7525 1.1843 0.5762 0.1836      1 ]]  .

The Perron eigenvectors of A and AA^T are, respectively,

    w=[
    [ 1.4008 0.8134 1.1503 1.3488      1 ]]  ^T and v=[
    [ 1.4667 0.8522 1.3142 2.2704      1 ]]  ^T.

The vector w is not efficient for A, while v is.



Consider the reciprocal matrix

    A=[
    [      1 1.6119 1.1855 2.1413 2.6124; 0.6204      1 1.6338 3.7767 2.1376; 0.8435 0.6121      1 0.2347 4.6488; 0.4670 0.2648 4.2608      1 0.2462; 0.3828 0.4678 0.2151 4.0617      1 ]]  .

The Perron eigenvectors of A and AA^T are, respectively,

    w=[
    [ 1.4842 1.5318 1.1940 1.0829      1 ]]  ^T and v=[
    [ 1.1507 1.3620 1.0590 0.7746      1 ]]  ^T.

The vector w is efficient for A, while v is not.



Consider the reciprocal matrix

    A=[
    [       1  1.2767  0.5382  1.9522  0.2486;  0.7833       1  0.5521  0.2793  2.2088;   1.858  1.8113       1    1.54  1.1093;  0.5122  3.5804 0.64935       1  0.7887;  4.0225  0.4527  0.9015  1.2679       1 ]]  .

The Perron eigenvectors of A and AA^T are, respectively,

    w=[
    [ 0.6713 0.7041 0.9907 0.8591      1 ]]  ^T and v=[
    [ 0.6131 0.5337 0.8907 0.8152      1 ]]  ^T.

Both vectors w and v are not efficient for A.


Note that, from Theorem <ref>, in Examples <ref>, <ref> and
<ref> 𝒞(A) is not contained in ℰ(A).


Consider the reciprocal matrix

    A=[
    [      1 0.4245   0.21 1.0186 0.7814; 2.3556      1 0.0168 5.6974 0.9107; 4.7619 59.524      1 0.4913 0.1877; 0.9817 0.1755 2.0354      1 1.5855; 1.2798 1.0981 5.3277 0.6307      1 ]]  .

The Perron eigenvectors of A and AA^T are, respectively,

    w=[
    [ 0.1623 0.3658 1.9915 0.5080      1 ]]  ^T and v=[
    [  0.3988  0.9565 45.6075  0.2369       1 ]]  ^T.

Both vectors w and v are efficient for A. However w+v is not efficient
for A, showing that 𝒞(A) is not contained in ℰ(A).




§ COLUMN PERTURBED CONSISTENT MATRICES


A∈𝒫𝒞_n is called a column perturbed consistent matrix
if A has a consistent (n-1)-by-(n-1) principal submatrix or, in other
words, if A is a reciprocal matrix obtained from a consistent matrix by
modifying one column and the corresponding row. (Of course, if a column
perturbed consistent matrix has rank 1, it is consistent.) Since an
(n-1)-by-(n-1) consistent matrix is positive diagonal similar to the
matrix J_n-1 of all 1's, and the columns of A may be permuted to be in
any desired order, it follows that A is monomially similar to a matrix of
the form

    [
    c]c|c
    J_n-1   [   x_1;     ⋮; x_n-1 ]
    [   1/x_1       ⋯ 1/x_n-1 ]   1
     ]  ,

with x_1≥⋯≥ x_n-1>0.


Suppose that A∈𝒫𝒞_n is a column perturbed consistent
matrix. Then, 𝒞(A)⊆ ℰ(A).



Taking into account Lemmas <ref> and Lemma <ref>, we may assume
that A has the form (<ref>) with x_1≥⋯≥ x_n-1>0.
Consider the linear combination of the columns of A: w=As, with

    s=[
    [   s_1;     ⋮; s_n-1;   s_n ]]  ,

s_i≥0, i=1,…,n (not all 0). We want to see that
w∈ℰ(A), or, equivalently, by Theorem <ref>, that G(A,w)
is strongly connected. Let ww^(-T)-A=[b_ij]. We will show that
b_i,i+1≥0 for i=1,…,n-1, and b_n,1≥0, implying that

    1→2→⋯→ n→1

is a cycle in G(A,w), and, thus, G(A,w) is strongly connected.

Let a_i^T denote the ith row of A. For i=1,…,n-2, we have

    b_i,i+1     =(  a_i^Ts)  (  a_i+1^Ts)
    ^(-1)-1
         =s_1+⋯+s_n-1+x_is_n/s_1+⋯+s_n-1+x_i+1s_n
    -1
         =s_n(x_i-x_i+1)/s_1+⋯+s_n-1+x_i+1s_n≥0.



We also have

    b_n-1,n     =(  a_n-1^Ts)  (  a_n^Ts)
    ^(-1)-x_n-1
         =s_1+⋯+s_n-1+x_n-1s_n/s_1/x_1+⋯
    +s_n-1/x_n-1+s_n-x_n-1
         =s_1+⋯+s_n-1-x_n-1(s_1/x_1+⋯
    +s_n-1/x_n-1)/s_1/x_1+⋯+s_n-1/x_n-1
    +s_n
         =s_1(1-x_n-1/x_1)+⋯+s_n-2(1-x_n-1
    /x_n-2)/s_1/x_1+⋯+s_n-1/x_n-1+s_n≥0.

Finally,

    b_n,1     =(  a_n^Ts)  (  a_1^Ts)
    ^(-1)-1/x_1
         =s_1/x_1+⋯+s_n-1/x_n-1+s_n/s_1+⋯+s_n-1+x_1s_n-1/x_1
         =s_1/x_1+⋯+s_n-1/x_n-1-1/x_1
    (s_1+⋯+s_n-1)/s_1+⋯+s_n-1+x_1s_n
         =s_2(1/x_2-1/x_1)+⋯+s_n-1(1/x_n-1-1/x_1)/s_1+⋯+s_n-1+x_1s_n≥0.





If A∈𝒫𝒞_n is a column perturbed consistent matrix, then both
the Perron vector of AA^T and the right Perron vector of A lie in
ℰ(A).



Applying Theorems <ref> and <ref>, gives the desired conclusion.




The efficiency of the Perron vector of a column perturbed consistent matrix
was obtained in <cit.>, but this gives a unified and simpler proof.

It may happen that, for a column perturbed consistent matrix
A∈𝒫𝒞_n, we have that ℰ(A) is not convex. For this
to happen it must be that n>3, as we shall see after the next example.


Let

    A=[
    [   1   1   1   1   1   4;   1   1   1   1   1   3;   1   1   1   1   1   2;   1   1   1   1   1   1;   1   1   1   1   1   1; 1/4 1/3 1/2   1   1   1 ]]  .

Let

    q=[
    [ 4 3 4 2 2 1 ]]  ^T

and

    p=[
    [ 3 4 4 2 2 2 ]]  ^T.

The vectors p and q are efficient for A. However, p+q is not efficient
for A, implying that ℰ(A) is not convex.


A reciprocal matrix A∈𝒫𝒞_n that is obtained from a consistent
matrix by changing two entries in symmetrically placed positions is a
particular case of a column perturbed consistent matrix, and was called in
<cit.> a simple perturbed consistent matrix. Thus, as a
consequence of Theorem <ref>, 𝒞(A)⊆ ℰ(A),
implying that the Perron vector of AA^T and of A are efficient. This
gives a much simpler proof of the known efficiency of the Perron vector of A
<cit.>. We note that any 3-by-3 reciprocal matrix is a simple
perturbed consistent matrix <cit.>.

A simple perturbed consistent matrix A∈𝒫𝒞_n is monomial
similar to a matrix, say S_n(x_1), as in (<ref>), with x_1≥1
and x_2=⋯=x_n-1=1. The efficient vectors w for S_n(x_1)
were described in <cit.>. Then, by Lemma <ref>, the efficient vectors
for A may be obtained.


<cit.> Let n≥3 and x≥1. Let w=[
[   w_1     ⋯ w_n-1   w_n ]]  ^T be a positive n-vector. Then w is efficient for
S_n(x)∈𝒫𝒞_n if and only if

    w_n≤ w_i≤ w_1≤ xw_n, for i=2,…,n-1.





If A∈𝒫𝒞_n is a simple perturbed consistent matrix
then ℰ(A) is convex.



Taking into account Corollary <ref>, we may assume that A=S_n(x)
with x≥1. Since a vector w in ℰ(A) is defined by a finite
set of linear inequalities on its (positive) entries, as described in Lemma
<ref>, it follows that ℰ(A) is convex.




The cone 𝒞(A) may be properly contained in ℰ(A), even
when ℰ(A) is convex, as we next illustrate.


For the simple perturbed consistent matrix S_5(4)∈𝒫𝒞_5, the
vector [
[ 4 2 3 4 1 ]]  ^T is efficient for A and is not in 𝒞(A), as the
middle entries are distinct.




§ EMPIRICAL EFFICIENCY OF THE SINGULAR VECTOR AND THE RIGHT PERRON
VECTOR


For each n≤25, we generate n-by-n matrices A_i∈𝒫𝒞_n, i=1,…10000, as follows. We first generate an n-by-n matrix
C_i with entries from a uniform distribution in (0.1,15) and then
construct the matrix A_i as C_i∘ C_i^(-T), in which ∘
denotes the Hadamard product and C_i^(-T) denotes the transpose of the
entrywise inverse of C_i. For each n, we count the number of matrices
A_i for which the Perron vectors of A_i and of A_iA_i^T are
efficient for A_i. We also generate a random positive vector for each
A_i, with entries from a uniform distribution in (0,5), and test it for
efficiency, as a baseline. The results are presented in Figure <ref>. We
can observe that, for n close to 12 and beyond, almost always the right
Perron vector and the left singular vector are efficient for A_i. Not
surprisingly, the random vector is efficient much less often.

















We then generate 10000 random matrices A_i∈𝒫𝒞_n, for each
n=4,5,7,9,12,15,20, following the same procedure as before and obtain for
each matrix the Perron vector v_P and the singular vector v_S. In
Table <ref> we compare the efficiency of these vectors for each matrix.
We can observe that, when just one of the vectors v_P or v_S is
efficient, v_S is the one that is efficient more often.









§ CONCLUSIONS


Motivated by the fact that the normalized outer product of the left and right
singular vectors (associated with the spectral norm) of a matrix A give the
best rank 1 approximation to A (in the Frobenius norm), we developed theory
for the Perron vector of AA^T for A∈𝒫𝒞_n and its efficient
vectors. This is compared to the right Perron vector for A. These vectors
are efficient for a reciprocal matrix A∈𝒫𝒞_n that contains a
consistent (n-1)-by-(n-1) principal submatrix, due to the shown fact that
any vector in the cone generated by the columns of A is efficient for A.
Then, simulations, generating random matrices A∈𝒫𝒞_n, test how
often each vector is efficient, as a function of n. Typically, the Perron
vector of AA^T performs better than the traditional right Perron vector of
A, though both perform well for large n. But as the singular vector is
nearly as easy to calculate, it should be preferred.



We declare there are no conflicts of interests.

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 Paper Title: A Search for High-Threshold Qutrit Magic State Distillation Routines.






































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====


theoremTheorem

conjectureConjecture





1cm







Department of Physics and Computer Science, Dayalbagh Educational Institute, Agra, India






§ ABSTRACT 
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We carry out a search for high-threshold magic state distillation routines for a highly-symmetric qutrit magic state known as the strange state. Our search covers a large class of [[n,1]]_3 qutrit stabilizer codes with up to 23 qutrits, and   is facilitated by a theorem that relates the distillation performance of a qudit stabilizer code to its weight-enumerators. We could not find any code with n<23 qutrits that distills the strange state with better than linear noise suppression, other than the 11-qutrit Golay code. However, for n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic. 














§ INTRODUCTION


Contextuality was identified as a necessary and possibly-sufficient condition for universal quantum computing in <cit.>. The argument of <cit.> is based on magic state distillation <cit.> for qudits of odd-prime dimension, and later extended to qudits of arbitrary odd dimensions in <cit.>, and continuous variable systems in <cit.>.[There are certain subtleties associated with state-independent contextuality for qubits and qudits of even dimension – for simplicity, we focus exclusively on qudits of odd-prime dimension in this paper.] The authors of these works showed that qudit states that do not exhibit contextuality with respect to stabilizer measurements have a non-negative discrete Wigner function <cit.>. The set of such states is known as the Wigner polytope. Because Clifford unitaries and stabilizer measurements are efficiently simulable for states in the Wigner polytope <cit.>, they thus cannot be distilled into pure magic states. 

In order to demonstrate that contextuality is not only necessary but also sufficient for qutrit quantum computation, one must demonstrate that a supply of qudits that do exhibit contextuality with respect to stabilizer measurements may be used to achieve universal quantum computation. In the language of magic state distillation, this translates into the question, can any qudit mixed state outside the Wigner polytope be distilled into a pure magic state?

For qudits of odd prime dimension p, the Wigner polytope is a convex polytope with p^2 facets that lives in the p^2-1 dimensional space of qudit density matrices.  In <cit.>, it was shown that no finite magic state distillation routine can distill all states that lie outside one of the faces of the Wigner polytope[Some distillation routines that distill qutrit states up to one of the hyperedges of the Wigner polytope were found in <cit.>.], generalizing the analogous result for qubit states that lie outside the stabilizer polytope <cit.>. However, the possibility remains that a sequence of magic state distillation routines, based on stabilizer codes of increasing length n may distill states arbitrarily close to a face of the Wigner polytope. Is there any evidence that such a sequence of magic state distillation routines exist?

The problem simplifies if one focuses on qutrits. There exists a qutrit magic state, first identified by Howard and van Dam <cit.>, sometimes known in the literature in the qutrit strange state |S⟩ <cit.>, that lies directly above one of the facets of the Wigner polytope[See <cit.> for another application of the strange state.]. As discussed in <cit.>, noisy |S⟩ states can be twirled via Clifford unitaries to lie on a line connecting a pure |S⟩ state to the maximally mixed state:

    ρ̂(ϵ)=(1-ϵ) |S⟩⟨S| + ϵ1/3,

with all noise converted to depolarizing noise, parameterized by ϵ. Any state ρ̂(ϵ) for ϵ <3/4 lies outside the Wigner polytope and exhibits contextuality with respect to stabilizer measurements. We then ask whether or not there exists a family of n-to-1 magic state distillation routines that distill the strange state with a threshold approaching ϵ = 3/4 as n →∞? [An additional consideration is the increase in overhead cost of distillation as n →∞.]

At the time <cit.> was published, and for several years thereafter, while some qutrit and qudit magic state distillation routines had been proposed <cit.>, no magic state distillation routine that distilled the Howard van Dam strange state was known. It was later discovered that an 11-qutrit CSS code based on the ternary Golay code can distill the |S⟩ state, with a threshold of ϵ_* = 0.38 <cit.>. Do there exist any other qutrit stabilizer codes that distill the strange state? If so, how do their thresholds compare to that of the 11-qutrit Golay code?[Recently, <cit.> reported the existence of 13-qutrit and 29-qutrit codes that could distill the qutrit strange state with high thresholds. However, we attempted to reproduce these results and found that neither of the codes reported in <cit.> are able to distill the qutrit strange state. See Appendix <ref>.] In this paper, we carry out a computational search over reasonably small qutrit error-correcting codes to help answer these questions. 

To facilitate our search, we prove a simple theorem connecting the performance of a stabilizer code for qudit magic state distillation its complete weight enumerator. For distillation routines for the qutrit strange state, this formula simplifies and depends only on the stabilizer code's simple weight enumerator. 

Our search is limited by practical considerations, as well as the existing classifications of qutrit error-correcting  codes in the literature – namely, the classification of qutrit stabilizer states in <cit.> and a classification of self-orthogonal classical ternary codes available on <cit.>. We carried out a search over all [[n,1]]_3 stabilizer codes with n ≤ 9, and a search over all [[11,1]]_3 stabilizer codes that can be obtained from a [[12,0,6]]_3 stabilizer state. For such codes, we demand transversality of a particular single-qutrit gate (the square of the qutrit Hadamard gate), which allows us to restrict our search to projection onto the trivial syndrome of each stabilizer code. We also searched over all [[n,1]]_3 CSS codes for odd n ≤ 23 that possess a complete set of transversal single-qudit Clifford gates. 

We found that none of the codes we searched with n<23 could distill the |S⟩ state with better-than-linear[We also found a few 9 and 11 qutrit codes that could distill the state with linear noise suppression.] noise suppression, other that the 11-qutrit Golay code of <cit.>. However, for n=23, we found over 600 CSS codes that could distill the strange state with cubic noise suppression – which is approximately 1/3 of all the codes we could construct from the ternary self-orthogonal codes listed in <cit.> – suggesting that for large codes, magic state distillation is somewhat generic. None of these 23-qutrit codes, however, had a threshold that exceeds that of the 11-qutrit Golay code.   

Let us point out that the new distillation routines we find here are mainly of theoretical interest. The success probabilities are quite low, and far better yields are obtained via triorthogonal codes <cit.> (see <cit.> for constructions of qutrit and qudit triorthogonal codes). Nevertheless, the CSS codes we study have a complete set of transversal Clifford gates, and may turn out to be useful for fault-tolerant quantum computation in other settings.  

Our paper is organized as follows. In section <ref> we briefly review some background material. In section <ref> we derive a relation between weight-enumerators and the performance of magic state distillation routines. In section <ref> we describe our search space and the results. In section <ref> we conclude with some brief discussion.In addition, in appendix <ref> we present a useful lemma, and in appendix <ref> we study a two qutrit codes, conjectured in <cit.> to distill the |S⟩ state, that we find do not distill the |S⟩ state.



§ PRELIMINARIES


Here we review some well-done definitions and results concerning stabilizer codes for qudits of dimension p where p is any odd prime.



 §.§ The Heisenberg-Weyl displacement group


The qudit version of the Pauli group is generated by

    X̂ = ∑_k |k+1⟩⟨k|,   Ẑ = ∑_k ω^k |k⟩⟨k|,
 and multiplication by ω= e^2π i/p, and is known as the Heisenberg-Weyl displacement group.  The conventions for Heisenberg Weyl displacement operators we will follow are those in <cit.>, 

    D̂(u,v) = ω^-2^-1 u vX̂^u Ẑ^v.

n-qudit Heisenberg Weyl-displacement operators are denoted as

    D̂(u⃗,v⃗) = D̂(u_1,v_1) ⊗D̂(u_2,v_2) ⊗…⊗D̂(u_n,v_n),

where u⃗=(u_1, u_2, …, u_n) and v⃗=(v_1, v_2, …, v_n).

It is convenient to combine u⃗ and v⃗ into a symplectic vector χ=(u⃗,v⃗), and write D̂(χ) =D̂(u⃗,v⃗). Then 

    D̂(χ)D̂(χ') = ω^2^-1  [χ, χ']  D_χ+χ',

where 

    [χ, χ']= u⃗·v⃗'- u⃗' ·v⃗

is the symplectic inner product (and · is the usual dot product). For commuting Heisenberg-Weyl operators [χ,χ'] vanish.


Note that [χ, χ]=0, so D_χ^m=D_m χ.

We arrange these into p+1 rays,

    D̂(0,1), D̂(1,0), D̂(1,1), …, D̂(1,p-1).

The set of p+1 bases formed by the eignespaces of these operators form a set of mutually unbiased bases.




 §.§ Qudit stabilizer codes

An [[n,k]]_p stabilizer code is defined by the simultaneous eigenspace of n-k independent commuting n-qudit Heisenberg-Weyl displacement operators, and has dimension p^k. We will assume that each of these operators are of the form given in (<ref>), with no additional overall phase. As such, to completely specify the stabilizer code, we also have to specify the eigenvalues of each stabilizer, which are of the form ω^a, where a can be taken to be an element of  ℤ_p. In this paper we will restrict our attention to codes defined by the 1=ω^0 eigenspace of each operator, for reasons we explain below. The set of commuting operators {D̂(u⃗_1, v⃗_1), …, D̂(u⃗_n-k, v⃗_n-k) } can be specified by a symplectic matrix

    H = ( [   u⃗_1   v⃗_1;   u⃗_2   v⃗_2;      ⋮      ⋮; u⃗_n-k v⃗_n-k ]),

such that the symplectic inner product of any two rows vanishes. These operators generate an abelian group of order p^n-k, which we refer to as the stabilizer group 𝒮. Because 𝒮 is abelian, there are no overall phases arising from multiplication and 𝒮 consists of all Heisenberg-Weyl Displacement operators of the form D̂(χ) where χ is in the row-span of H.

A stabilizer code can equivalently be thought of as an additive classical code over GF(p^2), that is self-orthogonal under the Hermitian inner product <cit.>. 

We use the notation N(𝒮) to denote all n-qudit Heisenberg-Weyl displacement operators that commute with 𝒮. 𝒮⊆ N(𝒮). In general, N(𝒮) will not form an abelian group, and multiplication of two different elements may give rise to Heisenberg-Weyl operators with overall phases. It is nevertheless useful to define N(𝒮)^* as N(𝒮) modulo overall phases (i.e., identify Heisenberg-Weyl operators that differ by an overall phase), and specify N(𝒮)^* as the row-span of a symplectic matrix. In this way N(𝒮)^* can be thought of as a classical additive code in GF(p^2), denoted as 𝒮^⊥, that is the set of all codewords orthogonal to 𝒮 under the Hermitian inner product. We use the notation 𝒮^⊥ and N(𝒮)^* interchangeably. 

Clearly, 𝒮⊂𝒮^⊥. Any L ∈𝒮^⊥ defines a coset of 𝒮 in 𝒮^⊥ which is of the form[In this expression, addition means addition as symplectic vectors (or equivelently, vectors in GF(p^2)^n), or multiplication as Heisenberg-Weyl displacement operators.] 𝒮 + L = { L + M | M ∈𝒮}.  All stabilizers in the same coset will commute with each other. 







 §.§ Weight enumerators

Let us briefly review the definition of the weight enumerator of a stabilizer code. 

Let 𝒮 denote the stabilizer group of an [[n,k,d]]_3 stabilizer code. The projector onto the codespace of 𝒮 with trivial syndrome can be written explicitly as,

    Π̂^0_𝒮 = 1/p^n-k∑_M ∈𝒮 M.


Let 𝒮̃ be a either a stabilizer code 𝒮, or one of its cosets in 𝒮^⊥. The complete weight enumerator of 𝒮̃ is a function of p^2 formal variables {y_ij}, for i, j = 0, …, p-1, defined as follows:

    w(𝒮̃; ω_ij ) = ∑_(x⃗|z⃗) ∈𝒮̃∏_i=1^n y_x_i,x_j


The simple weight enumerator of 𝒮̃ to be a function of two formal variables x and y, defined as follows:

    w(𝒮̃; x, y) =  w(𝒮̃; y_ij(x, y) ).
 
where 

    y_ij(x,y) =  x     (i,j)=(0,0) 
    
        y     (i,j) ≠ (0,0).

If we further set x=y=1, then the simple weight enumerator becomes equal to |𝒮̃|. These definitions coincide with the standard definitions of the simple and complete weight enumerators of a classical error-correcting code over the finite field GF(p^2). 





A MacWilliams identity relates w(𝒮;1,z) to w(𝒮^⊥; 1,z) <cit.>. This is:

    w(S^⊥;1,z) = (1+(p^2-1)z)^n/p^n-k w(S;1,1-z/1+(p^2-1)z).

It is conventional to define A_𝒮(z)=w(𝒮;1,z) and B_𝒮(z)=w(𝒮^⊥; 1,z). 
A_𝒮(z)-B_𝒮(z) is a polynomial in z with non-negative coefficients and is the simple weight enumerator of the set of all logical operators for the stabilizer code 𝒮. The lowest power of z that appears in A(z)-B(z) is z^d, where d is the distance of the code.



 §.§ Discrete Wigner functions

A discrete phase space formalism for qudits was formulated in <cit.>, that we briefly review here. 

Singe-qudit phase point operators are defined as,

    Â(0,0) = 1/p∑_x,zD̂(u,v),   Â(u,v) =    D̂(u,v)Â(0,0) D̂(u,v)^†.

These are normalized so that Â(u,v)=1.
Multi-qudit phase point operators are defined as 

    Â(u⃗,v⃗)=⊗_i Â(u_i,v_i),

and we will use the notation Â(0,0)^⊗ n = Â(0,0) ⊗Â(0,0) ⊗…⊗Â(0,0).

Any qudit density matrix ρ̂ can be written as a linear combination of the phase point operators with real, but possibly negative, coefficients. These coefficients define the discrete Wigner function of the qudit state:

    ρ̂ = ∑_u,v W( ρ̂; u,v)Â(u,v),    W( ρ̂; u, v) = 1/pρ̂Â(u,v)

Note that ρ̂ = ∑_u,v W( ρ̂; u,v) =1, and the discrete Wigner function defines a quasi-probability distribution.

The Clifford group acts covariantly on the discrete Wigner function, for qudits of odd prime dimension. Heisenberg-Weyl displacement operators act as translations in discrete phase, and a general Clifford unitary can be represented as a symplectic rotation followed by a translation. See <cit.>, and references therein, for more details. This allows one to use the discrete Wigner function to define an efficient classical simulation for Clifford unitaries and stabilizer measurements acting on qudit states with non-negative Wigner functions <cit.>, and develop resource theories of magic for qudits of odd prime dimension – see <cit.>.



§ DISTILLATION AND WEIGHT ENUMERATORS


In the magic state model of fault-tolerant quantum computing <cit.>, one begins with a quantum computer that can initialize qudits in the computational basis, perform Clifford unitaries, and carry out stabilizer measurements. These operations are conventionally assumed to be noise-free for simplicity. To obtain universal quantum computing we supplement this quantum computer with the ability to initialize ancilla qudits in certain non-stabilizer states known as magic states, which are noisy. A magic state distillation protocol is a way to distill an arbitrarily pure magic state from many noisy magic states using only Clifford unitaries and stabilizer measurements. For the purposes of determining the best attainable threshold <cit.>, any magic state distillation protocol for the qutrit strange state  can be thought of as a procedure that projects n noisy qudits onto the codespace of an [[n,k]]_p stabilizer code – if the projection is successful, one decodes the resulting qudits to obtain k hopefully-less-noisy magic states.[Sometimes, i.e., in the case of triorthogonal codes, error-correction is also possible prior to decoding, which can result in a substantial reduction in overhead. However, this is not possible for distillation routines for the qutrit strange state.] 





 §.§ A general formulation in terms of complete weight enumerators

Here we provide a general formulation for qudit magic state distillation in the language complete weight enumerators. These results follow in part from the formulation in <cit.>. Let us also mention that a similar formulation in terms of signed-weight enumerators for qubit magic state distillation was given in <cit.>.

Let 𝒮 be a group of commuting Heisenberg-Weyl operators acting on n qudits. (Recall that, each element of 𝒮 is assumed to be of the form given in equation (<ref>), with no additional overall phase.) The projector onto the joint eigenspace of the code, with eigenvalues 1=ω^0, is given by,

    Π̂^0_𝒮 = 1/|𝒮 |∑_M ∈𝒮 M.


lemmaLemma

    Let ρ̂ be a  single qudit mixed state described by the Wigner function W(ρ̂; x,z). Then the probability for successful projection onto the eigenspace of 𝒮 with trivial syndrome is given by the complete weight enumerator of 𝒮^⊥, w( 𝒮^⊥; { W(ρ̂;i,j) }), with the formal variables y_ij in the complete Weight enumerator replaced by the entries of the Wigner function of ρ:

    ( Π̂^0_𝒮ρ^n ) = w( 𝒮^⊥; { W(ρ̂;i,j) }) = ∑_(u⃗|v⃗) ∈𝒮^⊥∏_i=1^n W(ρ̂; u_i, v_i).
  





Let the single-qudit state ρ̂ be described by Wigner function W_in(ρ̂; u,v). The n-qudit Wigner function for ρ^⊗ n is given by, 

    W^(n)_in(ρ̂^⊗ n; u⃗,v⃗) = ∏_i W_in(ρ̂; u_i,v_i).

The probability for successful projection onto the codespace is,

    ν    =     (Π̂^0_𝒮ρ_in^⊗ n) 
        =    ∑_u⃗, v⃗(∏_i=1^n W_in(u_i,v_i))  ( Π̂^0_𝒮Â(u⃗,v⃗) ).

In Appendix <ref>, Lemma <ref>,  we show that 

    (  Π̂^0_𝒮Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮^⊥
    
       1     (u⃗|v⃗) ∈𝒮^⊥.

We therefore find that,

    ν  =  ∑_(u⃗|v⃗) ∈ S^⊥∏_i W_in(ρ̂; u_i,v_i).



In magic state distillation, we first project onto the codespace of 𝒮, which we assume for simplicity to be an [[n,1]]_p code, then decode, to obtain a new state ρ' = f_MSD(ρ). We can use the above lemma to compute the Wigner function of ρ' in terms of the complete weight enumerators of 𝒮^⊥ and its cosets.

Choose any representatives of the logical Pauli operators X̅ and Z̅, to define logical Heisenberg-Weyl displacement operators D̅(x,z) and logical phase point operators A̅(x,z). The Wigner function of ρ̂' is given by,

    W(ρ̂'; x,z) = 1/ν( p^-1A̅(x,z) Π̂^0_𝒮ρ̂^⊗ n).

Let,

    p^-1A̅(0,0) Π̂^0_𝒮 =  p^-2∑_u,vD̅(u,v) Π̂^0_𝒮 =  1/p^n+1∑_(u⃗|v⃗) ∈𝒮^⊥D̂(u⃗,v⃗) ≡Π̂^0_𝒮^⊥.

Although Π̂^0_𝒮^⊥ is not a projector onto the codespace of a stabilizer code, in Appendix <ref>, we also show that,

    W(ρ̂'; 0,0) = 1/ν( Π̂^0_𝒮^⊥ρ̂^⊗ n) = 1/ν( Π̂^0_𝒮^⊥ρ̂^⊗ n) = 1/ν w( 𝒮; { W(ρ̂,i,j) }).

In general,

    W(ρ̂';u,v) = 1/ν w( 𝒮+D̅(u,v); { W(ρ̂;i,j) }).

As a cross-check, note that ∑_u,v w( 𝒮+D̅(u,v); { y_ij}) = w(𝒮^⊥; {y_ij}), so the resulting Wigner function is correctly normalized.

The problem of computing the output state of a general qudit magic state distillation routine defined by a stabilizer 𝒮 is thus reduced to computing the complete weight enumerators of 𝒮, and its cosets in S^⊥.




 §.§ The strange state and simple-weight enumerators


We now restrict our attention to the special case where ρ̂ a twirled qutrit strange state |S⟩. For this particular magic state, we will show that the complete weight enumerators in the previous subsection can be replaced by simple weight enumerators.

The qutrit strange state is given by 

    |S⟩= 1/√(2) (|1⟩- |2⟩).
  
After twirling by symplectic rotations, as described in <cit.>, all noise is converted to depolarizing noise, and noisy |S⟩ states are described by the density matrix:

    ρ̂_S(ϵ)=(1-ϵ) |S⟩⟨S| + ϵ1/3.

This corresponds to the discrete Wigner function:

    W(ρ̂_ϵ; u,v) =  x     (u,v)=(0,0) 
     y     (u,v) ≠ (0,0)  = [ y y y; y y y; x y y ],

where the parameters x and y satisfy x+8y=1, and are related to ϵ via:

    3x     =     -1+4ϵ/3
    
    6y     =     1-ϵ/3.


Applying Theorem <ref> to ρ̂_S(ϵ), we find the following lemma.

The probability for successful projection of ρ̂(ϵ)^⊗ n, ν, onto 𝒮, is given by the simple weight enumerator of 𝒮^⊥, 
    
    ( Π̂_𝒮ρ̂_S(ϵ)^n ) = w(𝒮^⊥; x(ϵ),y(ϵ)),

where x, and y are given by eqs. (<ref>) and (<ref>).



Let us assume we distilling using a stabilizer code that has a complete set of transversal Clifford gates. The output state ρ̂'=f_MSD(ρ̂) of such a distillation protocol will also be of the form in equation (<ref>), with parameter ϵ'. (Alternatively, if the stabilizer code does not have a complete set of Clifford gates, one could still twirl the output state by symplectic rotations as well to bring it into this form.) Let W(ρ̂';0,0)=x' and W(ρ̂';i,j)=y' for (i,j) ≠ (0,0). Then using equation <ref>, we find,

    x'     =    w(𝒮; x,y)/w(𝒮^⊥;x,y)
    
    y'     =    1/8w(𝒮^⊥;x,y)-w(𝒮;x,y)/w(𝒮^⊥;x,y).

The formal variables x and y used to define the simple weight enumerator in equation <ref> are now reinterpreted as entries of the discrete Wigner function in equation <ref>.

Let us rewrite this expression in terms of the noise parameter ϵ. We find, 

    ϵ' = 33A(z(ϵ))+B(z(ϵ))/4B(z(ϵ).

where, 

    z(ϵ) = 3-ϵ/8 ϵ-6,

A(z)=w(𝒮; 1,z) and B(Z)=w(𝒮^⊥; 1,z).

The error suppression of the magic state distillation routine is determined by the largest power of ϵ that divides 33A(z(ϵ))+B(z(ϵ)). By contrast, the distance of the error-correcting code is given by the largest power of z that divides B(z)-A(z).



Expressing ϵ in terms of x and y via

    ϵ/3    =     x+ 2y,

we see that 

    ϵ'/3 = 3 w(𝒮;x(ϵ),y(ϵ))+w(𝒮^⊥; x(ϵ),y(ϵ))/4w(𝒮^⊥; x(ϵ),y(ϵ)).



It is also convenient to write these results in terms of the parameter z = y/x via 

    ϵ/3  =  1+2z/1+8z,

as:

    ϵ' = 33A(z)+B(z)/4B(z).

which implies

    z'=B(z)-A(z)/8A(z).




  §.§.§ Example: the [[11,1,5]]_3 Golay code

To illustrate the above formalism, let us apply it to the 11-qutrit Golay code of <cit.>. The 11-qutrit Golay code is an [[11,1,5]]_3 CSS-code formed using two copies of the (self-dual) classical ternary Golay code.

The weight enumerator of the 11-qutrit Golay code is computed (e.g., via Magma <cit.>) to be:

    A(z) = 1+528 z^6+7920 z^8+11000 z^9+23760 z^10+15840 z^11.

Using the MacWilliams identity, we find,

    B(z) = 1+528 z^5+528 z^6+15840 z^7+40920 z^8+129800 z^9+198000 z^10+145824 z^11.
 
Substitute these results into equation (<ref>), to obtain

    ϵ'     =  3 (48336 z^11+67320 z^10+40700 z^9+16170 z^8+3960 z^7+528 z^6+132 z^5+1)/145824 z^11+198000 z^10+129800 z^9+40920 z^8+15840 z^7+528 z^6+528 z^5+1
        =  (3021 ϵ ^11-24816 ϵ ^10+92180 ϵ ^9-203280 ϵ ^8+292710 ϵ ^7-283536 ϵ ^6+181764 ϵ ^5-71280 ϵ ^4+13365 ϵ ^3)/ 
       =(990 ϵ ^11-7920 ϵ ^10+27500 ϵ ^9-50490 ϵ ^8+37620 ϵ ^7+47256 ϵ ^6-172656 ϵ ^5+243540 ϵ ^4-204930 ϵ ^3 
        +106920 ϵ ^2-32076 ϵ +4374) 
       ≈    55 ϵ^3/18 + O(ϵ^4).


The threshold of the code, ϵ_*, is the critical value of ϵ such that, ϵ<ϵ_* implies ϵ'<ϵ. Using (<ref>), we find 

    ϵ_*     =1/45(-262 √(2/405 √(109)-2981)+2^2/3√(405 √(109)-2981)+31) 
       ≈ 0.387.

Interestingly, z_*=z(ϵ_*) satisfies a simple cubic equation, 11 z_*^3+12 z_*^2+3 z_*+1=0.



 §.§ Conditions for magic state distillation


There are two conditions that a stabilizer code must satisfy for it to qualify as a magic state distillation routine for the strange state. We first require that the probability of successful projection to be nonzero in the limit ϵ→ 0, which translates into the requirement

    B( z( ϵ=0) ) =B(-1/2) ≠ 0.

We also require that the noise suppression be better than linear.

Assuming equation (<ref>) is satisfied, the noise-suppression exponent, δ, of the magic state distillation routine, ϵ' = θ(ϵ^δ), is determined by the smallest power of ϵ that divides 3A(z(ϵ))+B(z(ϵ)). Let us write

    3A(z(ϵ)) + B(z(ϵ)) = C_0 + C_1 ϵ + C_2 ϵ^2 + …
 
Generically, we expect C_0 and C_1 will be non-zero. The necessary and sufficient condition for C_0 and C_1 to vanish are,

    ( 3A(z(ϵ)) + B(z(ϵ)) )|_ϵ=0    = 0 
    d/dϵ|_ϵ=0( 3A(z(ϵ)) + B(z(ϵ)) )     = 0.

Translated into z, these conditions become

    3A(-1/2)+B(-1/2)     =     0, 
    
       3A'(-1/2) + B'(-1/2)     =     0 .

As a check, observe that, for the weight enumerators of the 11-qutrit Golay code, A(-1/2)=2187/64, B(-1/2)=6561/64, A'(-1/2)=-8019/16 and B'(-1/2)=24057/16, these conditions are satisfied. 

The MacWilliams identity simplifies at z=-1/2 to:

    B(-1/2)= 3(-1)^n A(-1/2).
 
For n odd, (<ref>) is therefore automatically satisfied. We can also ask, when do we get cubic noise suppression? If condition in equation (<ref>) is satisfied, the condition for C_2 to vanish is that 

    3 A”(-1/2)+B”(-1/2)=0.
 
Taking derivatives of the MacWilliams identity, we find that,

    B'(-1/2)     =     (-1)^n+1(3 A'(-1/2)+8n A(-1/2) ), 
    
        B”(-1/2)     =    1/3 (-1)^n (9 A”(-1/2)+48 (n-1) A'(-1/2)+64(n^2-n)A(-1/2)).

When n is odd, then the MacWilliams identities above along with (<ref>) imply that condition (<ref>) is satisfied. Thus, cubic noise suppression is guaranteed for any odd-length stabilizer code whose weight enumerator satisfies the two conditions, (<ref>) and (<ref>). 







§ SEARCH FOR DISTILLATION ROUTINES


With the above results in place, a computational search for distillation routines for the qutrit strange state is straightforward. For each stabilizer code 𝒮 in our search space, we compute the simple weight enumerator A(z), and then B(z) using the MacWilliams identity. We then check if conditions (<ref>) and (<ref>) are satisfied.



 §.§ Equivalence of distillation routines

When are two stabilizer codes equivalent for the purposes of magic state distillation? 

Conventionally, one considers two quantum error-correcting codes to be equivalent if they differ by permutations of coordinates and by local Clifford operations. However, two stabilizer codes which differ by local Clifford transformations will in general give rise to different magic state distillation protocols. 


This observation increases the size of our search space substantially – to search for all magic state distillation routines associated with a given stabilizer code 𝒮, we need to search over all orbits of 𝒮 under local Clifford transformations, and all possible eigenvalues of the stabilizers. For each [[n,1]]_3 code, this could increase the search space[Any local Clifford operator can be written as a product of V_F̂D̂(χ) where V_F̂ is a symplectic rotation. Acting on a stabilizer code with a transversal Heisenberg-Weyl displacement operator can only change the eigenvalues of the stabilizers.  Symplectic rotations leave the eigenvalues unchanged but transform the stabilizers by a product of local SL(2,ℤ_p) transformations. Therefore, in principle, a search over magic state distillation routines seems to require a search over all local SL(2,ℤ_3) orbits of each [[n,1]]_3 stabilizer codes, with all possible allowed eigenvalues for each of the stabilizers.] by a factor as large as, |SL(2,ℤ_3)|^n 3^n-1=24^n 3^n-1. 

It is possible, however, to substantially reduce this search space size, by demanding that the both the stabilizer codes we study and the magic states we wish to distill possess certain symmetries. 

Let the magic state we wish to distill be invariant under a subgroup G of the single-qudit Clifford group. It is natural to require[One, in principle, can also consider magic state distillation without twirling the input states – see <cit.>.] our noisy input states first undergo a twirling procedure (as described in detail for qutrits in <cit.>) so that they are also invariant under G. Let C be a single-qudit Clifford-unitary, if a twirled noisy magic state ρ̂_M, satisfies Cρ̂_M (C)^†=ρ̂_M, then two stabilizer codes which differ via local Clifford transformations belonging to the subgroup of the Clifford group generated by by C are equivalent for distillation of the ρ̂_M magic state. The qutrit magic state |S⟩ is a simultaneous eigenvector of all symplectic rotations, as emphasized in <cit.>. Assuming noisy input magic states are twirled by applying a random symplectic rotation, two stabilizer codes are equivalent for magic state distillation if and only if they are related to each other by local symplectic transformations. This reduction in the size of search-space is only possible for the qutrit strange state, by virtue of its exceptional symmetry – and is not expected to be true for any other qutrit or qudit magic state. 

It is also natural to require that some, or all, of the unitaries in G be transversal gates for the stabilizer code used for distillation. This is not a sufficient condition for distillation, but it is true for all magic state distillation routines studied in the literature that we are aware of, other than triorthogonal codes (which possess a transversal non-Clifford gate) <cit.>. 

In this paper, we are interested in distilling the qutrit magic state |S⟩. The symmetry group of |S⟩, is the set of all symplectic rotations. Therefore, it is natural to demand that our code possess a complete set of transversal Clifford gates. However, we can instead impose a slightly less restrictive requirement, and demand only that Ĥ^2 be a transversal gate, where Ĥ denotes the single-qutrit Hadamard gate. Notice that, acting with (Ĥ^2)^⊗ n on a stabilizer code corresponds to multiplying each row of the symplectic matrix defining the code by -1, leaving the eigenvalues unchanged. Alternatively, its action can be thought of as changing the eigenvalues ω^a of any stabilizer to ω^-a. Demanding that (H^2)^⊗ n commute with the codespace is therefore equivalent to demanding that the eigenvalues of all stabilizers in the code must be +1=ω^0. This condition allows us to restrict our search to projection onto the trivial syndrome. 

Our search space thus consists of two classes of codes: a search over all small quantum error correcting codes, and a search over CSS codes with a complete set of transversal Clifford gates. For both classes, we restrict attention to projection onto the trivial syndrome, due to our requirement that (Ĥ^2)^⊗ n be transversal.  



 §.§ Small quantum error-correcting codes


Any n-qudit stabilizer state can be thought of as a [[n,0,d]]_3 stabilizer code, and any [n-1,1]_3 stabilizer code can be obtained by shortening an [[n,0]]_3 stabilizer code.  Using the graph-state formalism, <cit.> classified all [[n,0,d]]_3 stabilizer states up to local Clifford transformations for n≤ 10, and all[Danielsen has also classified all optimal 11-qutrit stabilizer states, but we could not obtain them.] 12-qutrit stabilizer states of maximum distance[The distance of a stabilizer state is the minimum weight of any non-trivial element of its stabilizer group.] 6. 

For each [[n,0]]_3 code, we computed all inequivalent ways of shortening it to obtain an [[n-1,1]]_3 stabilizer code. For each resulting [[n-1,1]]_3 code, we computed the weight enumerator A(z) and the performance for magic state distillation, as described in the previous section. We thus searched over all [[n,1,d]]_3 codes for m ≤ 9, and all [[11,1,d]] codes that can be obtained from shortening a [[12,0,6]]_3 stabilizer state.

Remarkably the only code we found that could distill the qutrit strange state with better-than-linear noise-suppression was the 11-qutrit Golay code. There were also a few [[9,1]]_3 and [11,1]]_3 codes that distilled the strange state with linear noise suppression, (i.e., ϵ' = αϵ with α<1), all with lower thresholds than the 11-qutrit Golay code.




 §.§ CSS codes

A maximal self-orthogonal classical ternary code has parameters [n, ⌊n/2⌋]_3. For n odd, therefore, using the CSS construction <cit.>, two copies of a self-orthogonal ternary code of odd length can be combined to form a [[n, 1, d]]_3 qutrit code, whose transversal gate set is the entire single-qutrit Clifford group. Explicitly, if G_c is the generator matrix of the self-orthogonal classical ternary code, the quantum CSS code is given by symplectic matrix:

    H = ( [ G_c   0;   0 G_c ]).

Any such CSS code is a natural candidate for a magic state distillation routine for the strange state.

Classical maximal self-orthogonal ternary codes up to size n=23 have been classified in <cit.>, and are conveniently available on a website maintained by Harada and Munemasa <cit.>.

We computed the weight enumerators of all CSS-codes constructed this way from the codes given in <cit.>. We found that no indecomposable[A code is said to be indecomposable if its generator matrix cannot be written as the direct sum of two smaller generator matrices.] 13, 15, 17 or 19-qutrit CSS codes were able to distill the strange state.

Somewhat surprisingly, however, we found a total of 646 inequivalent indecomposable 23-qutrit CSS codes[The 646 inequivalent CSS codes gave rise to 263 different simple weight enumerators.] that were able to distill the strange state with cubic noise suppression. A complete list of the 646 classical ternary codes that gave rise to these codes is included in MAGMA format <cit.> as supplemental data along with this arXiv submission.

There are a total of 1928 indecomposable maximally-self-dual [23,11]_3 codes, so the probability that a randomly chosen code will give rise to a quantum CSS code that distills the strange state is 0.335, which is very close to 1/3.  The probability that a randomly chosen maximally self-orthogonal [11,5]_3 code gives rise to a CSS code that distills the strange state is also 1/3. This seems to suggest that quantum error correcting codes that distill the qutrit strange state are actually quite common.

The thresholds that arise from these codes range from 0.063 to 0.318, and are plotted in a histogram in Figure <ref>. None of these thresholds exceed that of the ternary Golay code.



The code with the highest threshold was a [[23,1,5]]_3 code. It is formed from two copies of the classical [23, 11, 6]_3 code whose generator matrix is:

    G_C = [ 1 0 0 0 0 0 0 0 0 0 0 1 2 2 0 0 2 0 2 0 0 0 0; 0 1 0 0 0 0 0 0 0 0 0 2 2 1 1 2 1 2 1 2 2 0 1; 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 2 2 0; 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 2 0 1 0 1 0; 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 0 0 1 0 2 2 2 0; 0 0 0 0 0 1 0 0 0 0 0 2 2 0 0 2 1 0 2 0 0 0 0; 0 0 0 0 0 0 1 0 0 0 0 1 0 0 2 0 1 0 0 2 0 2 0; 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 1 0; 0 0 0 0 0 0 0 0 1 0 0 0 0 0 2 0 0 1 0 1 1 0 1; 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 2 0 2 1 2 2 0 1; 0 0 0 0 0 0 0 0 0 0 1 1 2 2 1 1 2 2 1 2 2 0 1 ].

The quantum CSS code has weight enumerator 

    A(z)     =  2079023616 z^23+6035662080 z^22+8258226816 z^21+7208904960 z^20+4504066560 z^19+2182781824 z^18
       + 790797312 z^17+252077184 z^16+52015680 z^15+14590080 z^14+2083104 z^13+628800 z^12+121824 z^11
        +58320 z^10+16120 z^9+4608 z^8+720 z^6+1.

It gives rise to a distillation performance 

    ϵ' ≈73 ϵ ^3/18+O(ϵ ^4),

plotted in Figure <ref>. 



The probability of successful projection onto the codespace for these codes is very low. We refer to the probability for 23 pure copies of the strange state to successfully project onto the codespace as the success probability of the code. The success probabilities range from 1/5159780352 ≈ 1.9 × 10^-10 to 1/35831808≈ 2.8 × 10^-8.  The highest probability of successful distillation is attained for the 23-qutrit code in equation (<ref>) is 1/35831808 – three other codes have the same success probability and very similar thresholds. It appears that the success probability is correlated with the threshold. A plot of success probability versus threshold for the 646 codes that distill the strange state is shown in Figure <ref>. 




§ DISCUSSION



    

Magic state distillation <cit.> is a somewhat mysterious approach to fault-tolerant quantum computing. <cit.> identified two magic states for qubits, which they denote as |H⟩ and |T⟩. For the qubit |H⟩ state of <cit.> and its qudit generalizations defined in <cit.> – magic state distillation is better understood, and systematic approaches to construct distillation routines with low overhead have been proposed in the literature via the formalism of triothogonal codes. 
However, for the |T⟩ state of <cit.> few distillation routines are known <cit.>, and the mechanism behind distillation remains unclear. The qutrit analogue of the |T⟩ state, which lies directly above a face of the stabilizer polytope, is the strange state |S⟩ <cit.> that we study here. The study of distillation of the strange state may in fact be more tractable than that of the qubit |T⟩ state, thanks to the simple relation between distillation performance and simple enumerators valid only for the qutrit strange state. We hope that, by enlarging the landscape of codes known to distill this state, will lead, in the future, to a better understanding of magic state distillation. 

In this work, we carried out a computational search and found that over 600 [[n,1,d]]_3 codes that distill the qutrit strange state, with cubic noise suppression. This seems to suggest that the ability to distill the qutrit strange state is a generic feature, at least of codes constructed to have a complete set of transversal Clifford gates. While none of these codes have thresholds that exceed that of the 11-qutrit Golay code, their existence does provide some hope that there may exist larger codes with thresholds better than the 11-qutrit Golay code. These results also do provide some support for the belief that a sequence of distillation routines of increasing size whose threshold approaches the limit set by contextuality exists, as required to prove the conjecture that contextuality is sufficient for qutrit quantum computation <cit.>.



§ ACKNOWLEDGEMENTS


SP thanks Prof. P.S. Satsangi for inspiration and guidance. We also thank Mark Howard and Amolak Kalra for comments on a draft of this manuscript. We acknowledge the support of MeitY QCAL, Amazon Braket and DST-SERB grant (CRG/2021/009137).





§ COMPLETE WEIGHT ENUMERATOR

Using the following notation

    [ y_20 y_21 y_22; y_10 y_11 y_12;    x y_01 y_02 ]

we can define the complete weight enumerator of a code as a polynomial in the above 9 variables. The simple weight enumerator is defined by seting y_ij=y.



§ NORELL STATE

Suppose we wish to distill the Norell state. The discrete Wigner function of the Norell state is of the form: 

    [ n_2 n_1 n_1; n_2 n_1 n_1;   x n_1 n_1 ]

For the mixed state

    ρ̂ = (1-ϵ_0 - ϵ_s) |N⟩⟨N|+ϵ s |s⟩ + ϵ_0 |0⟩,

we have:

    x = 1/3 (1-2 ϵ_s), n_1 = (1 - ϵ_0)/6, n_2=1/6 (3 ϵ_0+2 ϵ_s-1).


We define the Norell-weight enumerator W_N(M;x,n_1,n_2) of a code M, to be its complete weight enumerator, with the assignment y_22=y_21=y_12=y_11=y_01=y_02≡ n_1 and y_10=y_20=n_2. This weight enumerator is invariant under local Clifford transformations belonging to the Norell subgroup of the Clifford group.

The success probability for distillation by the code M is P=W_N(M^⊥; x,n_1,n_2). 

The output functions are:

    x'     =     W_N(M; x,n_1,n_2)/W_N(M^⊥; x,n_1,n_2) 
    
    n_1'     =     W_N(M+X̅;x,n_1,n_2)/W_H(M^⊥; x,n_1,n_2) 
    
    2n_2'    =     1-x'-6x_1'


For a tight distillation routine, we require that n_2'=0 when n_2=0. We can write:

    n_2'= W_N(M+Z̅;x,n_1,n_2)/W_H(M^⊥; x,n_1,n_2)

so this implies that we require

    0 = W_N(M+Z̅;x,n_1,0)

for a tight distillation scheme. This means there should be no words in M+Z̅ whose entries are only (0,0) or (1,i), or (2,i) which lead to a term in the weight enumerator proportional to  n_1 and x. 




§ HADAMARD STATES

Suppose we wish to distill the Hadamard state. The discrete Wigner function of the Hadamard state is of the form: 

    [ h_1 h_2 h_2; h_1 h_2 h_2;   x h_1 h_1 ]

These can be used to parameterize noisy input states. For the mixed state

    ρ̂ = (1-ϵ_-1 - ϵ_s) |H,1⟩⟨H,1|+ϵ_s |S⟩ + ϵ_-1|H,-1⟩,

we have:

    x     =    1/3 (1-2 ϵ_s), 
     h_1     =    1/6 (1-12 b) ϵ_-1+1/6 (1-6 b)ϵ_s+b, 
     h_2     =    (2 b-1/6) ϵ_-1+b ϵ_s-b+1/6

where: b=1+√(3)/12

We define the Hadamard-weight enumerator W_H(M; x, h_1, h_2) of a code M, to be its complete weight enumerator, with the assignment y_10=y_20=y_01=y_02≡ h_1 and y_11=y_12=y_21=y_22≡ h_2. This weight enumerator is invariant under local Clifford transformations belonging to the Hadamard subgroup of the Clifford group.

The success probability for distillation by the code M is P=W_H(M^⊥; x,h_1,h_2). 

The output functions are:

    x'     =     W_H(M; x,h_1,h_2)/W_H(M^⊥; x,h_1,h_2) 
    
    h_1'     =     W_H(M+X̅;x,h_1,h_2)/W_H(M^⊥; x,h_1,h_2) 
    
    4h_2'    =     1-x'-4h_1'


A tight distillation routine means that, h_1'=0 when h_1=0. This implies that h_2'=(1-x')/4



§ A DISTILLATION ROUTINE FOR THE HADAMARD STATE


For Hadamard states, we find the 17-qutrit CSS code, constructed from two copies of the following classical self-orthogonal ternary code:

    M_c=    (
    [ 1 0 0 0 0 0 1 2 0 1 2 0 2 1 0 2 1; 0 1 0 0 0 0 2 1 0 2 1 0 2 1 0 2 1; 0 0 2 0 1 0 2 1 0 1 2 0 0 0 0 0 0; 0 0 0 2 1 0 1 2 0 2 1 0 0 0 0 0 0; 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1;   ])

gave rise to an almost tight distillation routine for hadamard states. For this code,

    ϵ_s'     =    ϵ_s(128 ϵ_-1^2/9+O(ϵ_-1^3))+ϵ_s^2 (128 ϵ_-1^2/3+O(ϵ_-1^3))+O(ϵ_s^3) 
    ϵ_-1'     =    (ϵ_-1+O(ϵ_-1^3))+ϵ_s(ϵ_-1+O(ϵ_-1^3))+ϵ_s^2 (16/9+ϵ_-1+32 ϵ_-1^2+O(ϵ_-1^3))+O(ϵ_s^3)


We also find

    M_c' =  (
    [ 1 0 0 0 0 0 0 0 0 1 2 0 0 1 0 2 2; 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 2 2; 0 0 1 0 2 0 0 0 0 1 2 2 0 1 0 0 0; 0 0 0 1 2 0 0 0 0 2 1 1 0 2 0 0 0; 0 0 0 0 0 1 0 2 0 1 2 0 2 1 0 0 0; 0 0 0 0 0 0 1 2 0 2 1 0 1 2 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2;   ])

gave rise to an almost tight distillation routine for hadamard states. These are codes 1 and 4 in our scheme. 
For code 4, we have

    ϵ_S'     =    ϵ_S (52/289+2880 ϵ_-1^2/289+O(ϵ_-1^3))+ϵ_S^2 (12324/83521+2197440 ϵ_-1^2/83521+O(ϵ_-1^3))+O(ϵ_S^3) 
    ϵ_1'     =    (ϵ_-1+O(ϵ_-1^3))+ϵ_s(237 ϵ_-1/289+O(ϵ_-1^3))+ϵ_s^2 (300/289+56169 ϵ_-1/83521+10728 ϵ_-1^2/289+O(ϵ_-1^3))+O(ϵ_s^3)




These routines distill in the ϵ_s direction, but they do not distill in the ϵ_-1 direction. When ϵ_s=0, ϵ_-1'=ϵ_-1 exactly, and when ϵ_s ≠ 0, ϵ_-1' is slightly larger than ϵ_-1. These routines can be applied to move a state onto the line connecting |H,1⟩ to |H,-1⟩. In principle it can also be used to distill |H,-1⟩ states from anywhere on the distillable region of this line if ϵ_s ≠ 0. 

Both can be written as tensor networks, using the 4 qutrit perfect tensor, we conjecture.


We found that there are no 17-qutrit CSS codes using our construction, that distill the Norell state or the strange state. 










§ APPENDICES



§ PROJECTING PHASE POINT OPERATORS ONTO A STABILIZER CODE
 
In this appendix we prove a lemma which is needed for the proof of Theorem <ref> in the main text.


Let 𝒮 be a group of (conventionally-normalized) commuting n-qudit Heisenberg-Weyl operators.

    (  Π̂^0_𝒮Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮^⊥
    
       1     (u⃗|v⃗) ∈𝒮^⊥.
 



𝒮 can be generated by n-k independent Heisenberg-Weyl displacement operators, which we assume have been chosen.

Let 

    Π̂^0_𝒮 = 1/p^n-k∑_M ∈𝒮 M
 
denote the projector onto the ω^0 eigenspace of each M ∈𝒮. Let Π̂^s⃗_𝒮 denote the projector onto a non-trivial subspace, labeled by the vector of syndromes s⃗ of the n-k generators of 𝒮. 

Explicitly, we can write 

    Π̂^s⃗_𝒮 = 1/p^n-k∑_M ∈𝒮ω^f(s⃗,M) M,

for some phases f(s⃗,M) that depend on the syndrome s⃗. Clearly f(0,M)=0. For s⃗≠ 0, then it is easy to see that f(s⃗,M) takes on all the values j ∈ℤ_p, i.e.,  { 0, 1,  …, p-1}, and 

    |{ M | f(s⃗,M)=j }|=p^n-k-1.

(One way to prove this is to note that the operators ω^f(s⃗,M M form an abelian group, and there is a group homomorphism that maps ω^f(s⃗,M M →ω^f(s⃗,M).)

Note also that 

    Π̂^s⃗_𝒮 M = 
        0     M ∉𝒮
    
        p^k ω^f(s⃗,M^-1)    M ∈𝒮.

Therefore 

    (Π̂^s⃗_𝒮Â(0,0)^⊗ n)      =    1/p^n∑_u⃗,v⃗(Π̂^s⃗_𝒮D̂(u⃗,v⃗) ) 
        =    1/p^n∑_M ∈𝒮ω^f(s⃗,M^-1)p^k
        =    p^k/p^n· p^n-k-1( 1 + ω + ω^2 + …ω^p-1) =0    s⃗≠ 0 
    
        1    s⃗=0


Note that 

    D̂(u⃗,v⃗)^-1Π̂^0_𝒮D̂(u⃗,v⃗) = Π̂^s⃗_𝒮,

where the syndrome s⃗=0 if D̂(u⃗,v⃗) commutes with all M ∈𝒮, and s⃗≠ 0 otherwise. The condition that D̂(u⃗,v⃗) commutes with all M ∈𝒮 is equivalent to the condition (u⃗,v⃗) ∈𝒮^⊥. Therefore,

    Π̂_𝒮Â(u⃗,v⃗)
           =    ( D̂(u⃗,v⃗) Π̂_S D̂(-u⃗,-v⃗) Â(0,0)^⊗ n) 
        =     (Π̂_S^s⃗≠ 0Â(0,0)^⊗ n)     (u⃗|v⃗) ∉𝒮^⊥
     (Π̂^0_S Â(0,0)^⊗ n)     (u⃗|v⃗) ∈𝒮^⊥,

and the result follows using equation (<ref>).


Let 𝒮^⊥ be the dual of 𝒮. |𝒮^⊥|=p^n+k. Let us also point out that, if we define,

    Π̂^0_𝒮^⊥≡∑_u,vD̅(u,v) Π̂^0_𝒮 =  1/p^n+k∑_(u⃗|v⃗) ∈𝒮^⊥D̂(u⃗,v⃗),

then one can also show 

    (  Π̂^0_𝒮^⊥Â(u⃗,v⃗) )
           =     0     (u⃗|v⃗) ∉𝒮
    
       1     (u⃗|v⃗) ∈𝒮.



§ SOME CODES THAT DO NOT DISTILL THE STRANGE STATE


It was recently <cit.> proposed that two qutrit stabilizer codes – a [[13,1]]_3 and a [[29,1]]_3 CSS code – distill the strange state, and estimated their thresholds to be of 0.425 and ≈ 0.7, respectively. We computed the distillation performance of these codes exactly, via their simple weight enumerators, and found that, contrary to the estimates in <cit.>, these two codes do not, in fact, distill the strange state. 

The [[13,1 ,4]]_3 CSS-code of <cit.> generated from two copies of the [13,6,3]_3 maximal self-orthogonal ternary code with generator matrix is, in row-reduced form,

    M_13= [ 1 0 0 0 0 0 0 0 2 0 2 0 0; 0 1 0 0 0 0 0 2 2 1 1 2 0; 0 0 1 0 0 0 1 2 0 1 0 2 2; 0 0 0 1 0 0 1 0 2 0 1 2 2; 0 0 0 0 1 0 0 0 2 2 1 1 1; 0 0 0 0 0 1 2 1 2 0 1 1 0 ].

The weight enumerator of the [[13,1 ,4]]_3 code is

    A(z) = 1+8 z^3+600 z^6+720 z^7+4320 z^8+18320 z^9+61200
       z^10+151200 z^11+178144 z^12+116928 z^13.

This gives rise to a relation 

    ϵ' = 19683-157464 ϵ+612360 ϵ^2-1516320 ϵ^3+2708235
       ϵ^4-3744144 ϵ^5+4143960 ϵ^6-3674592 ϵ^7+2543865
       ϵ^8-1312920 ϵ^9+471264 ϵ^10-104384 ϵ^11+10713
       ϵ^12/2 (6561-52488 ϵ+204120 ϵ^2-495720
       ϵ^3+844425 ϵ^4-1078704 ϵ^5+1074888 ϵ^6-849744 ϵ^7+529395
       ϵ^8-251400 ϵ^9+85152 ϵ^10-18184 ϵ^11+1827
       ϵ^12),

plotted in Figure <ref>. As ϵ→ 0, we see that ϵ' → 3/2. The reason for this is that the pure state |S⟩^⊗ 13 is orthogonal to the codespace of the stabilizer code, so the code is completely unsuitable for magic state distillation. This can be seen from the fact that B(-1/2)=0. This feature was shared by 6 out of the 7 possible 13-qutrit codes that arise from our construction using the classical ternary codes listed in <cit.>.



The [[29,1,7]]_3 CSS code has weight enumerator

    A(z) =     1+40 z^6+4280 z^9+96 z^10+2832 z^11+196584
       z^12+198768 z^13+1773408 z^14+15542368
       z^15
        +91797024 z^16  +565547232 z^17+3037545272
       z^18+13979050848 z^19+55970778960
       z^20+192507694176 z^21
        +559711606992
       z^22+1361197350960 z^23+2723501140720
       z^24+4358977591776 z^25
        +5363568387600
       z^26+4767481212256 z^27+2724627154368
       z^28+751557878400 z^29.
 (Computing the above weight enumerator on the machines we had access to took approximately one day of computation time.)

The resulting relation between ϵ' and ϵ is ϵ' = 1937 ϵ/224+O(ϵ^2), and is shown in Figure <ref> below. The threshold for distillation is zero.



Therefore, at present, the 11-qutrit Golay code has the highest threshold for distillation of the strange state, which is still quite far from the threshold set by contextuality <cit.>. We have not yet carried out a search over 29-qutrit codes for distillation, as, to our knowledge a complete list of [29, 14]_3  self-orthogonal ternary codes is not yet available. 





ssg




 Paper Title: $q$-Binomial Identities Finder.



§ ABSTRACT 
This paper presents a symbolic computation method for automatically transforming q-hypergeometric identities to q-binomial identities. Through this method, many previously proven q-binomial identities, including q-Saalschütz’s formula and q-Suranyi’s formula, are re-fund, and numerous new ones are discovered. Moreover,
the generation of the identities is accompanied by the corresponding proofs.
During the transformation process, different ranges of variable values and various combinations of q-Pochhammer symbols yield different identities. The algorithm maps variable constraints to positive elements in an ordered vector space and employs a backtracking method to provide the feasible variable constraints and q-binomial coefficient combinations for each step.



S-SYNTH: Knowledge-Based, Synthetic Generation of Skin Images
    Andrea Kim* Niloufar Saharkhiz* Elena Sizikova* 

Miguel Lago Berkman Sahiner Jana Delfino Aldo Badano
    August 5, 2024
==========================================================================================================





§ INTRODUCTION


The study of Gaussian binomial coefficients, also known as q-binomial coefficients, along with their numerous related identities, is valuable in various mathematical and applied contexts due to their broad range of applications and deep theoretical implications. These fields include number theory <cit.>, Lie algebra <cit.>, quantum physics <cit.> and even privacy protection <cit.>. The main goal of this paper is to introduce an idea that is both “old” and “new” for generating more q-binomial identities. We refer to it as “old” because it roots in the traditional method of deriving q-binomial identities from basic hypergeometric identities. However, it is also “new” as it has successfully automated the generation of identities with the assistance of computers. Before delving into more details, let us start with some common notations in q series.


Throughout this paper, q always represents a complex number such that |q|<1. For a positive integer n, the q-Pochhammer symbol is defined as 

    (a;q)_n=∏_k=0^n-1(1-aq^k).

The q-Pochhammer symbol can be naturally extended to zero, infinite and negative products as follows.







    (a;q)_0=1,    (a;q)_∞=∏_k≥0(1-aq^k),    (a;q)_-n=(a;q)_∞/(aq^-n;q)_∞.



Let m and n be two integers. The q-binomial coefficient is defined as the following combination of q-Pochhammer symbols.

    m+nn_q=
    
    (q;q)_m+n/(q;q)_m(q;q)_n   if  m and n≥ 0,
    
        0    otherwise.


The q-binomial coefficients are set to zero if either m or n is negative, which makes the coefficients not always match (q;q)_m+n/(q;q)_m(q;q)_n. Consequently, q-binomial identities hold true only within a fixed range of variables <cit.>.




The first significant q-binomial identity is the q-analog of Vandermonde convolution, which connects q-binomial coefficients to the summation of products of q-binomial coefficients

[q-Vandermonde convolution, <cit.>]
For non-negative integers m and n,

    ∑_r = 0^min(n,k)q^r(m-k+r)mk-r_qnr_q=m+nk_q.


Identity <ref> is proved by Heine (or q-Gauss) Summation Theorem <cit.> or combinatorially proved by counting finite vector subspaces of certain objects <cit.>.

The most studied q-binomial identity is the following Saalschütz’s formula of q-binomial form.
[Gould, <cit.>]
For non-negative integers a and b,

    ∑_r = 0^min(a,b)q^(a-r)(b-r)x+y+rr_qx+a-ba-r_qy+b-ab-r_q=x+aa_qy+bb_q.



Identity <ref> is proved by applying q-Vandermonde convolution several times <cit.>, or combinatorially proved by the enumeration of ordered pairs of subsets of {1,2,3,…,v} <cit.>, or combinatorially proved by determining an explicit bijection between sets of integer partitions <cit.>.

Besides Identity <ref>, there exist plenty of q-binomial identities with “three in sum side” and “two in product side”, such as the following identity.

 [q-analog of Suranyi’s formula, <cit.>]
For non-negative integers k and n,

    ∑_r = 0^mq^r^2kr_qnr_qx+k+n-rk+n_q=x+kk_qx+nn_q.

where m=min(k,n,x+k+n)


Identity <ref> is proved by extensions of the ordinary Vandermonde theorem <cit.>.






From the proof techniques mentioned above, we can observe that, the basic hypergeometric series, also known as q-hypergeometric series, has been a fertile ground for discovering q-binomial identities. 
The most common way to generate Gaussian binomial identities is to make some unpredictable substitutions in q-hypergeometric identities and then to simplify them. Here, q-hypergeometric function is defined as follows.



    _rϕ_s ( [ a_1, a_2, …, a_r; b_1, b_2, …, b_s ]; q, z )
    = ∑_n=0^∞(a_1, a_2, …, a_r; q)_n/(q, b_1, b_2, …, b_s; q)_n( (-1)^n q^n2)^s-r+1 z^n

where (a_0,a_1,⋯,a_r;q)_n=∏_j=1^r(a_j;q)_n.
This approach, albeit straightforward, is particularly laborious. Determining combinations of variable substitutions often relies on chance, and the simplification process requires both keen insight and continuous experimentation. 
The only way to avoid relying on luck is to test all possible combinations, which leads to significantly increased manual calculations. 

To address the complexity of manual computations, computer algebra has been utilized in the study of many of the topics in q series <cit.>. For example, Kanade and Russell <cit.> employed symbolic computation to generate various possible Rogers–Ramanujan type identities. Again, with the help of computer algebra and the concept of linked partition ideal <cit.>, Chern and Li <cit.> reproved six generating function identities due to Kanade and Russell. For more computer application in q-series, please refer to <cit.>.
This motivates us to use computational tools to overcome the tedious work in transforming q-hypergeometric identities to q-binomial identities. Consequently, we introduce a Python-based method in this paper. (The source code is available at <https://github.com/Z798971901/q_binomial_identities_finder>). 
The method can be summarized into three main modules. 

    
  * Constraint Conditions Generator: Ensuring each step of transformation is valid by restricting the range of variables.
    
  * q-Binomial Coefficients Generator: Converting each q-Pochhammer symbol in the original q-hypergeometric identity into a square bracket form of positive integers, and then transform them to q-binomial coefficients.
    
  * Backtracking Framework: Trying all possible constraint conditions of variables and as many as possible transformations from square brackets to q-binomial coefficients.















Although not all q-hypergeometric identities can be transformed into q-binomial identities, when such a transformation is feasible, our identity finder can efficiently generate q-binomial identities. This method has two main applications.

First, our finder can generate q-binomial identities. In addition to known identities such as Identity <ref> and Identity <ref>, we can discover new ones by selecting less-studied q-hypergeometric identities as input. For example, feeding Bailey–Daum q-Kummer sum into our finder results in the following identity.

For two positive integers m and n,

    ∑_r = 0^nq^r+122m+r-1r_q2m+2nn-r_qm+nn_q^2=(1-q^n+1)2m+2n2n+1_q2n+1n_q.



Second, following each step of our algorithm can provide proofs for the output q-binomial identities. This is because our finder essentially simulates a rigorous human proof, with the only difference being that it assembles q-binomial coefficients faster and more comprehensively. This will be demonstrated in the subsequent sections.




























The remainder of this paper is organized as follows. Section <ref> introduces the necessary preliminaries related to square brackets and positive spans; Section <ref> provides a detailed description of our method; Section <ref> lists some of the identities discovered by our generator, which lead to Identity <ref>, Identity <ref> and Identity <ref>; and finally, Section <ref> concludes this paper with a discussion of potential future works.
















§ PRELIMINARIES



 §.§ Square Brackets

Square brackets serve as a unified language throughout our q-identities generator. To clarify their role and application, we will first define square brackets in detail, explaining how they function as an essential tool for translating q-Pochhammer symbols to q-binomial coefficients.


    Let k and n be two non-negative integers, and let m be an integer. We define the square bracket notation as follows.
    
    [m;k]_n := (q^km;q^k)_n

    For the sake of simplicity, we denote [m;1]_n by [m]_n, [m;k]_∞ by [m;k] and [m;1]_∞ by [m], respectively. Additionally, we call [m]_n the square bracket with subscript n and [m] the pure square bracket.


To represent more q-Pochhammer symbols using square brackets, we present the following lemmas.


Follow the notations in Definition <ref>. Then

    (-q^km;q^k)_n=[m;2k]_n/[m;k]_n,
    ∏_r=1^k-1(q^km+r;q^k)_n=[km;1]_kn/[m;k]_n.


This lemma directly follows from Definition <ref>, so we omit the proof. In most q-hypergeometric identities, the symbol “q” simply represents either q or both q and q^2. Consequently, in this paper, we only consider the case where k=2 in Eq. (<ref>).



Let m be an integer and let n be a non-negative integer. Then

    [m]_n=
    
        (-1)^nq^(mn+n2)[-m-n+1]/[-m+1]   if  m ≤ -n,
    
        0    if  -n < m ≤ 0,
    [m]/[m+n]   if  m > 0.

Moreover, [m]_n can be represented using square brackets of positive integers, or simply equals to zero.


An important fact is that the Gaussian binomial symbols can be also represented using square brackets of positive integers, or simply equals to zero.

Let m and n be two integers. Then

    m+nm_q=
    [m+1,n+1]/[m+n+1,1]   if  m and n≥ 0,
    
        0    otherwise.



Now our main purpose can be formally expressed by using the square bracket symbols of positive integers as to transform the following q-hypergeomtric identity

    ∑_n ≥ 0α(n)∏_k=1^∞[a_1,…,a_r_k;k]/[b_1,…,b_s_k;k]=∏_k=1^∞[c_1,…,c_t_k;k]/[d_1,…,d_u_k;k]

to as many as possible q-binomial identities

    ∑_n ≥ 0β(n)∏_k=1^∞A_1B_1_q^k⋯A_v_kB_v_k_q^k=∏_k=1^∞C_1D_1_q^k⋯C_w_kD_w_k_q^k.




 §.§ Positivity

Determining whether an item inside the square bracket is positive or negative might not pose a significant challenge to a human, but it can be a complex task for a computer at times. This is particularly true when the constraints of variables come into play, making it difficult to determine the polarity of an expression. Simply storing positive or negative results in memory also prevents computers from detecting contradictions in subsequent calculations. In Section <ref>, we will address this issue. For now, let us lay the groundwork by introducing the definition of ordered vector space from <cit.>.


Given a vector space V over the real numbers ℝ, and a strict partial order < on the set V. We say < is compatible with V, and (V,<) is an ordered vector space if for any u and v in V,

    
  * u<0, r is real and positive, implies ru<0;
    
  * u<0, v<0, implies u+v<0;
    
  * u<v if and only if u-v<0.


Consequently, 0 < u, or simply denoted as u > 0 if and only if -u < 0. Moreover, we denote ≤ (respectively, ≥) as the associated non-strict partial order relation of < (respectively, >). For any subset S of V, let S_+:={u∈ S:u≥0}. Then V_+ is a convex cone, with 0 being its vertex as well as its infimum.

Having defined the order concept of a vector space V, we now turn our attention to the specific case where V=ℝ^n. Each n-variable affine expression can be associated with a vector in V=ℝ^n+1 where the coefficients of the affine expression correspond one-to-one with the components of the vector. Specifically, the affine expression a_0+a_1x_1+⋯+a_nx_n can be associated with the vector (a_0,a_1,…,a_n) in V=ℝ^n+1. Obviously, this one-to-one correspondence is an isomorphism with respect to addition. Thus, the compatible order < can also be inherited. Once 
ℝ^n+1_+ is determined, we can obtain a polytope that defines the range of values for x's. To ensure the existence of integer solutions, we first add an infimum to ℤ^n_+\{0}. Specifically, We extend the definition of < by requiring u-1≥0 for any nonzero vector u in ℤ^n_+, where 1=(1,0,0,…,0). Then, we slightly modify the definition of positive span and frame as follows.




The positive span of a finite set of vectors S={v_1, v_2, …, v_k}⊂ℤ^n\{0,1} is defined as

    pos(S):= {λ_01+∑_j=1^kλ_j(v_j-1):λ_0∈ℝ_>0 and λ_j∈ℝ_≥0 for j=1,2,…,k}.

We say S is positively independent if v_i∉ pos(S\{v_i}) for i=1, 2, …, k.

A simple consequence is that such a positive span is a convex cone. Thus, all the elements in the positive span of positive vectors are positive.

Let C be a convex cone in ℝ^n. A finite set ℱ⊂ℤ^n\{0,1} is an integer frame of C if it is a positively independent set whose positive span is C. 


Regis <cit.> proposed several algorithms for determining positively independent sets and positive spanning sets in the real sense. Now, we transfer them to the integer setting as follows.
















§ METHODS


Roughly speaking, our method first express the given q-hypergeometric identities using the square bracket notation by Definition <ref> and Lemma <ref>. Then we ensure that each term inside the brackets is positive using Lemma <ref>. By grouping these bracketed expressions following the rules in Lemma <ref>, we construct q-binomial coefficients, which lead to the final identities. To be more specific, We illustrate these processes using a simple example.


For |c|<|ab|, the q-Gauss sum states:

    ∑_n≥ 0(a,b)_n(c/ab)^n/(q,c)_n=(c/a,c/b)_∞/(c,c/ab)_∞.

In order to put them in square brackets, we make some straightforward substitutions a=q^A, b=q^B and c=q^C where A, B and C are all integers such that C-A-B > 0. Then 

    ∑_n≥ 0q^n(C-A-B)[A,B]_n/[1,C]_n=[C-A,C-B]/[C,C-A-B].

Recalling our procedure from the beginning of this section, we must eliminate the subscripts n to ensure that the terms inside square brackets are positive, as per Lemma <ref>. However,in this case, whether these terms are positive or negative is uncertain. A reliable approach to find as many identities as possible is to consider any possible constraint conditions on these terms. We will explore this further in subsection <ref>. Here, instead, we assume that A and B are negative, while C is positive. Thus,

    ∑_n≥ 0q^n(C + n - 1)[-A-n+1,-B-n+1,n+1,C+n]/[-A+1,-B+1,1,C]=[C-A,C-B]/[C,C-A-B].

This is exactly the form of Eq. (<ref>). Next, by combining these bracketed expressions, we transform this identity into a binomial form. It is worth noting that such combinations are not unique, and we will discuss more practical combination methods in subsection <ref>. Here, we only demonstrate one possible combination. Following (<ref>),

    ⇒   ∑_n≥ 0q^n(C + n - 1)[-A-n+1,n+1]/[-A+1,1][-B-n+1,C+n]/[-B+1,C]=[C-A,C-B]/[C,C-A-B]
       ⇒   ∑_n≥ 0q^n(C + n - 1)-An_q[-B-n+1,C+n]/[-B+C,1][-B+C,1]/[-B+1,C]=[C-A,C-B]/[C,C-A-B]
       ⇒   ∑_n≥ 0q^n(C + n - 1)-An_q-B+C-1-B-n_q=[C-A,-B+1]/[C-A-B,1]
       ⇒   ∑_n≥ 0q^n(C + n - 1)-An_q-B+C-1-B-n_q=-A-B+C-1-B_q.

We begin with the sum side by combining the bracketed terms associated with n into q-binomial coefficients. The remaining bracketed terms independent of n can then be extracted from the summation and moved to the product side. In this example, the right side of Eq. (<ref>) precisely forms a q-binomial coefficient. Usually, similar operations to those on the sum side are also required on the product side.
In Eq. (<ref>), [-B+C] and [1] are introduced as intermediate terms in order to generate -B+C-1-B-n_q. This is feasible because [-B+C] and [1] are all positive. Unfortunately, it's not always guaranteed that the intermediate terms are positive. Therefore, we must verify the positivity of intermediate terms before incorporating them into our identity.




 §.§ Constraint Conditions Generator

Recall the process from Eq. (<ref>) to Eq. (<ref>) and the addition of intermediate terms. 
Each time we encounter a [m], whether it originates from our original identity or is newly introduced, we must ensure that m is positive to avoid scenarios where it equals zero making the identity trivial. Sometimes, this can be confirmed or contradicted based on prior knowledge. For instance, in Eq. (<ref>), C-A>0 holds due to the assumption that A<0 and C>0, whereas A-C cannot be positive under the same assumption. However, there are other instances where this cannot be inferred from prior knowledge alone. In such cases, we must force it to be positive to proceed with our process. As for the square bracket with positive subscript in Eq. (<ref>), say [m]_n, either m>0 or m<0 and -m-n+1>0 provides a feasible constraint condition due to Lemma <ref>. In the case of [m], things get simpler as only m>0 provides a feasible constraint condition. 





To help computers comprehend all the aforementioned processes, we express the variables as vectors. Continuing with Example <ref>, we write m=α_0+α_1n+α_2A+α_3B+α_4C as vector ϕ(m)=(α_0,α_1,α_2,α_3,α_4). Then by repeating Algorithm <ref> until all brackets in Eq. (<ref>) are considered or the output is empty, we will obtain all possible constraint conditions structured in a tree shape as follows. In this tree, each node represents a feasible constraint condition composed of inequalities involving integer affine expressions. Thus, each condition node corresponds to some positive integer vectors. Let C denote the positive span of these vectors. Then -C ∩ C=∅. Otherwise, no variables can satisfy this constraint.


>=stealth, 
         every node/.append style=font=, draw,
         edge from parent/.append style=->, shorten >=0pt, shorten <=0pt
         

level 1=[sibling distance=3cm, level distance=2cm]
level 2=[sibling distance=1.5cm, level distance=4cm]
level 3=[level distance=3cm]
level 4=[level distance=2.5cm]

[grow=right, edge from parent/.style=draw, ->, >=stealth, shorten >=2pt, shorten <=2pt, midway]
  
  C-A-B>0, 
    n≥ 0.
    child node -A, C-A-B>0, 
    0 ≤ n ≤ -A.
      child node -A, -B, C-A-B>0, 
    0 ≤ n ≤min(-A,-B).
        child node[draw=none] ⋯
            child node -A, -B, C>0, 
    0 ≤ n ≤min(-A,-B).
        
      
      child node -A, B, C-A-B>0, 
    0 ≤ n ≤ -A.
        child node[draw=none] ⋯
            child node -A, B, C-B>0, 
    0 ≤ n ≤ -A.
              
      
    
    child node A, C-A-B>0, 
    n≥ 0.
      child node A, -B, C-A-B>0, 
    0 ≤ n ≤ -B.
        child node[draw=none] ⋯
            child node A, -B, C-A>0, 
    0 ≤ n ≤ -B.
        
      
      child node A, B, C-A-B>0, 
    n ≥ 0.
        child node[draw=none] ⋯
            child node A, B, C-A-B>0, 
    n ≥ 0.
        
      
    ;






The feasibility of all nodes can be proven by induction.
First, the statement holds trivially for the initial case where there is only one constraint. Assume that the statement holds for one node with frame ℱ, namely, -pos(ℱ) ∩ pos(ℱ)=∅. Let m be a non-trivial integer affine expression and n be a positive integer. If ϕ(m-1) ∉ -pos(ℱ), then Definition <ref> and Definition <ref> imply that -pos(ℱ∪{ϕ(m)}) ∩ pos(ℱ∪{ϕ(m)})=∅. Otherwise, there exists a vector u∈-pos(ℱ∪{ϕ(m)}) ∩ pos(ℱ∪{ϕ(m)}), implying that there exist positive real numbers α_0 and β_0, and non-negative real numbers α_v's, β_v's, α and β such that 

    α_01+α(ϕ(m-1))+∑_v∈ℱα_v(v-1)=-β_01-β(ϕ(m-1))-∑_v∈ℱβ_v(v-1).

Thus, we have 

    (α_0+β_0)1+(α+β)(ϕ(m-1))+∑_v∈ℱ(α_v+β_v)(v-1)=0

where α+β cannot be zero as ℱ contains only positive integer vector. Hence, we have ϕ(m-1)∈ -pos(ℱ) which is a contradiction. Similarly, by these two definitions, we have if ϕ(m+1) ∉ pos(ℱ) and ϕ(-m-n) ∉ -pos(ℱ), then -pos(ℱ∪{ϕ(-m-n+1)}) ∩ pos(ℱ∪{ϕ(-m-n+1)}). Therefore, due to the design of Algorithm <ref>, the statement holds for all child nodes, which complete the proof.












 §.§ q-Binomial Coefficients Generator

To design an algorithm that transforms all square brackets to q-binomial coefficients and always stops after a finite number of steps, we eliminate the variables in the square brackets one by one as demonstrated in Example <ref>. Specifically, each time we will only focus on a target variable x, and then complete the following transformation.

    [a_1+m_1x,a_2+m_2x,…,a_r+m_rx]/[b_1+n_1x,b_2+n_2x,…,b_s+n_sx]→[c_1,c_2,…,c_t]/[d_1,d_2,…,d_u]∏_kf_k(x)g_k(x)_q

or abbreviated as

    ∏_a+mx∈𝒜[a+mx]/∏_b+nx∈ℬ[b+nx]→∏_c∈𝒞[c]/∏_d∈𝒟[d]∏_(f(x),g(x))∈𝒢f(x)g(x)_q

where m's and n's are positive integers, (a+mx)'s, (b+nx)'s, c's, d's, f(x)'s and g(x)'s are positive integer affine expression. Moreover, the coefficients of x in a's, b's, c's and d's are all zero. By Lemma <ref>, this transformation succeeds only if the sum of m's equals the sum of n's. For any a+mx in the numerator on the LHS and b+nx in the denominator on the LHS, the following transformation can reduce the number of square brackets containing x.

    [a+mx]/[b+nx]→b+nx-1a+mx-1_q[1]/[(b-a)+(n-m)x+1].

Transformation (<ref>) succeeds if the introduced term (b-a)+(n-m)x+1 is positive. This can be achieved if ϕ((b-a)+(n-m)x) ∉ -pos(ℱ) where ℱ is the frame corresponding to the current constraint condition. Following this transformation, the constraint condition should be updated using Algorithm <ref>.
Although, in theory, we can achieve Transformation (<ref>) using a finite number of Transformation (<ref>) by determining the order of a+mx and b+nx involved in each step, it is still extremely time-consuming for a computer to exhaust all these possible orders, especially when r or s is particularly large. Therefore, we propose a simpler and more efficient algorithm (see Algorithm <ref>) only considering the situation where m=n in Transformation (<ref>) and the following two combinations. 

    [a+mx,a'+m'x]/[(a+a'-1)+(m+m')x]=a+a'-2a+mx-1_q[1]; [a+mx,a'-mx]=a+a'-2a+mx-1_q[1,a+a'-1].

Because of this simplification, our method does not always cover all possible identities under a fixed constraint condition. However, for each repeat loop in Algorithm <ref>, there is more than one combination that meets the loop condition, which results in more than one valid transformation.



Our q-Binomial Identities Finder also provides an enhanced algorithm  in . It allows q-binomial coefficients to appear in the denominator, and non-q-binomial coefficients to exist in the equation. The former can be achieved simply by swapping the order of the numerator and denominator in Algorithm <ref>. For the latter, we exhaust all possible orders in Transformation (<ref>) and retain the parts where the target variable cannot ultimately be converted into a q-binomial coefficient. For example, 

    [x+1,x+2]/[2x+1,1]=[x+1,x+1]/[2x+1,1][x+2]/[x+1]=1/1-q^x+12xx_q.


















































































































 §.§ Backtracking Framework

Recall the Algorithms <ref> and <ref>. They both share a common framework for traversing feasible solutions. In Algorithm <ref>, we construct a tree structure: each time a new integer expression is introduced, if it meets the predefined conditions, the branch continues; otherwise, the branch is discarded. For Algorithm <ref>, we aim to pair elements from sets 𝒜 and ℬ such that after three repeat loops, all elements from both sets are used. Throughout the process, we can sequentially select elements from 𝒜 and ℬ, continually testing whether the required intermediate terms meet the conditions. If they do, the process continues; if not, that choice is discarded, and we backtrack to the previous step to choose a different combination. Therefore, we can use backtracking to implement Algorithm <ref> and explore feasible solutions for Algorithm <ref>. Backtracking is a method where solutions are constructed incrementally by making choices at each step and undoing them if they fail to satisfy the constraints. It is akin to exploring different paths and, upon encountering a dead end, backtrack to the last decision point to try a different route. Instead of detailing how backtracking is implemented for Algorithms <ref> and <ref> (details can be found in the  and  functions in  of the package), we will illustrate the backtracking framework using Algorithm <ref>.













    




    







                


























    

















§ RESULTS



By feeding various q-hypergeometric identities, we obtained a large number of q-binomial identities (refer to the Python package). In this section, we mainly focus on Identity <ref>, Identity <ref> and Identity <ref>.



 §.§ Identity <ref> and Identity <ref>

 
In this subsection, we will demonstrate some known q-binomial identities including Identity <ref> and Identity <ref>.
We start with our input q-hypergeometric identity.
 [q-Pfaff–Saalschütz Sum, <cit.>]
For positive integer N,

    _3ϕ_2 ( [   a, b, q^-N; c, abq^1-N/c ]; q, q )
    = (c/a,c/b)_N/(c,c/ab)_N


Identity <ref> has been combinatorially proven by Andrews and Bressoud <cit.>, Goulden <cit.> and Zeilberger <cit.>. Assume a=q^A, b=q^B and c=q^C for some integers A, B and C. Then feeding it to our finder leads to many q-binomial identities under different constraint conditions. In this subsection, we focus on two parallel constraint conditions.



  §.§.§ Condition I

For -A, B, C, N and A + B - C - N + 1 > 0, the total six outputs are as follows.


    ∑_r = 0^Mβ(r) B + r - 1B - C_qNr_qB - C - N-A - r_q = -A + C + N - 1-A_qB - 1A + B - C_q



    ∑_r = 0^Mβ(r) B + r - 1B - C_q-Ar_qA + B - CN - r_q = -A + C + N - 1N_qB - 1C + N - 1_q



    ∑_r = 0^Mβ(r) B + r - 1r_qC + N - 1N - r_qB - C - N-A - r_q  = -A + C + N - 1N_qB - C-A_q



    ∑_r = 0^Mβ(r) B + r - 1r_q-A + C - 1-A - r_qA + B - CN - r_q = -A + C + N - 1-A_qB - CN_q



    ∑_r = 0^Mβ(r) B + r - 1-A + C + N - 1_qC + N - 1N - r_q-Ar_q = B - CN_qB - 1A + B - C_q



    ∑_r = 0^Mβ(r) B + r - 1-A + C + N - 1_q-A + C - 1-A - r_qNr_q = B - 1C + N - 1_qB - C-A_q

where β(r) = q^(r+A)(r-N) and M = min(-A,N).

By Eq. (<ref>)-(<ref>), we note that under this constraints condition, the fact that -A and N are interchangeable in Identity <ref> is inherited. By making simple variable substitutions of (A, B, C, D), we can obtain some classic q-binomial identities which are listed in Table <ref>. 







  §.§.§ Condition II

For -A, -B, C, N and -A - B + C + N - 1 > 0, there are six outputs. We only list one of them as follows since -A, -B and N are interchangeable in Identity <ref> and this condition.








    ∑_r = 0^Mβ(r) -A-B+C+N-r-1N-r_q-B+C-1-B-r_q-Ar_q = -A+C+N-1N_q-B+C+N-1-B_q

where β(r) = q^r(C+r-1) and M = min(-A,-B,N,-A - B + C + N - 1). Letting (A,B,C,N)=(-k,-n,1,x) in Eq. (<ref>) leads to Identity <ref>. 








 §.§ Identity <ref> and Its Proof

Identity <ref> is derived by Bailey–Daum q-Kummer sum, which is stated as follows.
[Bailey–Daum q-Kummer Sum, <cit.>]
For positive integer N,

    _2ϕ_1 ( [ a, b; aq/b ]; q, -q/b )
    = (-q;q)_∞(aq,aq^2/b^2;q^2)_∞/(-q/b,aq/b;q)_∞


Our finder only accepts identity of the form given in Eq. (<ref>). Thus, we first set a=q^2A and b=q^B for some integers A and B. Then by Lemma <ref>, Identity <ref> is equivalent to the following identity.

    ∑_r ≥ 0 (-1)^rq^r(-B+1)[2A,B]_r/[1,2A-B+1]_r=[2A,-B+1][1,A-B+1;2]/[1,2A-B+1][A,-B+1;2]

Applying  to Eq. (<ref>) outputs Identity <ref>. In the algorithm, the square brackets [m;k] will be grouped according to k and then be processed in parallel. Moreover, following the procedure of our algorithm leads to the following proof of Identity <ref>.

For A>0 and B<0, Eq. (<ref>) implies

    ∑_r = 0^-B q^r+12[2A,-B-r+1,2A-B+r+1,r+1]/[2A+r,-B+1,2A-B+1,1]=[2A,-B+1][1,A-B+1;2]/[1,2A-B+1][A,-B+1;2]
       ⇒   ∑_r = 0^-B q^r+122A+r-1r_q2A-2B-B-r_q[2A-2B+1,1]/[2A-B+1,-B+1]=[2A,-B+1]/[1,2A-B+1]A-B-B_q^2^-1
       ⇒   ∑_r = 0^-B q^r+122A+r-1r_q2A-2B-B-r_qA-B-B_q^2 = [2A,-B+1,-B+1]/[2A-2B+1,1,1]
       ⇒   ∑_r = 0^-B q^r+122A+r-1r_q2A-2B-B-r_qA-B-B_q^2 = 2A-2B-2B+1_q[-B+1,-B+1]/[-2B+2,1]
       ⇒   ∑_r = 0^-B q^r+122A+r-1r_q2A-2B-B-r_qA-B-B_q^2 = 2A-2B-2B+1_q-2B+1-B_q[-B+1]/[-B+2].

Our finder will continue to transform [-B+1]/[-B+2] to (1-q)-B+11_q, but we simply write it as 1-q^-B+1 to obtain the following result.

    ∑_r = 0^-B q^r+122A+r-1r_q2A-2B-B-r_qA-B-B_q^2 = (1-q^-B+1)2A-2B-2B+1_q-2B+1-B_q

Thus, setting A=m and B=-n in Eq. (<ref>) leads to Identity <ref>. 


































        



        





















        



        




























§ FUTURE WORKS

Only a few identities are demonstrated in this paper. Our q-binomial identities finder remains hungry, waiting to be fed with more q-hypergeometric identities including multiple-term summation formulas <cit.>. Moreover, there exist many other variant generalizations of Gaussian binomial coefficients, including Fibonomial coefficients <cit.>, rising binomial coefficients - type 2 <cit.>, Gaussian q-binomial coefficients with two additional parameters <cit.>, a generalization of binomial coefficients, replacing the natural numbers by an arbitrary sequence <cit.>, and  the second quantized binomial coefficients <cit.> for quantum scenario. To put them in the form of square brackets and modify our finder accordingly can also lead to related identities.

Although our finder is capable of generating q-binomial identities and providing proofs derived from q-hypergeometric identities, it is insufficient in combinatorial interpretation and further application. Nowadays, the q-binomial identities have been followed by many applications across various fields. The generating function for sets of pairs of partitions that have an ordering relation between parts of the two pairs were studied in the form of q-binomial coefficients in <cit.>, which led to an infinite number of identities between a single and a multiple summation. Following this study, Foda et al. <cit.> obtained an infinite tree of q-polynomial identities for the Virasoro characters, by using the q-Saalschütz derived Burge transform, along with a combinatorial identity between partition pairs. Investigating the properties of q-binomial identities has also led to interesting results on supercongruences for q-analogs of the integer sequences, such as Apéry numbers <cit.>. Additionally, by using Identity <ref> and the Chinese remainder theorem for coprime polynomials, Wei et al. <cit.> derived a series of q-supercongruences related to the third power of cyclotomic polynomials. Moreover, binomial identities were even applied to the design of privacy schemes and the assessment of their privacy performance <cit.>. Thus, newly discovered identities should not be seen as the endpoint of research, but rather as the starting point for further exploration and discovery.
















    











§ ACKNOWLEDGMENTS

We thank Iskander Aliev for some helpful comments on integer positive space, and Dawei Niu for some helpful comments on q-hypergeometric identities. 
This work was supported by the National Natural Science Foundation
of China (Grant No. 12301004).


amsplain




 Paper Title: Smooth Points on Positroid Varieties.





















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jkf68@cornell.eduSmooth Points on Positroid Varieties
    Joseph Fluegemann
    August 5, 2024
====================================

§ ABSTRACT 


In the Grassmannian Gr_(k,n) we have positroid varieties Π_f, each indexed by a bounded affine permutation f and containing torus-fixed points λ∈Π_f. In this paper we consider the partially ordered set consisting of quadruples (k,n,Π_f,λ) (or (positroid) pairs(Π_f,λ) for short). The partial order is the ordering given by the covering relation ⋖ where (Π_f',λ') ⋖ (Π_f,λ) if Π_f' is obtained by Π_f by deletion or contraction. Using the results of Snider <cit.>, we know that positroid varieties can be studied in a neighborhood of each of these points by affine pipe dreams. Our main theorem provides a quick test of when a positroid variety is smooth at one of these given points. It is sufficient to test smoothness of a positroid variety by using the main result to test smoothness at each of these points. These results can also be applied to the question of whether Schubert varieties in flag manifolds are smooth at points given by 321-avoiding permutations, as studied in <cit.>. We have a secondary result, which describes the minimal singular positroid pairs in our ordering - these are the positroid pairs where any deletion or contraction causes it to become smooth.  


§ INTRODUCTION


 §.§ Motivation from Schubert varieties


In order to provide context of our smoothness problem for those familiar with Schubert varieties, we note both the similarities and differences compared to the case of smoothness for Schubert varieties. In flag manifolds, the smooth Schubert varieties were classified via a pattern avoidance technique by Lakshmibai and Sandiya <cit.>. Note that if we just restrict to the Grassmannian, it turns out that the smooth Schubert varieties are essentially trivial - for any Gr(k,n), the smooth Schubert varieties are just the ones that are subGrassmannians (isomorphic to Gr(k',n') for k'≤ k, n'≤ n). We emphasize that in this paper, we are not classifying smooth positroid varieties, as was done in <cit.>. Rather, we are only considering smoothness at the fixed points, one fixed point at a time. 

Inside a given Schubert variety X_w, for any fixed point v, the Schubert subvariety X_v is the B_--orbit closure of the point v, and the singular locus for any given Schubert variety is invariant under the action of B_-, so the singular locus must consist of a union of Schubert subvarieties. Within X_w, these Schubert varieties X_v on which X_w is singular are obviously closed under going up in the Bruhat order, since if X_w is singular along X_v then X_w will also be singular on any subset of X_v. Then, the natural question of what are the maximal (that is, minimal in Bruhat order) such Schubert varieties X_v  was answered by the 321-hexagon-avoiding permutations described by Billey and Warrington <cit.>. 

Another question of checking whether X_w is singular at a T-fixed point v was answered by Dale Peterson: consider s= #{ v r_α≥ w } where α is over positive roots. Since each v r_α≥ w produces a linearly independent vector in the tangent space, s ≤ T_v (X_w). Thus is s >  (X_w) then the tangent space is too large so X_w is singular at v. Conversely if the group is simply-laced, as it is in the case when the group is a torus, then s =  T_v (X_w) implies that X_w is smooth at v.

We answer the corresponding questions for positroid varieties, both the question of whether a positroid variety Π_f is smooth at a given fixed point λ, and also what the minimal singular positroid varieties are, under an ordering given by deletion and contraction. 




 §.§ The Results


First, we describe our poset (with more details given in Subsection <ref>). We make use of the following two subGrassmannians inside Gr(k,n):

Π_del,i := {V∈ Gr(k,n) |  V≤ Proj_i(^n)≅^n-1}≅ Gr(k,n-1)
Π_contr,i := {V∈ Gr(k,n) |  V≥_i }≅ Gr(k-1,n-1)


The T-fixed points on Gr(k,n) are given by the span of k coordinates lines, which we have denoted by λ, so there are n k such fixed points. Given such a T-fixed point λ in Gr(k,n) and a coordinate x_i, then either that fixed point λ uses x_i or it does not. This can be rephrased as: for each T-fixed point λ and each i∈{1,...,n}, either λ∈Π_del,i or λ∈Π_contr,i, but not both. We can intersect a positroid variety Π_f with one of these two subGrassmannians, and what we obtain is a smaller positroid variety Π_f' and a point λ' on it. Call this Π_f' the deletion or contraction respectively. In other words: 



(1) If we have a T-fixed point λ on a positroid variety Π_f and we have one of the coordinates x_i, then we can either delete or contract x_i, and we will get to another point λ' on a smaller positroid variety Π_f'.



(2) Furthermore, if the point λ we started out with is smooth on Π_f, then the point λ' after deletion or contraction will again be smooth on Π_f ∩Π_del,i or Π_f∩Π_contr,i (this will be proved as Proposition <ref>). 



Because of (1), we can make an (infinite) poset on the set of quadruples (k,n,Π_f,λ) with deletion and contraction as the covering relations. By (2), the set of singular quadruples is closed under going upwards. We can visualize the poset as:

 

Given this poset, we can consider the points along the boundary (the minimal singular pairs):

 

Since n decreases as one goes down in the ordering, we can only go down in the order a finite number of steps from any given point. By this finiteness of going down, a pair is singular if and only if it is greater than one of these minimal singular pairs in the ordering. We give these minimal singular pairs a name:

[Atomic Positroid Pair]
A positroid variety Π_f is atomic on an open patch U_λ if it is singular at λ, but any deletion or contraction (of any column) causes this positroid variety to be smooth (or empty) at λ. We call (Π_f, λ) an atomic positroid pair.



Next, we recall that an affine pipe dream shape is the boundary of an affine pipe dream <cit.>. More details can be found in Section <ref>. A quick example is provided by the following. 



Here, we are in Gr(3,7) and λ={1,2,4} (these are the full rank columns). We “collapse" these columns, such that they are now depicted by the vertical segments in the distinguished path (staircase line) running from the NW to SE corners. The affine pipe dream's shape is an infinite strip obtained by taking this distinguished path and placing an exact copy directly above it and another copy to the right. This continues forever in the NW and SE directions as depicted in the figure below (note that the numbers within the pipe dream shape can be ignored for now). 




If the numbers in λ are consecutive, say λ={i,i+1,i+2}, then the shape obtained will be an infinite string of rectangles touching in the northwest/southeast corners, as depicted in the following:

 

We remark that Grassmannian duality: Gr(k,n)∼ Gr(n-k,n) reflects the pipe dreams shapes across the line x=-y (and this also switches deletion and contraction).



Additionally, this is what the maximal rectangles look like in our affine shapes:

 


One goal we have is to quickly determine whether a positroid pair is singular at a given point. Our main method uses the following Theorem <ref>:


A positroid variety Π_f is smooth at the point λ (it meets λ and is smooth there) if and only if there is a single affine pipe dream representative <cit.> of the affine permutation f on the open set U_λ. Specifically Π_f meets the point λ if and only if there exists (at least) one affine pipe dream, and it is smooth if and only if that affine pipe dream is unique. Thus, we can take Π_f and look at pipe dreams for all choices of λ to test for smoothness. 


By applying the previous theorem, we can prove the Main Theorem <ref>. To state this theorem, we first need two definitions:


We have a partition in the northwest of a rectangle if the maximal connected set of crosses containing the cross in the northwest corner (it can be empty) is of the following type of shape: if each successive row as we go down consists of a consecutive crosses coming from the left edge of the rectangle such that the number of such crosses is nonincreasing (equivalently, if we take the axes as going right and down, given any cross at (a,b), all (i,j) with i≤ a and j≤ b must be crosses). A partition in the southeast is the same, but reflected across the southwest-northeast axis. Here is an example of a northwest partition:

 
A pipe dream in a rectangular shape reduces to SE/NW partitions if, after deleting all entire rows and columns of crosses, the result is a partition in the southeast and northwest corners. 

This is equivalent to the condition that all crosses must be contained in: (1) entire rows of crosses, (2) entire columns of crosses, (3) a partition in the northwest, (4) a partition in the southeast.

The following is an example of a rectangular pipe dream that reduces to SE/NW partitions:
 [Main Theorem]
If each maximal rectangle in the pipe dream for Π_f on U_λ reduces to SE/NW partitions, then Π_f is smooth at λ; otherwise, it is singular. 


Thus, if we check all maximal rectangles and, for each one, after removing all entire rows and columns of crosses, see that we are left with a partition in the northwest and one in the southeast, then our positroid variety is smooth at the given point; otherwise, it is singular. In other words, within each maximal rectangle, the only crosses that exist are contained in: (1) entire rows of crosses, (2) entire columns of crosses, (3) a partition in the northwest, (4) a partition in the southeast; in any given maximal rectangle, there can be many possible instances of (1) and (2), but only a single partition (3) and a single partition (4). 



Another method for determining from the notion of top and bottom pipe dreams. From <cit.>, we know that in the matrix Schubert variety case, the set of all pipe dreams for a given permutation has two polar opposites: the top and bottom pipe dreams. In Appendix B, we will show that this is also true in the affine pipe dream case. Thus, a second way to determine whether a pair is smooth is to find the top and bottom pipe dreams and see if they match. 



Finally, although not an algorithmic method like the previous two, we also have a condition for whether a positroid pair (Π_f,λ) is smooth or singular using atomic positroid pairs, which we defined above; this essentially follows from the definition of the partial order, and is the content of Proposition <ref>:


A point λ being singular on a positroid variety Π_f  is equivalent to the pipe dream for Π_f on U_λ being able to reach an atomic configuration via a series of deletions and contractions. 


We want to classify these atomic positroid pairs. This is given in Theorem <ref>:



The atomic positroid pairs have affine pipe dreams that look like the following:

 

Specifically, the atomic positroid pairs:

(1) Have pipe dreams in a square shape: so they are in Gr(k,2k) and on points where λ consists of k consecutive columns. 

(2) In this pipe dream, there is a single cross along the southwest to northeast (longest) diagonal, with the diagonals directly adjacent to this longest diagonal being free of any crosses.

(3) Besides this one cross, there are partitions of crosses in the northwest and southeast corners, and no others.




Finally, we remark that the techniques in the proof of the Main Theorem can be easily extended to the case of Schubert varieties in flag manifolds at points given by 321-avoiding permutations (Prop <ref>):


Let Fl(n) denote the variety of flags in ^n. Let w∈ S_n denote a permutation on n elements so that X_w denotes a Schubert variety in this flag manifold, and let v∈ S_n be 321-avoiding with v≥ w in Bruhat order. Then X_w is smooth at v,  if and only if there exists a pipe dream for w inside v's skew partition (defined above Prop <ref>) such that all maximal rectangles reduce to NW/SE partitions. (In this case, the pipe dream will be unique.) 


§ BACKGROUND INFORMATION


 §.§ Positroid Varieties


Our background for the following results on positroid varieties is <cit.>. 

[Bounded Affine Permutation]
Fix n∈ℕ. A permutation g:→ is called affine if it satisfies the periodicity condition g(i+n)=g(i)+n, ∀ i ∈ .

If the affine permutation also satisfies i≤ g(i)≤ i+n for all i, it is called bounded. We denote the set of bounded affine permutations of period n and average jump avg(g(i)-i)=k by Bound(k,n). 

Given that an affine permutation repeats with period n, it is sufficient to give a permutation by stating its action on any n consecutive numbers. This is called window notation. 

In window notation, g=(4,6,3,7,11,8) defines a bounded affine permutation of period 6 sending g(1)=4, g(2)=3,...,g(6)=8 and repeating, so g(7)=g(1)+6=10, etc.


The information contained in a bounded affine permutation f can be given a different, but equivalent, labelling called its siteswap, denoted Site(k,n).

[Siteswap]
The siteswap f∈ Site(k,n) corresponding to a bounded affine permutation g∈ Bound(k,n) is f(i):=g(i)-i, 1≤ i ≤ n. Notice that a siteswap f(i) is only defined for i∈[1,n].

The siteswap corresponded to g=(4,6,3,7,11,8) is  f=(3,4,0,3,6,2).
Remark: Siteswaps or bounded affine permutations represent (one-handed, bounded) juggling patterns. Namely, the affine permutation can be considered as a permutation taking the time at which a ball is thrown to the time at which that ball lands (so the siteswap gives the number of seconds after a ball is thrown that it lands). 

Let [i,j] := {i, i+1,...,j} n, so [i,j]= j-i+1     j≥ i 
 (n-i+1)+j     i>j. 



Define a_g,i,j:=#{[i,j]\{g(i+ℕ) }∩{≤ j}}. This can be visualized by drawing a permutation matrix for the bounded affine permutation g, as in the figures in Appendix A. For say g(ĩ)=j̃, the ĩ runs vertically downwards in the figure below, while j̃ runs horizontally to the right. 

We have: rank_g[i,j]=[i,j] - a_g,i,j = (j-i+1) - a_g,i,j. 

Now, we can define the positroid variety corresponding to a siteswap f (we apply n to the definitions using g above):

[Closed and Open Positroid Varieties]

    Π_f := GL_k ∖{M⊆ M_k,n| rankM=k  and  rank_M([i,j]) ≤ rank_f[i,j], i≤ j ≤ i+n}

    Π_f^0 := GL_k ∖{M⊆ M_k,n| rankM=k  and  rank_M([i,j]) = rank_f[i,j], i≤ j ≤ i+n}

where rank_M([i,j]) is defined cyclically.


The open positroid varieties Π_f^0 are open inside their closures (locally closed), and are all smooth.



 §.§ Deletion and Contraction


The Grassmannian is represented by full-rank k-by-n matrices (up to GL(k) action on the left): 


    {M}=
    {cccccc          n       
    c(ccccc)               
    
    k           *       
                   
    }


Then there is a circle action on the right via matrix multiplication by 

    S_i=(1,1,⋱,1,z,1,⋱,1)
 
and this scales the i-th column of M by z. 
 

 
There is a “Pascal recurrence" on Grassmannians:



Let S_i act on {M}=Gr(k,n) on the right. 

Let


    Π_contr,i := rowspan {(         0         
       *      0      *   
     0   ⋯   0   1   0   ⋯   0) }≅ Gr(k-1,n-1)
 

and


    Π_del,i := rowspan {(         0         
       *      0      *   
             0         ) }≅ Gr(k,n-1)


where the rest of the matrix is full-rank. 

Note that Π_contr,i corresponds to the siteswap (k-1)^n-kk^k-1n rotated so that the n is in the ith position, and Π_del,i corresponds to the siteswap (k)^n-k-1(k+1)^k0 rotated so that the 0 is in the ith position.


Then the fixed point set under the S_i action is: 


    Gr(k,n)^S_i= Π_contr,i⊔Π_del,i≅ Gr(k-1,n-1) ⊔ Gr(k,n-1)


and every T-fixed point is in one or the other.



The process called “contraction" has two steps. First we first intersect with Π_contr,i: Π_contr_i(f) := Contr_i(Π_f)= Π_f∩Π_contr,i. Here, we are defining contr_i(f) to be the bounded affine permutation of the positroid variety equal to Π_f∩Π_contr,i; this is possible by Proposition <ref> below, and the details of how to calculate it can be found in Appendix A. This restricts Π_f to the subset that can be written in the form of the matrix representative for Π_contr,i given in the proposition. Second, we do the projection that removes the last row and i-th column to bring it to a subset of Gr(k-1,n-1). This intersection with Π_contr,i=Contr_i(Gr(k,n)) turns out to be the largest positroid variety with f(i)=i+n contained in Π_f (the proof is below). As long as a given positroid variety Π_f contains at least one point whose k-plane uses the i-th coordinate, the contraction of this positroid variety will be nonempty. 

Remark: there is sometimes an abuse of terminology where “contraction" can also refer to only performing the first step, without the projection to Gr(k-1,n-1). We define contraction as performing both steps, and when only the first step is involved, we call it projectionless contraction. Similarly for projectionless deletion below. 
 
Remark: note that contraction (and similarly deletion, which we describe next) is an operation that is performed on the entire Grassmannian Gr(k,n) (in fact Gr(k,n)=Π_f' with f'=kk⋯ kk), but since we will be working with individual positroid varieties throughout this paper, our notation here emphasizes the effect of contraction (or deletion) on a specific positroid variety Π_f. 


 
Similarly, the process of “deletion" has two steps: we first intersect with Π_del,i: Π_del_i(f):= Del_i(Π_f)=Π_f∩Π_del,i where Π_del,i.  Then we do the projection that removes the i-th column to bring it to a subset of Gr(k,n-1). This intersection with Π_del,i=Del_i(Gr(k,n)) turns out to be the largest positroid variety with f(i)=i contained in Π_f. 



There is another description of the deletion and contraction of positroid varieties.
For positroid varieties of Gr(k,n), deletion and contraction of the ith column of a positroid variety Π_f can be described as:


The projectionless deletion Del_i(Π_f) of the i-th column of a positroid variety Π_f is equal to Π_f'^d(i):= the largest positroid variety Π_f' contained in Π_f such that f'(i)=0. (By largest, we mean that Π_f' is not properly contained inside another positroid variety Π_f” properly contained in Π_f.) 

The projectionless contraction Contr_i(Π_f) of the i-th column of a positroid variety Π_f is equal to Π_f'^c(i):= the largest positroid variety Π_f' contained in Π_f such that f'(i)=n.


Recall that the projectionless contraction was defined as the intersection: Contr_i(Π_f)=Π_f ∩Π_contr,i, and similarly the projectionless deletion is  Del_i(Π_f)=Π_f ∩Π_del,i.

Before we prove Prop <ref>, we prove another proposition that we will need in the proof. 


The intersections Contr_i(Π_f)=Π_f ∩Π_contr,i and Del_i(Π_f)=Π_f ∩Π_del,i are irreducible.

Because the positroid varieties form a stratification, intersections of positroid varieties are in general unions of positroid varieties, but we show now that in this case of Contr_i(Π_f)=Π_f ∩Π_contr,i, this intersection is irreducible (when it's nonempty), which means that Contr_i(Π_f) cannot split into multiple positroid varieties - it must be a single positroid variety. The intersection Π_f ∩Π_contr,i is irreducible because we have 

    Π_f ↩Π_f \Π_del,i↠Π_f ∩Π_contr,i
 
The latter map is a surjection because we have a right inverse given by taking a plane (point) in Π_f ∩Π_contr,i and taking its direct sum ⊕, a copy of  lying along the ith coordinate. 
Recall that irreducibility means the coordinate ring has no zerodivisors. Positroid varieties are irreducible, so Π_f is irreducible. If we look at Π_f \Π_del,i, ripping out a subvariety enlargens the coordinate ring by introducing denominators (localization), not zerodivisors; therefore, Π_f \Π_del,i is irreducible. Finally, if we consider Π_f \Π_del,i↠Π_f ∩Π_contr,i, we use the fact that the image of an irreducible variety is an irreducible variety to conclude irreducibility of Π_f ∩Π_contr,i (here, the image is everything due to surjectivity, so it is irreducible). This fact is true simply because the map of a variety onto its image corresponds to an inclusion of the coordinate rings in the contravariant direction, and a subring of a ring with no zerodivisors does not have zerodivisors either.  

The analogous statement for Π_f ↩Π_f \Π_contr,i↠Π_f ∩Π_del,i follows from Grassmannian duality and proves the irreducibility of Π_f ∩Π_del,i.

 (of Prop <ref>)

We make use of the fact that f'(i)=0 for a positroid variety Π_f' is equivalent to Π_f' being contained in Π_del,i; similarly, f'(i)=n for a positroid variety Π_f' is equivalent to Π_f' being contained in Π_contr,i. 

(⊇) We first show that Del_i(Π_f) ⊇Π_f'^d(i), that is, Π_f ∩Π_del,i⊇Π_f'^d(i). This is true because (a) Π_f'^d(i)⊆Π_f by definition (as the largest positroid variety contained withinΠ_f) , and similarly Π_f'^d(i)⊆Π_del,i follows from its definition as having f'(i)=0, in accordance with the fact that we just stated. The case of contraction is the same.

(⊆) Finally we show that Del_i(Π_f) ⊆Π_f'^d(i), that is, Π_f ∩Π_del,i⊆Π_f'^d(i). 
For contradiction, assume this is not true; that is, Π_f'^d(i) is properly contained within Π_f ∩Π_del,i: Π_f'^d(i)Π_f ∩Π_del,i. We will show the contradiction by making use of the irreducibility noted in Prop <ref> (namely, that while an intersection of positroid varieties like Π_f ∩Π_del,i can in general split into a union of positroid varieties: Π_f_1∩Π_f_2 = Π_f_3∪Π_f_4∪⋯∪Π_f_j, in this special case of Π_f ∩Π_del,i, the irreducibility implies that it is a single positroid variety: Π_f ∩Π_del,i = Π_g). In general, Π_g Π_f, so we would have Π_f'^d(i)Π_g Π_f, and this contradicts the defintion of Π_f'^d(i) as the largest positroid variety contained in Π_f such that f'(i)=0 (since Π_g = Π_f ∩Π_del,i also satisfies f'(i)=0). Of course, in the special case when Π_f ⊆Π_del,i, then Π_f ∩Π_del,i = Π_f, and the largest positroid variety Π_f' contained within Π_f satsifying f'(i)=0 is just Π_f itself, so Π_f'^d(i) = Π_f = Del_i(Π_f) and so everything holds automatically.

The proof for the contraction case is analogous.


§ AFFINE PIPE DREAMS
[Pipe Dreams for Matrix Schubert Varieties]
A pipe dream of size n is a diagram in an n-by-n square where each box can be filled by one of two types of tiles, elbows ­ and crosses , such that all crosses occur above the antidiagonal. Thus, we can think of the grid as a set of pipes that begin on the north and east edges and end on the west and south edges (with the south to east tiles being all elbows).

We called the pipe dream reduced if no two of the pipes cross twice. We will focus only on the reduced case in the following, so the adjective “reduced" should always be assumed. 

The following figure shows an example for all pipe dreams corresponding to the permutation π=2143 in S_4. The pipes running from the south edge to the east edge are uninteresting so are not shown. 


A cross-elbow move (in the following: “move," for short) in a pipe dream is an interchange of a cross tile and an elbow tile that preserves the connectivity of the pipes - in other words, that leaves the permutation invariant. We think of the cross as “moving" to the position where the elbow it is replacing was located. If we perform all possible moves in a pipe dream, and keep repeating, taking all the new pipe dreams obtained after performing moves and doing moves on these new ones, then we will obtain the set of all possible pipe dreams for that particular permutation (see <cit.>). If the pipe dream has no possible moves, so is the unique pipe dream for a particular permutation, then we call the pipe dream rigid. The notion of moves carries over without change to the case of bounded affine permutations which we define next (the chute and ladder moves of <cit.> are a subset of our moves), and will be used extensively below. 

In the previous figure, for the pipe dream on the very left, we see that the cross in the northwest corner has no moves, but the other cross (to the right) has 2 different places it could potentially move, which does not change the connectivity/permutation. Therefore, this gives us the 3 different pipe dreams. 


There is a set of affine pipe dreams associated to a pair (f, λ) of a bounded affine permutation f and a set of k columns λ∈[n]  k, which diagrammatically display the intersection Π_f ∩ U_λ. The shape or outer boundary of the affine pipe dream depends only on λ; it is constructed by picking a distinguished path corresponding to λ and using it to cut a matrix into two pieces, which are then shifted northeast and southwest infinitely. The details are in Chapter 4 of <cit.>, as well as in this section and the Appendix.


In this example, the vertical lines in the picture on the left occur right next to the 1's where the columns in λ are: these are columns 1, 2 and 4. On the right, after “collapsing" these columns, they are now depicted by the vertical segments in the distinguished path (the staircase line) running NW to SE. 



The affine pipe dream's shape is an infinite strip obtained by taking this staircase line and placing an exact copy directly above it and another copy to the right. This continues forever in the NW and SE directions as depicted in the figure below (note that later in this section we will explain the numbers within the pipe dream shape). Affine pipe dreams are diagrams drawn by filling in the squares inside the shape with cross and elbow tiles. 



The main theorem proven in <cit.> is that the restriction of a positroid variety to a point is stratified isomorphic to affine Kazhdan-Lusztig varieties. In the next section, we will use this to show that this implies that the equivariant cohomology class of a positroid variety restricted to a point is therefore given by a sum over these affine pipe dreams (for the shape of the point being restricted to). We will then show that by restricting our torus action to a circle acting by dilation, we obtain a corollary that the multiplicity of that point on the positroid variety is equal to the number of pipe dreams for that positroid variety in that shape. The details are below. In any case, because <cit.> proves this correspondence using these pipe dreams shapes, and since this correspondence is all we need from these shapes to prove the theorem on the relationship between smoothness and pipe dreams in the next section, we can use them for the results we prove below. However, it is not difficult to prove that any periodic pipe dream shape, which when filled in entirely with crosses produces the permutation of a point, must be of the form above; the proof is included in a companion thesis <cit.>. 

The numbers within the pipe dream shape come from thinking of the pipe dream as a rotated wiring diagram, and subsets of these numbers form words in a Coxeter group; for example, the numbers along the northeast diagonal coming out from the “1" spot along the lower boundary are also all labeled “1" since any cross placed along this diagonal should produce an s_1 in the word - that is, it should transpose the first and second elements, or equivalently in the picture, switch the pipes emerging from the first and second positions. 



Now we show that for these affine pipe dreams, it is still the case that if a particular affine pipe dream is rigid (it has no possible moves), then it is the only one that produces its corresponding bounded affine permutation. 



For any affine pipe dream Ω with bounded affine permutation g, every other affine pipe dream Ω' that also has bounded affine permutation g can be reached via Ω by performing a sequence of moves on Ω.


Before we give the proof, we state the following definition of a word, taken from <cit.>, and a couple of facts to be used in the proof.

[<cit.>]
Let Π denote the set of permutations on n elements, and let Σ denote the set of transpositions s_1,...,s_n where s_i switches i and i+1. A word of size m is an ordered sequence  Q=(σ_1,...,σ_m) of elements of Σ. An ordered subsequence P of Q is called a subword of Q. 

 Prepresentsπ∈Π if the ordered product of the simple reflections in P is a reduced decomposition for π.

Pcontainsπ∈Π if some subsequence of P represents π.

The subword complex Δ(Q,π) is the set of subwords Q\ P whose complements P contain π.



In Appendix B, we do an example to illustrate how a subword P_f,λ for the siteswap f at the point λ is obtained from the word Q_λ of the affine patch around the point λ. Here, we just state the following facts. 

[Corollary 1 in <cit.>] 
Define the affine pipe dream complexΔ(λ,π) as is the simplicial complex with vertices labeled by entries (i, j) in the periodic strip and faces labeled by the elbow sets in the affine pipe dreams for π with shape defined by λ. It is isomorphic to the subword complex Δ(Q_λ,f).
[KM04, Theorem 3.7] The topological realization of any subword complex Δ(Q,π) homeomorphic to a ball or sphere; in particular, every ridge (codimension 1 facet) is contained in one or two facets. The dual graph (whose vertices are the facets of Δ(Q,π) and whose edges are the ridges of Δ(Q,π)) is connected.
[Definition # 2 of pipe dreams moves]
We defined pipe dream moves above in terms of exchanges of tiles on a particular pipe dream. In the context of subword complexes, moves correspond to going to a ridge and then going to another codimension-1 facet. This follows from the fact the notion give above of a move in terms of exchanging a cross and a near-miss can actually be thought of as adding a cross in the new location before removing the original one, and the ridges in the complex are where we have nonreduced pipe dreams.

By Proposition <ref>, there exists a subword complex Δ(Q_λ,f) for which the set of all pipe dreams for a siteswap f on a patch λ are the codimension-1 facets of this complex. By Theorem <ref>, since subword complexes are homeomorphic to balls or spheres, they are connected in codimension-1. Therefore, any two pipe dreams for the same siteswap on the same patch must be connected by a sequence of moves. 




We conclude this section with a picture of what Π_del,i looks like on an open patch. Recall that this is the positroid variety which has siteswap (k+1)^k0k^n-k-1 rotated so the 0 is in the i slot. It is the largest positroid variety whose siteswap has a 0 in the i-th slot. It will be easy to see that the pipe dream must look like the following: 



Here is the proof. The siteswap is (k+1)^k0k^n-k-1. In the pipe dream above, the crosses are placed along the i-th slot, so that the only difference from the pipe dream with all elbows (producing the siteswap kk⋯ kk), is that the k crosses along that vertical line cause the k slots immediately prior to i to move one slot ahead of what they would be in the all-elbow pipe dream. This is precisely the (k+1)^k part immediately prior to the 0 in the siteswap. Thus, this pipe dream produces the siteswap we wanted, and in fact it is the unique such pipe dream since there are no moves in the pipe dream. 

Another way to prove this is that, by the proof above, not only must the column to be deleted lie along a horizontal segment, but the entire column must be filled with crosses so that it ends in the minimal spot (in this case: itself, given by the siteswap notation of 0 steps forward). Any additional crosses lowers the dimension of the variety. Π_del,i was defined as the maximal positroid variety with a 0 in the i-th slot, so it must be depicted by this pipe dream. 



The case of  Π_contr,i (which has siteswap (k-1)^n-kk^k-1n rotated so the n is in the i slot) is analogous, with the crosses proceeding from the vertical line segment labeled i in a horizontal line from the left to the right. 



The next question to be asked is then what the deletion or contraction of a positroid variety looks like. We know what Π_del,i and Π_contr,i look like, from what we have just done. Previously, we defined the deletion or contraction of a positroid variety Π_f as the intersection of Π_f with Π_del,i or Π_contr,i. So the question now is what this intersection looks like on an affine patch. We have the following proposition:



The pipe dreams of the i-th deletion of a positroid variety Π_f (that is, del_i(Π_f)=Π_f ∩Π_del,i) on the open patch U_λ are given by taking the pipe dreams for Π_f on U_λ with the maximal number of crosses along the ith column, and then filling in this ith column with crosses.

Here is a quick pictorial representation of Del_4(Π_f) (assuming the blank parts of the pipe dream are all elbows): 

 

That is, we take the pipe dream(s) with the largest number of crosses along that column, and fill in the rest of this column with crosses (denoted by the red crosses in this example here).


We can read out the new (deleted) affine permutation by following the pipes along this new pipe dream; however, note that the affine permutation del_i(f) is not dependent on which patch U_λ we draw the pipe dream on. Thus, if the goal was simply to compute del_i(f), rather than drawing the pipe dream (on a particular patch U_λ), we could do this more efficiently using an algorithm that we present in Appendix A.  

The case of contraction is analogous with the crosses filled in horizontally (in a row rather than a column of crosses).






We take Del_i(Π_f)=Π_f ∩Π_del,i and intersect everything with the vector space U_λ:


    Del_i(Π_f)∩ U_λ=(Π_f ∩ U_λ) ∩ (Π_del,i∩ U_λ).
 

In other words, we get the components of (Π_f ∩ U_λ) ∩ (Π_del,i∩ U_λ) by taking the components of the pipe dreams for Π_f and the components of the pipe dreams for Π_del,i and intersecting them and looking at the maximal element. Indeed, as we showed in the proof of Prop <ref>, the intersection Del_i(Π_f)=Π_f ∩Π_del,i is irreducible so is a single maximal positroid variety. We then intersect this positroid variety with U_λ, which is just looking at the pipe dreams for Del_i(Π_f) on the shape defined by λ. 

Furthermore, since we know that the maximal intersection will have the fewest number of crosses (since each cross represents a variable that is set to zero - thus an extra condition that lowers the dimension of the variety), and since we know that deletion involves filling in the column entirely with crosses, we must choose to intersect with the pipe dream(s) for (Π_f ∩ U_λ) that already had the greatest number of crosses along this column so that filling in this column involves adding the minimal number of additional crosses.



§ POSITROID VARIETIES AND SMOOTHNESS


We have already seen how to study positroid varieties on an open cover using affine pipe dreams. We have also seen how easy it is perform deletion and contraction on positroid varieties by using these affine pipe dreams. The main goal of this section is to prove Theorem <ref>, which will provide a means of studying smoothness or singularity of positroid varieties on this open cover using pipe dreams. 



One can check whether a positroid variety Π_f is singular (or smooth) just by testing singularity at the fixed points λ.


Fact (1): Borel's Fixed Point Theorem: If a solvable group (such as a torus) acts on some nonempty proper (such as projective) variety, then the fixed point set is also nonempty.



Fact (2): The torus T acts on the singular locus.

Proof: Whenever a group G acts on a variety, the singular locus will be G-invariant (i.e. a group of symmetries will leave the singular locus alone). 



Fact (3): Positroid varieties are proper. 
Positroid varieties are closed in Grassmannians which are closed in projective space (in the Plücker embedding).



Fact (4): The singular locus of any variety is closed. 




Suppose Π_f has singular point(s), i.e. the singular locus (Π_f)_sing is nonempty. Then we know that (Π_f)_sing is closed in Π_f by (4). Since Π_f is proper by (3) and (Π_f)_sing is closed in Π_f, we conclude that (Π_f)_sing is also proper. By (2), the torus acts on the singular locus, so we can apply (1) Borel's Fixed point Theorem to it, to conclude that the singular locus includes at least one fixed point (under the torus action).  

Denoting the fixed points of the torus action with superscript T, we summarize:

    (Π_f)_sing≠∅⇔ (Π_f)_sing^T ≠∅


In what follows, we will check these by checking on the affine open cover of the U_λ described earlier. Clearly, this will include all the torus-fixed points {λ}.





 §.§ Smoothness Theorem

A positroid variety Π_f is smooth at the point λ (it meets λ and is smooth there) if and only if there is a single affine pipe dream representative <cit.> of the affine permutation f on the open set U_λ. Specifically Π_f meets the point λ if and only if there exists (at least) one affine pipe dream, and it is smooth if and only if that affine pipe dream is unique. Thus, we can take Π_f and look at pipe dreams for all choices of λ to test for smoothness. 


We will spend most of the rest of this section proving this theorem. We begin by developing some of the background facts.

 
The multiplicityMult(x∈ X) is the degree of the tangent cone.

[<cit.> Exercise 5.3a] Mult(x∈ X)=1 if and only if X is regular (or smooth) at x. 



Thus, proving that “smoothness means one pipe dream" has become “multiplicity 1 means one pipe dream"; the next theorem shows that alternatively we have “coefficient 1 in equivariant cohomology means one pipe dream." For an introduction to equivariant cohomology, especially in this context, see for example sections 1.2, 2.1 of <cit.>. Note that the idea of using equivariant cohomological techniques to compute multiplicities of points was used in for example <cit.>.

<cit.>
Let V be a vector space, and X ⊆ V be a cone, i.e. a subvariety that is invariant under the dilation action of ^× on V (that is, ^× acts with all weights 1 on V). Denote the equivariant cohomology H_^×^*(V) by [h]. Then the multiplicity of X at 0∈ V is the coefficient of h^ X: 

    [X]_^× = ( X)h^ X∈ H^*_^×(V)


Remark: We use the notation [X]_^× to disambiguate from [X]_T ∈ H^*_T(V), which we will see later. 

In our case, this means that we want to show that [Π_f ∩ U_λ] = 1 · h^codim X∈ H_^×^*(U_λ) is equivalent to Π_f having a unique pipe dream on U_λ. More generally, we will show that the multiplicity counts the number of affine pipe dreams. 

The proof goes by showing the following sequence of equalities:


    (*) [Π_f ∩ U_λ⊆ U_λ] (1)= [X_v(f)∩ X_∘^v(λ)⊆ X_∘^v(λ)] (2)= [X_v(f)]|_v(λ)(3)=∑_R ∏_r∈ Rβ̂_̂r̂∈ H_T^*(U_λ) ≅[y_1,...,y_n]


The following subsections will individually go through equalities (1), (2), and (3) in this equation labeled (*), as well as a last part (4) which maps from the T-equivariant cohomology to a circle S within the torus T.

Note on notation: f is a siteswap and v(f) its corresponding bounded affine permutation, which indexes a T-fixed point on the affine flag variety GL_n/B (note: GL_n denotes n× n invertible matrices where the entries are Laurent series in one variable, and this is infinite-dimensional). Then X_∘^v(λ) is the B-orbit through the point v(λ)B/B; X_∘^v(λ) := B v(λ) B/B. And  X_v(f) is the B_--orbit closure through v(f); X_v(f) := B_- v(f) B/B. 







  §.§.§ Equation (*) Part (1)


Equality (1) follows from the work of <cit.> Section 4.2, which says we have a T-equivariant commutative diagram:


U_λ[r,leftarrow,"≅"]      X_∘^v(λ)
Π_f ∩ U_λ[r,leftarrow,"≅"] [u,hookrightarrow]     X_v(f)∩ X_∘^v(λ)[u,hookrightarrow] 


On the level of T-equivariant cohomology, this gives rise to:

g^*: H_T^*(U_λ)   ≅   H_T^*(X_∘^v(λ))
[origin=c]90∈      [origin=c]90∈
[Π_f ∩ U_λ ]   ↦   [X_v(f)∩ X_∘^v(λ) ]

Since U_λ and X_∘^v(λ) are contractible, we have canonical isomorphisms: H_T^*(U_λ) ≅ H_T^*(pt) ≅ H_T^*(X_∘^v(λ)). 
    H_T^*(pt) [dl,leftrightarrow,"≅"'] [dr, leftrightarrow,"≅"]    
 H_T^*(U_λ)  [rr,"g^*"]        H_T^*(X_∘^v(λ))  
Using these isomorphisms to identify each with H_T^*(pt), the polynomials [Π_f ∩ U_λ] and [X_v(f)∩ X_∘^v(λ) ] are equal. 

This completes the justification of equality (1). Next we prove equality (2): [X_v(f)∩ X_∘^v(λ)⊆ X_∘^v(λ)] = [X_v(f)]|_v(λ).  




  §.§.§ Equation (*) Part (2)


The right side of this equality is the restriction of the class [X_v(f)] to the point v(λ)B/B (in the following, we will denote this as simply v or v(λ)). Naively, we might consider this via the diagram: 

    X_v(f)[d, hookrightarrow] 

v  [r, hookrightarrow, "ι"]      G/B 


We factor this inclusion as follows:

    X_v(f)∩ X_∘^v(λ)[d, hookrightarrow]  [r]     X_v(f)[d, hookrightarrow] 

v  [r, hookrightarrow]     X_∘^v(λ)[r, hookrightarrow]      G/B 


The bottom row induces maps in cohomology:

    H^*_T(v)  ←  H^*_T(X_∘^v(λ))  ← H^*_T(G/B)

The goal is to show that under this induced map:

    [X_v(f)] |_v  [X_v(f)∩ X_∘^v(λ)]   [X_v(f)]


We know that the composition of the two maps will take [X_v(f)] to [X_v(f)] |_v because cohomology is functorial so the pullback on cohomology of the composite map v → X_∘^v(λ)→ G/B is the composite of the pullback; in other words, once we know where v maps to, given by the map v↪ G/B, this determines the map H^*_T(G/B) → H^*_T(v) regardless of how we might factor the map in the middle, and this defines [X_v(f)] |_v as the image of this map. 

The harder thing is we need to show that this map hits [X_v(f)∩ X_∘^v(λ)] in the factorization through the middle space X_∘^v(λ). This will follow from transversality of X_v(f) and X^v(λ)_∘<cit.>. We can apply Kleiman transversality because X_∘^v(λ) is a B-orbit, and and X_v(f) is B_- invariant, so then we conclude that under the map g: X_∘^v ↪ G/B going to g^*: H_T^*(G/B) → H_T^*(X_∘^v), g^*([X_v(f) ]) = [X_v(f)∩ X_∘^v(λ) ].

Finally, to prove equality (2), we need to show that  [X_v(f)∩ X_∘^v(λ)] doesn't just map to [X_v(f)] |_v, but that they are equal: we could see this already from the discussion preceding the triangle commutative diagram above, since X_∘^v(λ) is equivariantly contractible to v, so H_T^*(v) ≅ H_T^*(X_∘^v(λ)). 

This completes the proof of equality (2). 




  §.§.§ Equation (*) Part (3)

 
Equality (3) is the AJS/Billey formula <cit.>, which gives the restriction of a Schubert class to a fixed point. We state it here:


Let Q be a word in the simple roots of a Kac-Moody group (see <cit.> Prop 11.1.11 for the Kac-Moody case). We define β_i as follows:

    β_i := (∏_j=1^i-1 r_Q(j))· (x_Q(i) - x_Q(i)+1) =  r_Q(1)⋯ r_Q(i-1)· (x_Q(i) - x_Q(i)+1), i∈ [m]

Let ∏ Q be a word in the letters Q(i) so ∏ Q is an element of a Coxeter group (in our case, this will be a product of transpositions representing a bounded affine permutation), and [X_w]|_∏ Q denote the restriction of the Schubert class [X_w] to the point in the (affine) flag variety represented by ∏ Q. Then the AJS/Billey formula says:


    [X_w]|_Π Q= ∑_R⊆ Q 
     R reduced∏_r∈ Rβ̂_̂r̂



Our work so far has proven:

    [Π_f ∩ U_λ]_T = ∑_R⊆ Q 
     R reduced∏_r∈ Rβ̂_̂r̂

We state a proposition for what the right hand side looks like.


    [Π_f ∩ U_λ]_T = ∑_R⊆ v(λ) 
    ∏ R=v(f) 
     R  reduced∏_r∈ R  [(y_i)_r - (y_j)_r]


The AJS/Billey formula produces a sum of products of (y_i-y_j): the terms multiplied together to form a product are indexed by r∈ R, and each r corresponds to a cross at (j,i) in a pipe dream representative for the given positroid variety on the given patch. Specifically, the i corresponds to the column and the j to the row in the pipe dream. The labelling is given by the numbering of the distinguished path explained in Appendix B; thus, the rows j will come from the vertical boundary line segments, which correspond to λ, where the columns i will be from the horizontal boundary line segments found in [n]\λ. 


First, we note that we can think about the letters or transpositions in the affine pipe dream shape as SW to NE diagonals where all the letters are the same. See the following figure:



The important thing to note is that each s_i commutes with all other s_j except those in its neighboring adjacent diagonals s_i-1 and s_i+1. This implies that the invariant information of a word can be depicted in a heap (see e.g. <cit.> for details), since a heap diagram only keeps track of the information about the order in which adjacent s_i's are read. Any two words with the same heap will give the same permutation (c.f. for example <cit.>). 

The natural heap our affine pipe dream shapes provide can be obtained by rotating the pipe dream 45 degrees counterclockwise (c.f. the “bottom pipe dream" of Appendix B): 



There is a question of how to read the letters to form a word. This heap corresponds to a southwest (denoted SW for short) rule for reading, where we always only choose a (letter in a) square s_i if all squares southwest have already been chosen. In other words, regardless of how we choose to read the letters within the affine pipe dream, any choices of order that satisfy the SW rule will all be equivalent because they have the same heap (they will be related by commuting moves, since all non-commuting moves will be the same, being determined by the SW rule).




Thus, in order to prove the proposition, we just need to show 2 things: (1) that the way of reading the word in <cit.>, which is the way that we read it in this paper, satisfies the SW rule, and (2) we can choose an order satisfying the SW rules that makes it apparent that when we calculate the root β_i=∏_j=1^i-1 r_Q(j) (x_Q(i) - x_Q(i)+1), it equals x_row-x_col. 

For (1), it will be easiest to follow the argument if an example is presented. We give an example in Gr(4,10): (After this proof is completed, we continue this example in more detail.) 



Thus, the word would be: Q(a)Q(b)Q(c)Q(d)Q(e)Q(f)Q(g)Q(h)Q(i)Q(j)Q(k)Q(l)Q(m)Q(n)Q(o)Q(p)Q(q)Q(r)Q(s)Q(t)Q(u)Q(v)Q(w)Q(x).

The way of reading the word in <cit.>, which we use in this paper, satisfies the SW rule because we start in the left corner or the lowest row, which has a boundary to its south. We always read left to right in a row, with all letters to the south already chosen since we read starting from lower rows. This proves that we satisfy the SW rule.  

(2) Thus, our work in part (1) shows that β_i=∏_j=1^i-1 r_Q(j) (x_Q(i) - x_Q(i)+1) equals any product ∏ r_Q(l) (x_Q(i) - x_Q(i)+1) where ∏ r_Q(l) satisfies the SW rule. 

We make one observation, which is the affine pipe dream diagram has width k for any row, and height n-k for any column, so this is in particular the maximum width and height if we go west and south from any square in the pipe dream:



This means that if we go left from s_i, the leftmost square will be a maximum of k-1 steps left, and if we go south, the southernmost square will be a maximum n-k-1 steps down, so the entire region between where it hits the left boundary and south boundary (highlighted in yellow in the previous figure) will have no overlaps, and will contain n-1 diagonals s_j n, s_j+1 n,...,s_j-2 n. 
 
Now, for any square x, we consider just the word for which x is the very northeastern corner, and otherwise, the word contains the squares within the area to its southwest, e.g.:



We read the word start from the lowest row, row by row upwards, and left to right within each row, as depicted a,b,c,d,... in the example figure above. This satisfies the SW rule. This implies that β_i=∏_j=1^i-1 r_Q(j) (x_Q(i) - x_Q(i)+1) has the form: 

    β_i    = ⋯r_Q(i)+m⋯r_Q(i)+2⋯r_Q(i)+1r_Q(i)-l⋯ r_Q(i)-2  r_Q(i)-1(x_Q(i) - x_Q(i)+1) 
       = ⋯r_Q(i)+m⋯r_Q(i)+2⋯r_Q(i)+1r_Q(i)-l⋯ r_Q(i)-2(x_Q(i)-1 - x_Q(i)+1) 
       = ⋯r_Q(i)+m⋯r_Q(i)+2⋯r_Q(i)+1 (x_row - x_Q(i)+1) 
       = ⋯r_Q(i)+m⋯r_Q(i)+2⋯  (x_row - x_Q(i)+2) 
       = x_row - x_col


Note that the red reflections correspond to the row highlighted red in the figure above, and the green reflections to the column highlighted green in the figure above. Given the way the word is read, the red reflections will act first, bringing x_Q(i) all the way to x_row, where row is the label on the vertical line segment to the very left of x. For the remaining reflections, only the green ones will cause any change, and they will act by bringing x_Q(i)+1 all the way to x_col, where col is the label on the horizontal line segment to the very south of x. By the observation above, each diagonal of s_i's only occurs at most once in this region, so there will be no interference between the letters in the region highlighted red and the letters in the region highlighted green.
 

We continue the example in Gr(4,10) that we introduced in the proof above. Below, we will calculate β_i=∏_j=1^i-1 r_Q(j) (x_Q(i) - x_Q(i)+1), showing that it equals x_row-x_col. 
For this example, let us choose Q(i) to be the letter circled in green, where we have highlighted in orange ∏_j=1^i-1 r_Q(j) = r_Q(1) r_Q(2) r_Q(3)⋯ r_Q(i-1): 



We obtain:

    β_i    = r_Q(1) r_Q(2) r_Q(3)⋯ r_Q(i-1) (x_Q(i) - x_Q(i)+1) 
       = r_Q(1) r_Q(2) r_Q(3) r_Q(4) r_Q(5) r_Q(6) (x_Q(7) - x_Q(7)+1) 
       = r_6 r_7 r_8 r_3 r_4 r_5 (x_6 - x_7) 
       = r_6 r_7 r_8 r_3 r_4  (x_5 - x_7) 
       = r_6 r_7 r_8 r_3  (x_4 - x_7) 
       = r_6 r_7 r_8 (x_3 - x_7) 
       = r_6 r_7  (x_3 - x_7) 
       = r_6 (x_3 - x_8) 
       = x_3 - x_8


and this is precisely x_row-x_col if we draw lines west and south in the figure, as we wanted to show.


For the second example of Q(i) below the distinguished path, given the way the word is read in <cit.>, it is more natural to consider the lower section being placed above the distinguished path to form the word we have highlighted in orange below:



Here, we would have: 

    β_i    = r_Q(1) r_Q(2) r_Q(3)⋯ r_Q(p-1) r_n-k r_n r_Q(p+2)⋯ r_Q(i-1) (x_Q(i) - x_Q(i)+1) 
       = r_6 r_7 r_8 r_3 ⋯ r_7 r_1 ⋯ r_6 r_10 r_1 ⋯ r_5 r_9 (x_10 - x_1) 
       = r_6 r_7 r_8 r_3 ⋯ r_7 r_1 ⋯ r_6 r_10 r_1 ⋯ r_5 (x_9 - x_1) 
       = r_6 r_7 r_8 r_3 ⋯ r_7 r_1 ⋯ r_6 (x_9 - x_2) 
       = r_6 r_7 r_8 r_3 ⋯ r_7 (x_9 - x_3) 
       = r_6 r_7 r_8 (x_9 - x_4) 
       = x_6 - x_4

and this also equals x_row-x_col, as we wanted to show.




We just need to show one more thing. Note that the AJS/Billey formula produces an element of the n-dimensional torus T-equivariant cohomology H_T^*(V)≅[y_1,...,y_n]. However, the theorem from <cit.> involves H_^×^*(V)≅[h]. Thus, our final step will be to produce a map H_T^*(V) → H_^×^*(V). We will show that this map takes the rightmost term 
∑∏_r∈ Rβ̂_̂r̂ and maps it to (# pipe dreams on  U_λ)· h ∈ H_^×^*(V)≅[h]. Then the proof will be complete. 




  §.§.§ Equation (*) Part (4)


Finally, it remains to be shown, under the map from H_T^*(U_λ) to H_^×^*(U_λ), that the right side of the AJS/Billey formula corresponds to the number of pipe dreams.

Recall that we showed above the isomorphisms H_T^*(U_λ) ≅ H_T^*(pt) ≅ H_T^*(X_∘^v(λ)). However, to use Rossmann's Theorem <ref>, we need to have H_T^*(V) for a vector space V. We will get this via an isomorphism of U_λ with the vector space M_k,n-k, the space of k-by-(n-k) matrices. This is depicted in the following diagram, which shows M_k,n-k being identified with another space of matrices with the identity in columns λ, which we can call (M_λ)_k,n-k; the row span of (M_λ)_k,n-k is equal to U_λ⊆ Gr(k,n). 


M_k,n-k[r,"∼"]    {(   1   0   
   0   1   *
   ⋮   ⋮   ) }[rr,"∼"]  [drr,shorten >=3ex,"row span"']        U_λ↶ T ↩^×[d,hookrightarrow, shift right=5ex]  
[ v_1⋯ v_n-k] [u,""] [r,mapsto]    (      1   0   
 v_1     v_2    0   1   [u,""] 
      ⋮   ⋮   )        Gr(k,n)


We also recall that the right side of AJS/Billey is: ∑_R⊆ Q, R reduced∏_r∈ Rβ̂_̂r̂, where Q corresponds to the word representing v(λ), and then the sum is over all R, which are words representing the siteswap f. In other words, there is precisely one summand in this formula for each pipe dream representative of Π_f ∩ U_λ. We will show that our choice of map H_T^*(V) → H_^×^*(V) takes each of these summands to 1· h^codimX∈ H_^×^*(V)≅[h], so the result will be (# pipe dreams of Π_f  on  U_λ)· h^codimX. Then we will be done. 


The map H_T^*(V) to H_^×^*(V) is entirely determined by the map ^×→ T. We describe the relevant maps explicitly in a moment, but first we give the general theory. 

Suppose we have a map of tori S→ T. Then we get the corresponding Lie algebra map s→t as well as the dual map s^* ←t^*. Then we have the following commutative diagram:

t^* [d,hookrightarrow]  [r]    s^* [d,hookrightarrow] 

Sym(t^*) [d,leftrightarrow,"≅"] [r]      Sym(s^*) [d,leftrightarrow,"≅"]

H_T^* [r]     H_S^*


Note that if we pick a basis for t^*, then Sym(t^*) becomes a polynomial algebra, and these basis elements are degree-2 generators (the cohomology is in even degree). Also note that H_T^* := H_T^*(pt) = H_T^*(V) since a vector space V is contractible. Recall that an application of Theorem <ref> required that ^× acts with all weights 1. 


We satisfy the condition of the circle ^×⊆ T acting on U_λ with all weights 1 due to our choice of map ^×→ T, which is: 
    z↦ (z,1,1,z,...,1)
 with the entries z in the coordinates of [n]\λ.

Suppose we are in Gr(2,4) with λ={1,3}. Then we can represent U_λ, and the action of ^× on the right as:

    (1     *     0     * 
     0     *     1     *) (1,z,1,z).

This scales all the *'s, which give coordinates of U_λ≅𝔸^k(n-k)=𝔸^4, by z^1. Thus, ^× acts via dilation on U_λ, as required.

Alternatively, we can consider the full action of T on U_λ and find the action of ^× inside T using:

    (z_1^-1,z_3^-1) (1     *     0     * 
     0     *     1     *) (z_1,z_2,z_3,z_4)

    = (z_1^-1,z_3^-1) (z_1     z_2*     0     z_4* 
     0     z_2*     z_3     z_4*) = (1    z_2/z_1*     0    z_4/z_1* 
     0    z_2/z_3*     1    z_4/z_3*).


We see that in order to maintain our matrix representative in the form (1     *     0     * 
 0     *     1     *) we must “unscale" on the left, which results in the coordinates having factors z_col/z_row, where the number row refers to: within this number's row, find where the 1 is, and record the column of this 1.

Note that for a general embedding of ^× in T acting with arbitrary weight, we have z_i=z^y_i. Then z_col/z_row= z^y_col-y_row. Since we want the action to be by dilation, we need all these y_col-y_row=1. We just saw that we could do this via the map z↦ diag(z,1,1,z,...,1)=diag(z^1,z^0,z^0,z^1,...,z^0) with the z's in the non-λ spots. In other words, by setting 

    y_i=  0     i∈λ
     1     i∉λ,

then y_col-y_row=1-0=1 since the col corresponds to the columns not in λ and the row correspond to the unit vectors in the matrix which are where λ is. 


If the T-action on the tangent spaces to the fixed points on a flag manifold G/P contains dilation, G/P is called cominuscule. We have just recovered the fact that Grassmannians are cominuscule (something not true for full flag manifolds).


Let's now write the remaining maps explicitly.
^×→ T, z↦ diag(1,z,z,1,....,1,z) where the z's exist precisely in the places in the diagonal where λ is not gives us:
s→t, 1 ↦ [1,0,0,1,...,0] the map of Lie algebras, where we have just taken the derivative of the map above and extracted the diagonal.


    s^* ←t^* : (1
    0
    0
    1
    ⋮
     0)


Finally, once we go to Sym(t^*) → Sym(s^*) this is just a map of polynomial rings, where this map (1     0     0     1   ⋯    0)^T sends all variables y_i in Sym(t^*) corresponding to the numbers in λ to 0, and all other variables to h. 

Recall Proposition <ref>, which noted that 

    [Π_f ∩ U_λ]_T = ∑_R⊆ v(λ) 
    ∏ R=v(f) 
     R  reduced∏_r∈ R  [(y_i)_r - (y_j)_r]
 where i is the column and j is the row of a cross in a pipe dream. 


Now, we can apply to this our choice of map λ̃:^×→ T sets all y_row=0, y_col=h∈[h], so since each summand in the AJS/Billey formula is a product of codim(Π_f) many β's, each summand will be 1· h^codim Π_f. As we already showed above, there is a single summand for each pipe dream representative of Π_f on U_λ, so the AJS/Billey formula is equal to (# pipe dreams of Π_f  on  U_λ)· h^codim Π_f. This completes the proof of the uniqueness (smoothness) part of Theorem <ref>.

Finally for the proof of the existence part, that λ∈Π_f is equivalent to there existing at least one affine pipe dream, this follows from the fact that each affine pipe dream for Π_f on U_λ is a subword for Π_f in the word for λ in accordance with what we have already described in Section <ref>. In addition, it is a general fact of words in Bruhat order that a point λ being contained in a Schubert variety defined by f is equivalent to the word v(λ) > v(f) in Bruhat order, and furthermore, this is equivalent to v(f) being a subword in a reduced for for v(λ). 






 §.§ The Ordering and Smoothness


We presented the following figure in the introduction. In the ordering (given by deletion and contraction) of positroid varieties and points, the smooth versus singular pairs can be divided in the following way:

 

In other words, smooth positroid varieties remain smooth under deletion and contraction, while singular positroid varieties if obtained from another positroid variety via deletion or contraction must have come from a singular positroid variety. We can state this in the following proposition:


If λ is smooth on Π_f, then del_i(λ) is smooth on Π_del_i(f) and contr_i(λ) is smooth on Π_contr_i(f). Or this can be stated contrapositively: if del_i(λ) is singular on Π_del_i(f) or contr_i(λ) is singular on Π_contr_i(f), then λ is singular on Π_f.


By e.g. <cit.>, if the group G is reductive, then whenever G acts on a regular set X, the fixed point set X^G is also regular. In our case, G=T is the torus, so the conclusion follows. 

In particular, if Π_f is smooth at p, then Π_f^^× is smooth at p, and Π_f^^×=(Π_f∩Π_del,i) ∪ (Π_f∩Π_contr,i) = Del_i(Π_f) ∪ Contr_i(Π_f). 


We can give a combinatorial proof of this proposition using the machinery of affine pipe dreams:



Without loss of generality, let us do deletion on the ith column in a rigid pipe dream (the case of contraction will be analogous but for a row). 

The column will alternate between consecutive sets of crosses and consecutive sets of elbows. It will be helpful for the proof to look at the following figure which zooms in on the ith column, where the consecutive sets, for simplicity, have been drawn as groups of 3:

 

Note that the colors on the two pipe dreams in the figure are the same; for each one, reading from top to bottom, the left side (of column i) goes: light red - light orange - light green - light blue - light purple; and on the right side: dark purple - dark blue - dark green - dark orange - dark red. The fact that we have flipped the order of the coloring (while doing dark instead of light to distinguish them) is intentional: the pipe dream has a northwest-southeast symmetry, coming from the fact that the pipes (and the moves) go southwest to northeast. Thus, it is sufficient to make arguments on the just left side (light) colors, so for ease of reference, we call the left/light colors just: red, orange, green, blue, purple; whereas if we need to distinguish the colors on the right side, we will call them: dark purple, dark blue, cyan (or dark green), dark orange, dark red. The green to cyan pipes in the middle are where we will focus. After all, only the elbow tiles are changed in the deletion. 

Therefore, what we are doing here is picking some random block of consecutive elbow tiles in the column, and showing that, even though those tiles are changed (turned into crosses), no new moves can be created. If we can show this for the green to cyan tiles in this diagram, then this applies to any other elbow tile in this column that gets deleted, and the proof will be complete.

If we need to further distinguish the pipes within each block, we can call them, for example: red 1, red 2, red 3, going from top to bottom so that red 1 is the highest. 

We first make several observations. 

(1) There can be no crossing among any of the green pipes anywhere because they are elbows/near-misses along the ith column; any crossing would then provide a move (specifically, the cross could then be moved to the ith column), contradicting rigidity. For the same reason, there can be no crossing among the cyan pipes. (1a) Note, however, that the orange pipes have no such restriction among themselves.

(2) The green pipes cannot cross any of the orange pipes. The reason is that, by (1), in order for any of the green pipes to cross any of the orange pipes, green 1 would have to cross (first). However, since green 1 crosses all of the orange pipes along column i, this would create a double crossing. 

(3) The green pipes cannot cross any of the red pipes. The reason is that, by (1), in order for any of the green pipes to cross any of the red pipes, green 1 would have to cross red 3 (first). However, since green 1 and red 3 have a near miss, this would create a move, contradicting rigidity. The same argument show that the purple pipes cannot cross any of the green pipes. 

(4) The green pipes can cross the blue pipes. 

(5) By symmetry, all these points are valid for the right side of column i as well: just replace “color" by “dark color."

We can recap all of this succinctly. The only crossings among pipes that are allowed are: (I) within each of block pipes coming from consecutive crosses along the ith column, and (II) between a block of consecutive elbows and the block of consecutive crosses directly below them (on the left side of column i; or, on the right side, by symmetry, between consecutive elbows and consecutive crosses directly above)

We now show that doing the deletion does not create any new moves. 

The deletion only changes the connectivity of the elbow tiles. In other words, the orange and blue pipes do not change, so there cannot be any new moves coming from the orange or blue pipes. Let us focus in on the green pipes, and ask the question of whether there can be any new moves coming from the change in the connectivity of the green pipes.

A move consists of 2 pipes having a cross and a near-miss. Obviously, many new crosses are created along column i; however, these crosses cannot participate in any move because the ith column pipe runs straight up and down vertically, crossing every pipe along the ith column, and thus has no near-misses. Therefore, a new move must come from a crossing of 2 pipes on the left side of column i, for which these 2 same pipes have a near-miss on the right side of column i (or vice-versa switching left and right, but it is sufficient to just consider the former case). Since this is a new move, it did not exist before.  The cross and near-miss cannot have both been on the same side (left or right); otherwise, this move would have existed prior to the deletion and thus not only not be a new move, but the original pipe dream would have had a move, contradicting rigidity.

Thus, the question now becomes, (a) what possible crosses involving the green pipes on the left are there, and (b) what near-misses involving these pipes are there on the right. The work we did above now comes in handy. We know that the green pipes cannot cross among themselves by (1), and in fact cannot cross any pipes except the blue ones by (II). This answers (a). Now we consider the question of (b). Since the green pipes can only potentially cross the blue pipes, the question is whether it is possible for a potential green-blue crossing on the left, after we do the deletion which causes these pipes to turn into dark green - dark orange pipes on the right, to involve a dark green - dark orange near-miss on the right.

This is not possible for the following reason. Pipe dark green 3 crosses all three of the dark orange pipes. Since the pipe dream is rigid, that means that dark green 3 cannot have a near miss with any of the dark orange pipes. By (1) above, none of the dark green pipes can cross each other, which mean that none of the dark green pipes can have a near miss with any of the dark orange pipes. This completes the proof.



However, our work above allows us to say something further about this last point. In order for dark green 3 to not have a near miss with any of the dark orange pipes, we must have another color pipe come between the dark green and dark orange pipes. We saw above in (4) or (I) that, on the left, the elbow pipes could cross with the cross pipes in the block below, which, on the right, corresponds to elbow pipes crossing with the cross pipes above. This means that one, or any (since the dark blue can cross among themselves by (1a)), of the dark blue pipes can come down by crossing the dark green pipes to create separation from the dark orange pipes; or, dark red pipe 1 would have to come up (the other dark red could also come up, but dark red pipe 1 would have to come up first by (1)).

Thus, if there were no moves prior to the deletion, then there are no moves after the deletion. The contrapositive statement is that a singular positroid variety Π_f' coming from another positroid variety Π_f via deletion or contraction implies that Π_f is itself singular, since a smooth positroid variety being deleted to a singular positroid variety is impossible.

Finally, we note here that we can give another name to going down in this ordering, a concept which can also be applied to (affine) pipe dreams.


(1) We say that (Π_f_1,λ_1)d/c-reduces (for deletion/contraction) to (Π_f_2,λ_2) if a series of deletions and contractions of (Π_f_1,λ_1) results in (Π_f_2,λ_2).

(2) The notion of “d/c-reduces" in particular applies to (affine) pipe dreams: Let's denote the set of affine pipe dreams corresponding to (Π_f_1,λ_1) by: PD_(Π_f,λ). This becomes a bijection if we consider the elements in (Π_f_1,λ_1) up to equivalence by the “moves" defined in Definition <ref>. Furthermore, we have a notion of deletion and contraction for individual pipe dreams, that is, applying deletion and contraction to elements δ∈ PD_(Π_f,λ) (see the end of Xection <ref>).

Thus, we can consider, more specifically, whether a given δ_1 ∈ PD_(Π_f_1,λ_1) goes to δ_2 ∈ PD_(Π_f_2,λ_2) under a series of deletions and contractions on affine pipe dreams described in Section <ref>. Obviously, if we consider these affine pipe dreams only up to equivalence under moves, then the relations will be the same as (1) above. However, since it will be easier to work with individual δ∈ PD_(Π_f,λ) in many of the following proofs, in the rest of this paper when we use the term “d/c-reduces," we will use it to refer to this latter notion (2). 


§ PROOF OF MAIN THEOREM


Recall the following figure from Section <ref>, which displays an example of maximal rectangles. Maximal rectangles are defined simply as rectangles within the pipe dream whose southwest and northeast are also corners of the pipe dream shape.


A pipe dream in a rectangular shape reduces to SE/NW partitions if, after deleting all entire rows and columns of crosses, the result is a partition in the southeast and northwest corners. 

This is equivalent to the condition that all crosses must be contained in: (1) entire rows of crosses, (2) entire columns of crosses, (3) a partition in the northwest, (4) a partition in the southeast.

The following is an example of a rectangular pipe dream that reduces to SE/NW partitions:
  [Main Theorem]
Let δ∈ PD_(Π_f,λ) be an affine pipe dream for Π_f ∩ U_λ. If each maximal rectangle in δ reduces to NW/SE partitions, then Π_f is smooth at λ; otherwise, it is singular. 


This theorem consists of showing two parts:
 
(1) Showing that smoothness can be determined just by looking at the maximal rectangles in an affine pipe dream. This is proven in Proposition <ref> below. 

(2) Showing that the thing that needs to be looked at on each maximal rectangle is that each maximal rectangle is a NW/SE partition. This is proven in Theorem <ref> below. 

However, first we give the full proof, referencing both Proposition <ref> and Theorem <ref> as needed. 



We want to show that smoothness of (Π_f,λ) is equivalent to δ having no moves in any maximal rectangle. But we already have Theorem <ref>, which says that smoothness of (Π_f,λ) is equivalent to its pipe dream having no moves (anywhere), and, by Proposition <ref>, all moves exist within maximal rectangles. Thus, if all maximal rectangles have no moves, then the pipe dream for (Π_f,λ) has no moves. The other direction, that the pipe dream for (Π_f,λ) having no moves implies that all its maximal rectangles have no moves, is automatic. 

Therefore, smoothness of (Π_f,λ) is equivalent to δ having no moves in any maximal rectangle.
Thus, we just have to show that rigid rectangles reduce to NW/SE partitions and the proof will be complete, and this is shown in Theorem <ref>.

Every cross-elbow move exists within some maximal rectangle. More precisely, every pair consisting of a cross and a near-miss where this cross can move is contained within a maximal rectangle. 

Trivially, this implies that the portions of the two pipes involved in the move, starting from the cross and running all the way to the near-miss, are entirely contained in this maximal rectangle.

In fact, the proof below shows the stronger statement that if we consider a cross and the set of all places where it can move, these will all be contained in a maximal rectangle.  

The very notion itself of nonrigidity is that there is a cross to the southwest and a near-miss of the pipes in that cross somewhere to the northeast (or vice-versa with southwest and northeast switched). In the sorts of shapes given by affine pipe dreams we are considering, such a cross and then near-miss combination must exist within a full rectangle because all the jagged (or convex) parts of the shape are in the northeast and southwest.

This is depicted in the red rectangle in the following figure: 


    < g r a p h i c s >





Next, in order to prove Theorem <ref>, we will first need the following proposition, which itself requires a lemma:


In a reduced pipe dream, any time there is an elbow followed by m consecutive crosses to its east and n to its south,  then the n× m box formed by having the easternmost and southernmost crosses as corners must be entirely filled in with crosses. The same is true with “east and south" replaced by “west and north." That is: 

 
The following figure provides the proof of this lemma.
 
Consider the red dot in the first figure on the very left. It cannot be an elbow, or else the two green dots would be a double crossing (non-reduced); so the red dot must be a cross. So once we have filled that tile in with a cross, move the red dot to the right, where it again must be a cross, by a similar argument. We continue inductively until the entire first row is filled in with crosses. Then we move on to the 2nd row, and apply the same argument to fill it in with crosses. We continue on, row by row, until the n× m rectangle is filled in with crosses. 

A rectangular pipe dream is rigid if and only if: (1) no cross has both an elbow to its north as well as to its east, and (2) no cross has both an elbow to its south as well as to its west.



This is essentially a “double-Le condition," and the proof is similar to statements in Section 19 of <cit.> or Section 4.3.3 of <cit.>. As hinted in <cit.>, one of these conditions basically corresponds to the bottom pipe dream (see Appendix B), so the other would correspond to the top pipe dream. 





First we prove that if a pipe dream satisfies this double-Le condition, then it is rigid. We show the contrapositive, that if a pipe dream is not rigid, then it must violate this condition. The proof is essentially the following picture:



Let's assume that this pipe dream is not rigid. This means that, as shown at the top of the figure, there exists two pipes that cross somewhere and later have a near-miss, either to the northeast of the cross as in the figure, or to the southwest (not depicted, but analogous). This requires the two pipes to each have a “first bend," circled in red in the representative example bottom left figure, because if they never bent, then they would continue on indefinitely north and east and never cross. Thus, a pipe dream with a move either contains a cross with elbows both to its north and east (as shown in this figure), thus violating the north/east Le condition, or it has a move with the figure reflected across the x=-y line; that is, with a violation of the south/west Le condition. 

.7712mm Now we prove the other direction: if a pipe dream violates this double-Le condition, then it must not be rigid. Without loss of generality, let's again assume that it violates the north/east Le condition (the south/west case is analogous). This means that there is a cross with elbows somewhere to its north and east, and we can then assume the elbows to be the closest in these directions. That is, we have:
.200mm
    < g r a p h i c s >
 .05in


.7712mm Furthermore, once they bend at an elbow, they cannot keep moving along straight lines, as depicted in the dotted lines in the following figure, or else they would cross again, which is violation of reducedness:
.200mm
    < g r a p h i c s >
 .05in


.7712mmBecause of this, at least one of the pipes must have another bend. In fact, we must have each pipe bending a 2nd time because otherwise we would be able to apply Lemma <ref> to this rectangular region of interest to arrive at a nonreduced pipe dream, like the following figure: 
.200mm
    < g r a p h i c s >
 .05in



.7712mm Thus, putting in these additional bends, and filling in the parts to the west and south of the bends with crosses via an application of Lemma <ref>), the pipe dream looks something like:
.200mm
    < g r a p h i c s >
 .05in

In other words, by imposing a violation of the double Le condition and reducedness (simply having a valid pipe dream), we have concluded that we have two pipes that cross, that each bend, and then bend again, and everything to the southwest of the rectangle formed by the places where their second bends are must be filled in with crosses.

.7712mm Next, we apply the assumption of rigidity we made for the sake of contradiction. In the following figure, if the elbow on the right were to bend inwards, then we would have the red elbow depicted, and there would be the possible move depicted by the two red dots, which is not allowed by rigidity:
.200mm
    < g r a p h i c s >
 .05in


.7712mm So we must have the green cross below. But then, inductively, we can make the same argument for the spot next to it, for if it was an elbow, then we would have the possible move with the two red dots:
.200mm
    < g r a p h i c s >
 .05in


.7712mm Thus, it is clear that by this argument, we must fill in everything along the same lines as the elbows with crosses, to get the following:
.200mm
    < g r a p h i c s >
 .05in



.7712mm If we look at the red dots, we see that the configuration that we started with (with two northwest southeast elbows and a cross at one of the corners) is here, only smaller. We can keep doing this inductively, but due to the finite size, we cannot go on forever. We must reach a point where we have a move, giving us the contradiction:
.200mm
    < g r a p h i c s >

If a pipe dream is rigid within a rectangular shape, then every time there is a cross with an elbow to its south, then it must have crosses extending all the way west. We can visualize it as follows:

 

The same statement is true with “west" and “south" reversed, as well as replaced with “north" and “east". 



We note that the previous Prop <ref> was only valid within rectangular shapes. In nonrectangular shapes, a violation of the Le condition is not sufficient to conclude nonrigidity. A counterexample is provided by the following figure, where the two red dots represent a Le-violating condition, yet the portion of this pipe dream depicted is perfectly rigid. 
 

The following corollary, however, gives a condition for rigidity in any pipe dream shape (not just rectangular ones).

 [of the proof]

An affine pipe dream is rigid if and only the following two types of rectangular pipe crossings do not appear anywhere:

 
Note that the arguments above about reducedness above imply that the inside of these rectangles is filled in completely with crosses. 
 Remark 1. This corollary does not state that in a nonrigid pipe dream, that every nonrigid pair of pipes must be of this form. In fact, two pipes that cross and bend inwards at some point before having a near-miss is indeed possible; however, even in this case, the nonrigid pipe dream must contain nonrigid pairs of pipes of the form depicted in this corollary. The following figure gives an example, where the red and blue types have a move that is not of the type stated in the corollary, but there are 3 pairs of pipes that do have a move of the stated type, which are depicted in the 3 pairs of (different shades of) green dots: 

  Remark 2. First, we recall that the main theorem gives a criterion for testing smoothness (equivalent to rigidity) which is checkable on maximal rectangles. The fact that we only need to check on maximal rectangles is evident from this corollary, since this sort of pipe dream configuration can only exist within rectangular shapes. Of course, this is just a special case of Proposition <ref>.


The proof of Corollary <ref> is already in the proof of the Proposition <ref>. One direction is obvious: a box configuration as depicted implies nonrigidity. Conversely, if we have something that is nonrigid, then we have a cross and then a near-miss for a pair of pipes, like in the previous figure. Then these pipes must bend somewhere. The south and west portions of these bends must be filled in with crosses via the Lemma, as explained in the previous proof. Then, if there are any elbows in the remaining portion, they will form a rectangular configuration like that depicted. If not, then we arrive at a smaller box, as explained previously. 


Note that this rule, that neither of the two box configurations above occur, encompasses the sorts of moves considered in <cit.>. Specifically, they considered moves of the following form:

 

These can be viewed as size (1,j) or (j,1) boxes of the form in the previous corollary. 


All rigid rectangles reduce to NW/SE partitions.


What we will prove below is that every cross (schematically shown by circling the crosses in red) is contained in one of the four configurations below.



The proof is mostly constructive, and proceeds by going through the following diagram:

  


We can think of these as two steps, which we can call NE and SW. These come from the previous theorem which shows that any cross within a rigid diagram cannot have elbows both to its north and east, or both to its south and west. Thus, to ensure that both the NE condition and SW condition are satisfied, we can think of doing 2 steps: ensuring that the NE condition is satisfied, and ensuring that the SW condition is satisfied. 




Looking at the diagram, we start with a single cross, which we mark by giving it a circle. The top (separating the columns) then shows the result of applying the NE condition to this cross. Any time we have a series of three crosses coming out from the circled cross, that pictorially depicts an unbroken line of crosses stretching all the way to the wall. Thus, for example, the leftmost configuration at the top shows the cross (circled in red), with a line of crosses stretching north all the way to the wall. This satisfies the NE condition, since it can no longer have elbows both to its north and east. If we then apply the SW step to this configuration, then we obtain diagrams (a), (b) and (c). Diagram (a) is obtained by satisfying the SW rule by extending south, (b) by ending west, and (c) by extending both south and west. Diagrams (d)-(f), and (g)-(i) follow the same pattern of elongation starting from their diagram satisfying the SW condition.



Now, each (a), (e), and (i) is already contained in an entire row or column of crosses (or union thereof), so they already satisfy the conditions of the theorem. Thus, we have to deal with the rest. We do the argument for two of these: (c) and (d). The others follow by analogous arguments, and are left to the reader.

.7712mm (c) looks like the following:
.200mm
    < g r a p h i c s >
 .05in


.7712mm The crosses along the vertical line all satisfy both NE and SW conditions. However, note that the empty space is unknown: at the moment, it can be either crosses or elbows. Thus, if we look at the crosses boxed in grey in the next figure, they do not satisfy the NE condition if the empty space is all elbows. 
.200mm
    < g r a p h i c s >
 .05in



.7712mm In order to have these crosses satisfy the NE condition, we can extend them east, or extend them north. If we extend them east, then this extra set of crosses is sufficient for satisfying the NE condition for all the crosses boxed in grey:
.200mm
    < g r a p h i c s >
 .05in



.7712mm However, if we cannot extend east (that is, if there is any elbow to the east of the cross circled red), then we must extend north individually for each of the crosses boxed in grey:
.200mm
    < g r a p h i c s >
 .05in



In fact, we have shown something slightly stronger than the theorem. The configuration we started with here already had the cross circled in red contained within an entire column of crosses. Thus, what we showed is that if a cross is contained within an entire column or row of crosses, and the perpendicular direction has a line of crosses extending all the way to the wall in only one of the two possible directions, then it must be part of a northwest or southeast partition. 



Now let us take a look at (d):




Here we have boxed the crosses in pink and light blue. The pink crosses satisfy the SW condition but not the NE. The light blue crosses satisfy the NE condition but not the SW. Following the arrows through the figure, where the pink arrows represent resolving the SW condition, and analogously for the light blue arrows (the details are the same as the description for the case of configuration (c) above), we obtain 6 final configurations. Some are redundant, so the unique ones are labelled (w), (x), (y), and (z). 



It is clear that by symmetry, all other cases in (a)-(i) will just be certain reflections/rotations of what we have already shown. This proves the theorem. 



We are trying to show that if the point λ is smooth on Π_f, then λ is also smooth at any deletion or contraction of Π_f. 

Since we assume that λ is smooth on Π_f, we can apply Theorem <ref>, which says that the pipe dream for Π_f on U_λ is rigid. (This implies that any subset of the pipe dream (for example, any maximal rectangle) is also rigid, since no two pipes containing any cross and near-miss combo in the whole pipe dream implies the same thing for any subset; but we will not need this fact for this proof, and this can already by deduced from the Main Theorem since rectangles that reduce to NW/SE partitions are rigid). This also implies that if we do deletion or contraction by filling in entire rows or columns of crosses, as explained in Prop <ref>, we do not need to (since in fact we cannot) do any moves prior to the filling in - rather, in this smooth/rigid case, deletion (contraction) of column (row) i simply involves taking the ith column (row) and brute-force changing every elbow tile in this column (row) into a cross tile. In the second (combinatorial) proof of Proposition <ref> above, we did a detailed argument to show that this “brute-force filling in" did not create any new moves, but now we show that we can bypass that argument by applying the Main Theorem.

Recall the definition of “reducing to NW/SE partitions": this means having a rectangular shape where the northwest and southeast were partitions, and additionally, any number of rows or columns could be filled in with crosses; everything else is elbows. The Main Theorem says that (in this smooth case) all maximal rectangles reduce to NW/SE partitions; now, taking any one of these maximal rectangles (which reduce to NW/SE partitions) and filling in a row or column with crosses is still precisely a shape that reduces to NW/SE partitions. Therefore, it is still smooth. 



§ ATOMIC POSITROID PAIRS


Next, we describe the singular pairs that are lowest in the order: recall that we called these “atomic positroid pairs." By definition, any deletion or contraction of an atomic positroid pair results in a smooth positroid pair (see the figures in the introduction). The order we defined on positroid pairs has the property that for any given pair, as we go down in the order, we can only delete or contract a finite number of times. Because of this, any singular pair lies over an atomic pair, i.e. these minimal elements generate the order ideal.



The atomic positroid pairs have affine pipe dreams that look like the following:

 

More precisely, the affine pipe dreams looks like the following, with each square labeled “A" being the same, and taking the form of the previous figure:

 

Specifically:

(1) The atomic positroid pairs have pipe dreams in a square shape: so they are in Gr(k,2k) and on points where λ consists of k consecutive columns. 

(2) In this pipe dream, there is a single cross along the southwest to northeast (longest) diagonal, with the diagonals directly adjacent to this longest diagonal being free of any crosses.

(3) Besides this one cross, there are partitions of crosses in the northwest and southeast corners, and no others.



We begin by proving that for an atomic positroid pair, in each of its pipe dreams there must be a single cross that can move. Suppose for contradiction that there exist two crosses that we call cross A and cross B, both with moves. Recall that by the definition of atomicity, this implies that any deletion or contraction must eliminate both crosses. We show that in general we can always eliminate one of the crosses while preserving the other one through a deletion or contraction - unless the pipe dream stretches infinitely in both the northeast and southwest directions, which gives us the contradiction. 



In detail: suppose that cross A is at (i_1,j_1) and cross B at (i_2,j_2). The argument below will use the fact that, if a deletion
eliminates both crosses, then they have to be in the same column (the case of contraction: they have to be in the same row). Without loss of generality, at least the first or second coordinate must differ, or else cross A and cross B are the same cross, so let's take j_1=j_2: they are along the same column, and let's take i_1 to be above i_2 in the section of the pipe dream we are looking at. Now, clearly, deleting along this column eliminates both crosses, but since they are on different rows with cross A above cross B, we could contract cross A, which would leave cross B unchanged - that is, unless cross B has a move northeast to a new position (i_1,j_3) along the same row as A but to its east. Now, we have solved the issue of contraction along i_1 eliminating both crosses, but the problem is that B is in a new easternmost column j_3 so we can delete j_3 to only eliminate B - that is, unless  A has a move northeast to a new position (i_3,j_3) that is in the same column as B but north of B. But now we run into the issue that cross A has a new possible contraction at i_3. Thus, continuing on, we would have to do this infinitely in the northeast (as well as analogously in the southwest ) direction in order to preserve atomicity, which is impossible since our pipe dream shapes are bounded. 



Now that we have proven that there is a single cross with a move, let us consider the possible positions of this cross: it must exist along the main diagonal of a square-shaped pipe dream; this is because, if this cross's moves missed any row or column, then we could delete this skipped row or column while leaving the cross (and its moves) intact, contradicting atomicity. Furthermore, this means that we have a clear diagonal along which this cross moves; in other words, the diagonals above and below this diagonal are free of any crosses, otherwise such a cross would hinder the free movement of this cross along the main diagonal (this relies on the fact just proven that no other cross can move): this proves (2). Since the pipe dream shape includes every row and column along a southwest to northeast diagonal, and no other rows/columns, and because in the shapes we are considering the non-convex portions only occur along the southwest and northeast corners, the boundaries of the pipe dreams shape must extend straight from the southeast and northeast corners: thus, the shape must be a rectangle. But since we already showed that every row and column must intersect the main diagonal, the shape in fact must be square. This proves (1). Since we already proved that this is the only cross that can move, everything else must be rigid, and this proves (3). 

A point λ being singular on a positroid variety Π_f  is equivalent to the pipe dream for Π_f on U_λ being able to reach an atomic configuration via a series of deletions and contractions (that is, d/c-reduces to an atomic configuration). 


If (Π_f,λ) d/c reduces to an atomic configuration, since atomic positroid pairs are singular by definition (minimally singular), and since we already proved in Prop <ref> that the order on deletion and contraction is such that, for a singular element, everything higher up in the order is also singular, (Π_f,λ) must be singular. 

In the other direction, Π_f is finite dimensional, so a finite number of deletions and contractions will take it to the point λ, which is smooth in itself. Thus, every path going down in the order from a pair (Π_f, λ) becomes smooth. Suppose Π_f∈ Gr(k,n). Then there are k contractions (which can be applied to the columns in λ) and n-k deletions (for the other columns, which cannot be contracted since U_λ contains only the T-fixed point λ). Furthermore, suppose Π_f is singular. Then these n deletions and contractions are either (1) all smooth, in which case (Π_f, λ) is atomic, or (2) at least one is singular, which let's call (Π_f',λ'). We can apply the same argument to (Π_f',λ'), which will have n-1 possible deletions and contractions - either it is atomic, or it has a deletion or contraction that remains singular. In this way, we can view each point in the ordering as spawning a tree of children below. By Prop <ref>, every singular point only has singular parents, and so singular points are connected. Because all children become smooth eventually, these connected subtrees of singular points eventually end, with the last point being an atomic pair. See the next figure for an example.


Here, we provide an example of a part of the ordering. In particular, we show the positroid variety given by siteswap f=342333 containing point λ={1,2,5}. This point is singular, which we have denoted by putting this positroid pair inside a blue rectangle. 

 

We consider the set of all its deletions and contractions, denoted by the lines, and labeled for example, by “2c" meaning that we contract the 2nd column or “5d" meaning that we delete the 5th column. We stop doing further deletions and contractions once we reach a smooth point, since by Proposition <ref>, anything lower in the order is still smooth. We see that the pair (342333,{1,2,5}) has exactly 2 deletions or contractions that are still singular, which are contraction of the 5th column or deletion of the 6th column. We can continue to delete or contract and obtain something singular until we get to f=2312, λ={1,2}. Any further deletions or contractions result in a smooth point. Thus, by definition, this is an atomic pair (in fact, the only atomic pair that dŗeduces from (342333,{1,2,5})). And indeed, if we draw out the pipe dream, we see that (2312, λ={1,2}) has a pipe dream that matches the form depicted in Theorem <ref>:

 

We prove one more proposition in this section before making some final remarks.


Every atomic positroid pair comes from within a maximal rectangle. More precisely, if we have a pipe dream δ and there exists a series of deletions and contractions that take it (d/c reduces) to an atomic configuration that we call δ_atomic, the process of deletion and contraction will have removed every tile outside of some maximal rectangle R contained in δ (and most likely, some tiles within R as well). 

This can be seen via an argument based on the following figure:

 

On the left side, we see that any pair of tiles that are southwest/northeast of each other must be contained within a maximal rectangle: this follows purely from the shapes we are considering. However, if they are southeast/northwest of each other, this might not be the case. In this case, if northwest tile A has coordinates (i_1,i_2) and southeast tile B has coordinates (j_1,j_2), then they are not contained in a maximal rectangle if either (i_1,j_2) or (i_2,j_1) is not contained in the shape. In such a case, there can be no series of deletions or contractions that put A and B into the same atomic pipe dream. The reason is that we cannot delete row i_1 or contract column i_2 without eliminating tile A, and we cannot delete row j_1 or contract column j_2 without eliminating tile B. Therefore, we cannot eliminate the spaces (i_1,j_2) or (i_2,j_1): they will always exist, so there will always be a convex hole, meaning that tiles A and B will never be able to fit into a square shape. This completes the proof of the fact that every atomic positroid pair comes from within a maximal rectangle.


Let δ∈ PD_(Π_f,λ) be an affine pipe dream for the positroid variety Π_f on U_λ. We recap some of the statements that have been proven in this thesis.


    δ has a move within some maximal rectangle
       δ has a move (is not rigid)
       Π_f is singular
       δ d/c-reduces to an atomic pair
       δ d/c-reduces to an atomic pair in some maximal rectangle


The first equivalence comes from Proposition <ref> (one direction is trivial) or alternatively Corollary <ref>, the second equivalence comes from Theorem <ref> (as well as Prop <ref>), the third equivalence is the content of Proposition <ref>, and the last equivalence was just shown in Proposition <ref>. It might be hoped that, due to the apparent similarity of the first and last statements (both involving a condition within a maximal rectangle), that an equivalence between these two statements could be proved directly, which would produce a different proof of Proposition <ref> using atomicity, and thereby yield an alternate proof of the Main Theorem <ref>. Unfortunately, the proof is more difficult than it seems. In one direction, going from δ having a move within some maximal rectangle to showing that δ d/c reduces to an atomic pair in that same maximal rectangle can be done by arguments involving deleting and contracting all rows and columns except for the positions where two pipes have a cross and near-misses (the argument could be made a bit easier by making use of Corollary <ref>). The other direction is more difficult because the process of deletion or contraction occurs by doing all possible moves (in order to make sure that all pipe dreams for a given bounded permutation with the maximum number of crosses along the column/row to be deleted/contracted are obtained). This means that immediately upon invoking the notion of d/c-reduction, we are no longer just considering the single affine pipe dream δ but rather a possibly very large set of affine pipe dreams. In particular, we can no longer focus in on a single maximal rectangle (e.g. the maximal rectangle inside which we hope the atomic pair arises or the maximal rectangle where we hope to prove there exists a move) because we have to consider a set that includes pipe dreams where a cross-elbow move has shifted to being outside that single maximal rectangle we originally intended to focus on. 







§ SKEW PARTITIONS AND SCHUBERT VARIETIES


A skew partition can be denoted ν\ρ for two partitions ν and ρ where ρ is completely contained in ν: see the picture below. It will be easier to think of the partition as being rotated by 45 degrees, so the pipes enter along the bottom and exit through the top:

 
Let π∈ S_n. The following are equivalent:

(1) π is 321-avoiding.

(2) T ↷ X_0^π contains dilation.

(3) π has a unique reduced word up to commuting moves (a unique heap).

(4) The heap is a skew partition.


We prove the equivalence of (1) with (2), (3) and (4). 

First, we show (1) ⇔ (2). Recall that we called (2) the condition of being cominuscule, and this is the property that there exists a circle inside the torus that acts by dilation. This means that this circle acts with all weights one. Since the roots of X_0^π correspond to all inversions in π (that is, a pair 1≤ i<j ≤ n such that π(i)>π(j) implies a root x_j - x_i), this implies that we can simultaneously set all x_j - x_i=1, for all  1≤ i<j ≤ n such that π(i)>π(j).
Note that, dually, there is a map from the weight lattice of the torus to the weight lattice of the circle, which is the integers. Since we want all our roots to go to 1, this means that all the roots have to lie on a hyperplane, specifically the hyperplane of things that go to 1 under this dual map. 

For the (⇒) direction, if π is not 321-avoiding, then there exists i<j<k such that π(i)>π(j)>π(k). This would imply that we would need to find x_i, x_j, x_k satisfying the 3 equations: x_j-x_i=1,  x_k-x_j=1,  x_k-x_i=1, and clearly this is inconsistent, since if we add the first two equations we get x_k-x_i=2.

For the (⇐) direction, suppose π is 321-avoiding. This means that for any 1≤ k≤ n, if we find a j<k such that π(j)>π(k), then there will be no i<j such that π(i)>π(j). Therefore, we can set x_k =1 and x_j=0 so that x_k-x_j=1. Note that k cannot descend to the right (that is, we will be unable to find a k<l such that π(k)>π(l), because then j,k,l would form a 321-triple), and similarly j cannot ascend to the left (since as we already noted, i<j such that π(i)>π(j) would result in i,j,k forming a 321-triple); this proves that we can assign 0's and 1's to all x_i to satisfy all roots equalling 1 consistently, since x_i participates in each descent only as the higher or lower number, not both. 

None of these arguments relied on the permutation being finite, so the equivalence of (1) and (2) also holds in the affine case Ŝ_n.

Next, we show (1) ⇔ (3). First, note that (3) is equivalent to the statement that there are no braid moves in any reduced word for π, where the braid moves is defined as s_i s_i+1 s_i = s_i+1 s_i s_i+1. This follows from Tits' Theorem, that any word for a permutation can be obtained from any other word via (1) commuting moves or (2) braid moves. Since we allow commuting moves, the statement of (3) is that there are no braid moves. Second, note that (1) is equivalent to there not existing any pattern of the form pictured below:
 
The reason is because if π is not 321-avoiding, then it must contain a set of 3 pipes i,j,k such that π(i)>π(j)>π(k). Another way to state this pattern is that there is a set of 3 pipes such that each one intersects the other two. 

For (⇒), if (3) is not true, then it contains a braid move, so its reduced word (up to commuting moves) allows for a braid move; that is, there is a combination s_i s_i+1 s_i or s_i+1 s_i s_i+1. These two patterns correspond to the figure above, and since reducedness implies that double-crossings are not allowed so the pipes cannot uncross (stay in this position relative to each other), the permutation is not 321-avoiding. 

For (⇐), we noted above that (1) is equivalent to not being able to find a set of three pipes such that any one intersects the other two, as in the figure above. Note that if we do spot such a pattern, we can search for the smallest one; in other words, we can make sure that there are no pipes that, for example in the figure on the left, intersect the blue and black pipes to the right of their intersection point and to the left of the red pipe, or another pipe northeast of the blue pipe and below the red-black intersection. This implies that, for this set of 3 pipes forming the smallest triangle, there will be no other pipes entering the triangle, so that in their heap diagram, they will be read as s_i+1 s_i s_i+1 if it looks like the first diagram, and s_i s_i+1 s_i if it looks like the second diagram. 

Next, we show (1) ⇔ (4)

For (⇒), we assume that π is 321-avoiding. Assume for contradiction that the heap is not a skew partition. Consider the following figure:

 

The fact that it is not skew implies that there is an inward, nonconvex opening, either along the left or the right. There must be a first such opening (this would be a place where a bottom boundary of form (1) NW-SE touches a boundary of form (2) SW-NE such that the NW corner of (1) joins the SW corner of (2)). We displayed this opening with the 2 boundaries labeled a and b. The (1) NW-SE boundary is labeled c, and since of course the non-skew partition must close up, c must touch another boundary of form (2) SW-NE on its right, which we have labelled d. Along the edge b, there must be a first pipe entering (hitting the red dot), and we have colored this pipe in red and labelled it 2. After it hits a, it must continue northeast until it hits the top boundary. There must be a first pipe emerging from c, which we have colored green and labeled 1. It must also continue northeast, parallel to 2, until it hits the top boundary. Finally, perpendicular to pipes 1 and 2 must be a pipe 3 that we have labelled in blue. Clearly, in order to cross them perpendicularly, it must start later than both 1 and 2, and since it crosses perpendicularly moving left, it must end earlier than both 1 and 2. Therefore, pipes 1, 2, and 3 form a 321 triplet. Since these arguments were general, only assuming the most general property of being non-skew, this proves that π is not 321-avoiding. 

For (⇐), we show that skew partitions are 321-avoiding. We noted above that being 321-avoiding is equivalent to not having the diagram above where every one of 3 pipes intersects the other 2. 
Assume that we have a skew partition. Consider pipes coming in along the bottom edge. They enter in 2 types of ways: along diagonal lines that go (1) NW-SE, and diagonal lines that go (2) SW-NE:

 

In the figure, we have also displayed with some example pipes, that the (red) pipes coming from the (1) NW-SE bottom boundary move northeast, and the the (red) pipes coming from the (1) SW-NE bottom boundary moves northwest. All pipes coming from (1) NW-SE are parallel, and never cross. Additionally, reading from left to right, they are in the same order along the bottom as the top. The same is true for the (2) SW-NE pipes. Thus, in a skew partition being all pipes can be divided into 2 classes, each of which consists of lines parallel to all other pipes within its class. Let's call the classes A and B. All pipes in class A are parallel, so can only intersect pipes of class B. All pipes in class B are parallel, so can only intersect pipes in class A. Let's pick an arbitrary pipe a, which, without loss of generality, is in class A. If a intersects a pipe, this intersected pipe must be a pipe b in class B. Now, in order to violate the 321-avoiding condition, we must have a pipe that intersects both a and b. However, every pipe must be in class A or class B. If it is in class A, then it can only intersect b and not a; if it is in class B then it can only intersect a and not b. Thus, we have a contradiction. Note that this direction generalizes easily to the affine case as well. 



Let Fl(n) denote the variety of flags in ^n. Let w∈ S_n denote a permutation on n elements so that X_w denotes a Schubert variety in this flag manifold, and let v∈ S_n be 321-avoiding with v≥ w in Bruhat order. Then X_w is smooth at v,  if and only if there exists a pipe dream for w inside v's skew partition (defined above Prop <ref>) such that all maximal rectangles reduce to NW/SE partitions. (In this case, the pipe dream will be unique.) 

We assumed that v∈ S_n is 321-avoiding, so we can use Proposition <ref>. By (3) of this proposition, v has a unique heap, and by (4) of the same proposition, this heap is a skew partition, so v=ν/ ρ for some partitions ν and ρ. Then w be given by a subword inside ν / ρ. We can furthermore apply equation (*) in the beginning part of Subsection <ref> on the Smoothness Theorem; part (1) is the only part that deals with positroid varieties in the Grassmannian, so parts (2), (3) and (4) of equation (*) relate the T-equivariant cohomology [X_w∩ X_∘^v⊆ X_∘^v] inside the affine flag variety to the number of pipe dreams via the AJS/Billey formula. This allows us to take advantage of the combinatorial content of the Main Theorem <ref>, specifically Proposition <ref> that all moves exist within maximal rectangles and Theorem <ref> that all rigid rectangles reduce to NW/SE partitions. Therefore, if all maximal rectangle within the pipe dream for w inside ν/ ρ reduce to NW/SE partitions, then the pipe dream for w inside ν/ρ is rigid.

Getting the multiplicity as the coefficient of h^ X required the use of Theorem <ref> from Rossmann, which required the existence of a circle within T acting by dilation (cominuscule), and this is exactly the content of Proposition <ref> part (2). Therefore, the algebraic geometry statement of the Main Theorem applies as well; in other words, we can conclude that if all maximal rectangles for the pipe dream of w inside ν/ ρ reduce to NW/SE partitions, then X_w is smooth at v.



One then wonders whether the corresponding statement is true in more generality in the affine case as well. Unfortunately, it is not true. The issue is that it is possible for two pipes A and B that cross and near-miss combination on a cylinder to be such that, when drawn as an infinitely long strip on a page, the near-miss for example involves pipes A and C instead of pipe B. In particular, this implies that Proposition <ref> no longer holds. Such a counterexample requires a tile from which we can move both south and west, as well as north and east (in perpendicular direction) and reach the same letter s_i on the same square; this occurs for the case of n even when there is a tile from which one can move an additional n/2 tiles in the NE and NW directions or SE and SW directions (or n-1/2 and n+1/2 in the odd case). However, in the case of <cit.>, the shapes are bounded of height k and width n-k, so the maximum a tile could move would be k-1 squares NW and n-k-1 squares NE. Thus, there is always a rectangle containing any cross and near-miss combination of pipes. We hope to comment more about the affine case in a companion paper <cit.> to be released soon, which will focus more on computations and examples, as well as connections to several other notions in the literature.  




§ APPENDIX A


In this appendix, we include the details of how to calculate the siteswap of del_i(Π_f), the positroid variety obtained from Π_f by deleting the i-th column. <cit.> already gave an algorithm for doing deletion and contraction on decorated permutations and Grassmann necklaces; here we describe a pictorial way to do this for siteswaps. 



It is perhaps easiest to see what is going on diagrammatically:

 

Or, step by step:

 

The L-shaped box on top represents a successive process of inversions that move the 1 in the ith row left, until it lands on the left diagonal line (which is the (i,i) position in the ĩ-j̃ coordinate system). Thus, the last inversion performed will involve the 1 that is directly north of this (i,i) spot to bring it south. The case of contraction is analogous, except that the goal is to move the 1 in the ith row rightwards, until it lands on the (i,i+n) position. At every step, we look at the previous row for deletion (or next row for contraction), and see if there is a 1 lying between the value i on the j̃ axis, and where the 1 on the i-th row currently is; if there is, we do the inversion. This is simply the statement that we are doing the minimal number of weak Bruhat moves to get from our original affine permutation to one that has a 0 in the i-th slot. This results in the largest positroid variety contained in the original one that has a 0 in the i-th slot.  



Here is a more algorithmic way of writing what we have just described in words: 

Let g denote the bounded affine permutation, and f its corresponding siteswap. For deletion, start with i and g(i), we want g(i)=i since this means f(i)=0. Test i-1, i-2,... successively and if necessary perform the following operation until the goal g(i)=i is reached. So we begin by testing i-1: if i<=g(i-1)<g(i), then transpose i and i-1: define the new g(i):=g(i-1) and g(i-1):=g(i). (Whenever such an inversion is performed, g(i) is a little closer to i.) Continue with i-2 (i.e. the new condition to be tested is: i<=g(i-2)<g(i)) and so on until g(i)=i is reached. 



For contraction, start with i and g(i), we want g(i)=i+n since this means f(i)=n. Test i+1, i+2,... successively and if necessary perform the following operation until the goal g(i)=i+n is reached. So we begin by testing i+1: if g(i)<g(i+1)<i+n, then transpose i and i+1: define the new g(i):=g(i+1) and g(i+1):=g(i). (Whenever such an inversion is performed, g(i) is a little closer to i+n.) Continue with i+2 (i.e. the new condition to be tested is: g(i)<g(i+2)<i+n) and so on until g(i)=i+n is reached. 



The proof that this is the correct process follows right from the proof of Proposition <ref>. There, we showed that the deletion of the ith column of a positroid variety Π_f is a single positroid variety Π_f', so is described by a siteswap: Π_f'=Π_del_i(f). We showed there that this Π_del_i(f) is the largest positroid variety Π_f' contained in Π_f such that f'(i)=0. We know that for Π_f' to be a positroid variety contained in Π_f is equivalent to f' being reachable by a series of Bruhat moves from f, and each such weak Bruhat move decreases the dimension of the positroid variety by 1. Since we want the largest such positroid variety, we want the minimal number of such moves. This is exactly what the diagram in this Appendix does: it depicts pictorially the minimal number of Bruhat moves to get f'(i)=0; thus, it produces the largest positroid variety contained in Π_f with f'(i)=0. 




§ APPENDIX B


Here we give a self-contained introduction to affine pipe dreams. We do an explicit example to illustrate the process of taking in a pair (Π_f, U_λ) and associating a reduced subword to it. Also, in the introduction, we noted that one way to test whether a positroid variety has a unique pipe dream on an open patch (and thus is smooth there) is to draw the top and bottom pipe dream for that positroid variety on that open patch. If they are the same, then the pipe dream is unique so the positroid variety is smooth there. If the pipe dreams are different, then the positroid variety is singular there. <cit.> notes in Section 4.3.1 that the top pipe dream comes from the lexicographically first reduced word, and the bottom pipe dream comes from the lexicographically last word. We noted that the details would be included in this appendix. 



Let's take an example of a siteswap 2225377 (bounded affine permutation 3,4,5,9,8,13,14) inside Gr(3,7), and let's test it on λ={2,5,7}. 



Consider the following figure:

 

(1) In the first figure on the very left, the distinguished path (illustrated in red) is formed by moving horizontally starting from the northwest corner of the matrix until a 1 in one of the identity columns is encountered, at which point the distinguished path moves down. Thus, the distinguished borders the north and east of each 1 and otherwise is horizontal. 

(2) In the middle figure, we collapse the identity columns to obtain the distinguished path. The tiles will be filled by transpositions s_i. Here, we have highlighted in blue the southwesternmost square. This will be s_n (in the depicted example, s_n = s_7). 

(3) Finally, we take one copy of the distinguished path and move it to the right (depicted also in red in the figure on the very right). We can continue adjoining distinguished paths (depicted in green) to the northwest and southeast of these two red distinguished paths, forming an infinite strip, which is the boundary of the affine pipe dream. Note the labelling along the boundary: it simply starts with “1" at the top line segment of the distinguished path and increases moving downwards. Thus, we see why the blue dot will be the transposition s_7 (or just “7" in the next figure): a cross where the blue dot is will switch the 7-th and 8=1( 7)-th pipes coming from the bottom boundary.

Filling in the rest of the tiles, we get: 

 

If we just look at one block, denoting the transpositions by s_i again, it is:



To get the bottom pipe dream, we have to pick a way of reading this. There are many equivalent ways of reading it, corresponding to the southwest rule in the proof of Prop <ref>, but if we want to read consistently row-by-row or column-by-column in each block, the choices are depicted below: 



In other words, if we choose the direction of red followed by green, then the ordering will be s_5, s_2 s_3 s_4, s_7 s_1 s_2 s_3, s_6 s_7 s_1, s_5 (note that this is the standard choice for the word Q_λ given in <cit.>). If we apply this word to (k+1,k+2,...k+n), then we get the bounded affine permutation corresponding to +n for the λ columns 2,5,7 and +0 for all other columns; explicitly: Q_λ(4,5,6,7,8,9,10)=(0,7,0,0,7,0,7). This shows that this pipe dream shape represents the neighborhood U_λ.



Here's how reading it this way produces a word for siteswap 2225377 (bounded affine permutation 3,4,5,9,8,13,14):



1     2     3     4     5     6     7     8(=1) 

3     4     5     9     8     13     14



Since the big word is s_5, s_2 s_3 s_4, s_7 s_1 s_2 s_3, s_6 s_7 s_1, s_5, we find the first s_i (starting from the left with s_5) where there is a transposition with f(i)>f(i+1). Then we can apply s_i to switch f(i) and f(i+1). Here, the first one is s_4 which switches the 98 to 89:



1     2     3     4     5     6     7     8(=1) 

3     4     5     9     8     13     14    

3     4     5     8     9     13     14     10 



We continue on until we get the most general permutation:



1     2     3     4     5     6     7     8(=1) 

3     4     5     9     8     13     14    

3     4     5     8     9     13     14     10 

7     4     5     8     9     13     10     14 

4     7     5     8     9     13     10     11 

4     5     7     8     9     13     10     11 

4     5     7     8     9     10     13     11 

6     5     7     8     9     10     11     13 

5     6     7     8     9     10     11     12 



Keeping track of which transpositions we used, we get:

 

If we put crosses where the circles are, and elbows elsewhere, then we get the bottom pipe dream.



In contrast, the top pipe dream corresponds to the pipe dream that we would obtain if we applied the same process as above, but where we make a few changes: 

(1) We must use the inverse permutation. We write this as the numbers g^-1(i) +n.
 
(2a) We could change change the s_i's such that they correspond to the labelling along the top boundary of the pipe dream shape rather than the bottom (this is depicted in the bottom row of the following figure). We will then read the word in the opposite ordering to the one above; in fact, any ordering obeying a northwest rule rather than a southwest rule (see the proof of Prop <ref>) will work. If we want to read in a consistent direction within each block (above or below the distinguished path), the choices are shown in the colored arrows in the next figure. 

(2b) Alternatively to (2a) (but essentially equivalent), we could use Grassmannian duality to reflect the pipe dream shape across the northwest-southeast line, before going through with the same process of finding the bottom pipe dream; obviously, the changes of the s_i's to match the (originally) top boundary happens automatically here. Finally, after doing this, we have to reflect back across the northwest-southeast line to get the top pipe dream.  

 

§ ACKNOWLEDGEMENTS

The author would like to thank his advisor Allen Knutson for many helpful discussions and for suggesting the approach. 


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