Paper Title: Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability.



Grothendieck Shenanigans]Grothendieck Shenanigans:

Permutons from pipe dreams via integrable probability





[
    
	A. H. Morales,
	G. Panova,
	L. Petrov,
	D. Yeliussizov
    
============================================================



§ ABSTRACT 
	We study random permutations arising from reduced pipe dreams. Our
	main model is motivated by  Grothendieck
	polynomials with parameter β=1 arising in
	K-theory of the flag variety. The probability weight of a
	permutation is proportional to the principal specialization
	(setting all variables to 1)
	of the
	corresponding
	Grothendieck polynomial. By mapping
	this random permutation to a version of TASEP (Totally
	Asymmetric Simple Exclusion Process), we describe the
	limiting permuton and fluctuations around it as the order
	n of the permutation grows to infinity. The fluctuations
	are of order n^1/3 and have the Tracy–Widom GUE distribution,
	which places this algebraic (K-theoretic) model
	into the Kardar–Parisi–Zhang universality class.

	We also investigate non-reduced pipe dreams and make
	progress on a recent open problem on the asymptotic number
	of inversions of the resulting permutation. Inspired by
	Stanley's question for the maximal value of principal
	specializations of Schubert polynomials, we resolve the
	analogous question for β=1 Grothendieck polynomials,
	and provide bounds for general β.




§ INTRODUCTION





 §.§ Overview



Algebraic Combinatorics established itself as a field that uses combinatorial methods to understand algebraic behavior in problems ranging from Group Theory to Algebraic Geometry. It started with stark exact formulas, like the celebrated hook-length formula for the dimension of irreducible modules of the symmetric group S_n; beautiful interpretations, like the Littlewood–Richardson rule for the structure constants of representations of the general linear group GL_n; powerful and intricate bijections, like the Robinson–Schensted–Knuth correspondence. However, exact answers only go so far, leaving room for questions like  “approximately how many”, “what are the typical objects”,  and “what is the typical behavior”. These questions lead us into the realm of Asymptotic Algebraic Combinatorics, which aims to answer them with the help of tools originating outside of Combinatorics. In the present work, we employ Integrable Probability, a rapidly evolving field focused on developing and analyzing interacting particle systems and random growth models possessing a certain degree of structure or symmetry.

Many significant questions concerning exact formulas and combinatorial interpretations in cohomology and K-theory of the flag variety remain open. Understanding their asymptotic behavior is thus even more natural. Stanley <cit.> asked the most basic question on the  principal specializations of Schubert polynomials 𝔖_w
(representing cohomology classes of the flag variety): does the following limit exist

    lim_n→∞1/n^2log_2 max_w ∈ S_n𝔖_w(
    	1,…,1 _n),

and if so, what is it and for which permutations w is this achieved.
This question (including the existence of the limit) is still open.
In <cit.>, a lower bound of about 0.29 was established for layered permutations. An upper bound of about 0.37 comes from a remarkable connection with Alternating Sign Matrices and the six-vertex model (see <Ref>).

A natural one-parameter generalization of Schubert
polynomials are the Grothendieck polynomials
𝔊_w^β, which represent K-theoretic
classes of the flag variety. Extending Stanley's questions, we
would like to understand the asymptotic behavior of maximal
 principal specializations max_w∈ S_n𝔊_w^β(1^n)
of Grothendieck polynomials. This
question was first touched on in
<cit.>, <cit.>. Establishing a family of
such maximal permutations would shed light on the
Schubert questions as β→ 0.

Thanks to a combinatorial model for Grothendieck and
Schubert polynomials, these questions have very natural
statistical mechanics interpretations.
Namely, both polynomials are partition functions of tilings
into crosses and elbows of a size n triangle (staircase) shape,
which result in a configuration of n “pipes”. Such pipe configurations
are often called pipe dreams.
In the Schubert case, the only valid
tilings are the ones where no two pipes cross more than
once. This is a global (long-range interaction) condition,
which makes the model much less tractable.
In the Grothendieck case
for β=1, all tilings are allowed, but the pipes must be
resolved (reduced) to obtain a permutation. This model can be mapped to a
colored stochastic six-vertex model
(and furter to TASEP, the Totally Asymmetric Simple Exclusion Process).
These interacting particle systems have only local (short-range) interactions, and are
amenable to techniques from Integrable Probability.



In this work,
we investigate the asymptotics of
β=1 Grothendieck random permutations
and characterize their limit shape which is described by a permuton.
In principle, the typical
permutation (sampled from the limiting permuton) should give the
asymptotically maximal principal specialization
𝔊_w^β=1(1^n), but we do not address this question here.
We also study fluctuations of Grothendieck random permutations around the limiting permuton.
They
are of order n^1/3 and asymptotically have the Tracy–Widom GUE distribution.
This distribution was
first observed in the fluctuations of
the largest eigenvalue of Gaussian
random matrices
with unitary symmetry. By now, having Tracy–Widom fluctuations is an indication that a
model is within the
Kardar–Parisi–Zhang (KPZ) universality class <cit.>, which includes a wide range of
random growth models and interacting particle systems.



We also study the natural, yet not as algebraically
motivated, model, where the staircase is tiled with crosses
with probability p and elbows with probability 1-p, and
the pipes follow the tiles without any reductions or
reassignments. Using simple random walks, we
show that the resulting random permutation is close to the identity, and, moreover,
that the total
displacement
∑_i=1^n|i-w_i|
and length (number of inversions)
of this permutation
are of order n^3/2. This partially resolves a recent open problem
posed by
Colin Defant.



Returning to the original question, we also investigate the asymptotics
of max_w ∈ S_n𝔊_w^β(1^n).
When
β=1, this maximum is known to be O(2^n2). Using the
correspondence with
2-enumerated Alternating Sign Matrices
(equivalently, the six-vertex model with domain wall boundary conditions
and free-fermion weights; or the model of uniform domino tilings of the Aztec diamond),
we show that a large family of layered
permutations also achieve this asymptotic maximum. For
general β, we establish certain bounds for the maximal
 principal specialization.



 §.§ Grothendieck polynomials



The nonsymmetric Grothendieck polynomials generalize the Schubert and the Schur polynomials,
and capture the K-theory of the flag variety.
They can be defined or interpreted in a number of ways,
including via
divided difference operators <cit.>,
pipe dreams <cit.>, bumpless pipe dreams
<cit.>,
<cit.>,
and solvable lattice models
<cit.>, <cit.>.

We denote by S_n the set permutations of
{1,2,…,n} that we write in the one-line notation
w=w_1w_2⋯ w_n unless indicated otherwise.
We also denote the image of i under w by w(i), and use the notation
w_i=w(i) interchangeably when this does not lead to confusion.
The longest
permutation n n-1… 2 1 is denoted by w_0=w_0(n).
For a permutation w of length ℓ, we denote by R(w)
the set of reduced words of w, that is, tuples
(r_1,…,r_ℓ) such that s_r_1⋯ s_r_ℓ
is a reduced decomposition of w, where s_i=(i, i+1) are the
simple transpositions.

The (single) Grothendieck polynomials were introduced
by Lascoux <cit.> to study the K-theory of
flag varieties. Their original recursive definition is as
follows. Let π_i:ℤ[β][ x]→ℤ[β][ x] denote the isobaric
divided difference operator:

    π_i f(1+β x_i+1) f-(1+β x_i) s_i·  f/x_i-x_i+1,

where s_i acts on a polynomial f by permuting x_i and x_i+1.
The Grothendieck polynomials 𝔊_w^β are recursively
determined by the following conditions:

	
  * For the longest permutation, we have
		𝔊_w_0(n)^β=(1+β)^n2
		x_1^n-1x_2^n-2⋯ x_n-1.
	
  * For all w∈ S_n and i=1,…, n-1
		such that
		ℓ(w s_i)=ℓ(w)+1,
		we have
		𝔊_w^β =
		π_i 𝔊_ws_i^β.

Setting β=0 in 𝔊_w^β, we obtain a Schubert
polynomial 𝔖_w which represents cohomology
classes of Schubert cycles in the flag variety <cit.>.



 §.§ Pipe dreams and permutations



Grothendieck polynomials can also be defined combinatorially
via pipe dreams (equivalently, rc-graphs).

A pipe dream
of order n
is a tiling of the
staircase shape
(having n-1 boxes in the first row,
n-2 boxes in the second row, and so on,
with boxes left-justified)
by tiles of two types:
bumps
-2pt
    < g r a p h i c s >

and
crossings
-2pt
    < g r a p h i c s >
.
The n-th diagonal below the staircase is
equipped with half bumps
-2pt
    < g r a p h i c s >

(whose boundary we do not draw).
Each of the boxes can have a tile of any type, so there
are 2^n2 pipe dreams of order n.
See <Ref>, left, for an example of a pipe dream of order 6.



A pipe dream (a tiling of the staircase shape) forms
a collection of strands (or pipes) labeled 1 to n from
the row where they start.
A pipe dream is called reduced if
any two pipes cross through each other at most once.


	
	Given a pipe dream D
	that is not necessarily reduced, the reduction of D
	is a unique reduced pipe dream D' obtained as follows:
	starting at the bottom left tile traverse the pipe dream upwards along columns and to the right. For each encountered crossing,
	replace it with a bump if the pipes have already crossed in the already traversed squares.
	The labeling of pipes together with a reduction
	is indicated by colored paths in
	<Ref>, right.



	
	One can associate a permutation w(D)∈ S_n to a pipe dream
	of order n as
	follows. If D is reduced then w(D)^-1_j is the column where
	the pipe j ends up in.[Throughout
	the paper, the column coordinate j increases from left to right,
	and the row coordinate i increases from top to bottom.]
	Equivalently, the column j contains the exiting pipe of color
	w(D)_j. Note that to capture the column number for the pipe j,
	we need the inverse permutation w(D)^-1.

	If D is not reduced, then w(D)
	is the permutation associated to the reduction D' of D.



	Alternatively, the permutation w(D)
	from a non-reduced pipe dream
	can be defined using the Demazure product. We
	refer to <cit.> for details.


Let (n) and (n) be the sets of pipe
dreams and reduced pipe dreams of size n.
For each w∈ S_n,
let (w) and
(w) be, respectively, the sets of pipe dreams and reduced pipe dreams D such that w(D)=w. Note that
#(n)=2^n2, whereas there is no simple
known formula for #(n)
<cit.>. Given
a pipe dream D, let (D) be the set of
coordinates (i,j)
of the cross tiles.
The weight of D is the monomial
(D)∏_(i,j)∈(D) x_i.
For example, for the pipe dream in <Ref>, left, we have
(D)=x_1^3 x_2^2 x_3^2 x_4 x_5.
The following
result is due to
<cit.>, <cit.>, and <cit.>.

	
	For any w∈ S_n, we have

    𝔊_w^β(x_1,…,x_n )  = ∑_D ∈(w)β^#(D)-ℓ(w)(D).

In particular, setting β=0 forbids non-reduced
pipe dreams, so the Schubert polynomial is

    𝔖_w( x) = ∑_D∈(w)(D).





 §.§ Grothendieck random permutations from reduced pipe dreams



Fix p∈[0,1].
Equip the set of all pipe dreams
with a probability measure by independently
placing the tiles in each box:

    -6pt
        < g r a p h i c s >
      with probability p,
    	    -6pt
        < g r a p h i c s >
      with probability 1-p.

In particular, for p=1/2, we have the uniform measure on the set of pipe dreams.

By reducing this random pipe dream as in <Ref>,
we obtain a
random permutation 𝐰∈ S_n
which we call the Grothendieck random permutation (of order n and with parameter p;
we suppress the dependence on n and p in the notation).
The name is justified by a connection with the polynomials 𝔊_w^β=1.
Indeed,
we have for any w∈ S_n:

    (𝐰=w)
    	=
    	∑_D ∈(w) p^(D)(1-p)^elbow(D)
    	=
    	(1-p)^n2𝔊_w^β=1( p/1-p,…, p/1-p),

where the first equality is simply the definition of the measure (<ref>),
and the second immediately follows from <Ref> with β=1.
In <Ref> below, we explain how the
same distribution on permutations arises from
the six-vertex model with domain wall boundary conditions and free-fermion weights (corresponding
to 2-enumeration of Alternating Sign Matrices).

When p=0 or p=1, we almost surely have
𝐰=id or 𝐰=(n,n-1,…,1 ), respectively.
These cases are trivial, and in the rest of the paper we assume that p∈(0,1).
Individual samples of
Grothendieck random permutations with p=4/5 and p=1/2
and a plot constructed from averaging over many samples
are given in
<Ref>.





 §.§ Asymptotics of Grothendieck random permutations



For w∈ S_n,
define its height function by

    H(x,y)#(
    		{ w^-1(x),w^-1(x+1),…,w^-1(n)  }∩{ y,y+1,…,n  }),

where 1≤ x,y≤ n. In terms of the pipe dream as in <Ref>, right,
H(x,y) is the number of pipes of color ≥ x which exit through the positions
≥ y at the top. In the example in <Ref>, right,
we have H(4,3)=2. See also <Ref>
for an interpretation in terms of the permutation matrix of w,
where H(x,y) gives the number of 1s in the rectangle with lower left corner at (x,y).

Let now 𝐰∈ S_n be the Grothendieck random permutation with parameter p∈(0,1)
which we assume fixed.
This makes the height function H(x,y) random, too.
We show that H(x,y) satisfies the law of large numbers,
and characterize its asymptotic fluctuations:


	
	
		
  * There exists a limiting height function 𝗁^∘ such that
			
    lim_n→∞ n^-1 H(⌊ n𝗑⌋,
    				⌊ n𝗒⌋)=𝗁^∘(𝗑,𝗒),
    				    
    				(𝗑,𝗒)∈[0,1]^2,

			where the convergence is in probability.
			The function 𝗁^∘(𝗑,𝗒)
			is
			explicit, see
			(<ref>).
			It is continuous and depends only on p.
			The graph of 𝗁^∘
			is given in
			<Ref>, right.
		
  * The fluctuations of H(x,y) around 𝗁^∘
			belong to the KPZ universality class:
			
    lim_n→∞( H(⌊ n𝗑⌋,
    				⌊ n𝗒⌋)-n𝗁^∘(𝗑,𝗒)/𝗏(𝗑,𝗒) n^1/3≤ r )=F_2(r),     r∈ℝ,

			where F_2
			is the cumulative distribution function of the Tracy–Widom GUE distribution,
			and 𝗏(𝗑,𝗒) is given in (<ref>).
	

<Ref>
justifies
the simulations in
<Ref>. We prove it in
<Ref>
by realizing H(x,y) as an observable of a
discrete time
TASEP with parallel geometrically distributed jumps started from the densely packed
(step) initial configuration.
Here we omit the technical details of this coupling, and refer to
<Ref> in the text.
By employing known techniques
from Integrable Probability and passing back to the
Grothendieck random permutation, we obtain the limit shape and fluctuations.



The law of large numbers in the first part of
<Ref>
means that the Grothendieck random permutations 𝐰∈ S_n
converge to a deterministic permuton.
Recall that permutons are Borel probability measures μ
on [0,1]^2 with uniform marginals, that
is,
μ( [0,1]× [a,b] )=
μ( [a,b]× [0,1] )=b-a.
For details on permutons
and their connection to pattern frequencies in permutations, we refer to
<cit.>,
<cit.>, and the survey
<cit.>.
Our limiting Grothendieck permuton is completely determined by the limiting height function
via

    μ( [𝗒,1]× [𝗑,1] )
    	=𝗁^∘(𝗑,𝗒).

As a corollary of the permuton convergence,
one can obtain law of large numbers for
arbitrary pattern counts in Grothendieck random permutations.
The simplest example is the number of inversions:


	
	Let
	𝐰=𝐰(n)∈ S_n be the Grothendieck random permutations with
	p=1/2.
	We have
	
    lim_n→∞( 𝐰(n) )/ n2=1-π/4.


The limit in (<ref>)
exists for all p∈(0,1) and is expressed through certain integrals
of algebraic functions.
We are able to evaluate it in a closed form only for p=1/2.
We refer to <Ref> and its proof for more
discussion.


	
	In <Ref>, we connect
	Grothendieck random permutations
	to
	2-enumerated Alternating Sign Matrices
	(bijectively corresponding to the six-vertex model with
	domain wall boundary conditions and free-fermion weights, and also to
	domino tilings of the Aztec diamond).
	The connection between Grothendieck polynomials and the
	six-vertex model goes back to
	<cit.>, see also <cit.>.
	Equivalences between 2-enumerated Alternating Sign Matrices,
	six-vertex models, and domino tilings of the Aztec diamond
	are well-known in statistical mechanics and integrable probability,
	see, for example,
	<cit.>, <cit.>, and
	<cit.>, <cit.>, <cit.>.

	Via this connection,
	our asymptotic results (<Ref>)
	can be recast as results about a random permutation coming from the free-fermion
	six-vertex model
	with domain wall boundary conditions.
	Moreover, there is a vast amount of asymptotic results
	about domino tilings and free-fermion six-vertex models,
	and in <Ref>
	we show how some of them lead to further asymptotic properties
	of Grothendieck random permutations.




 §.§ A deformation and random non-reduced pipe dreams



Let us discuss a deformation of the Grothendieck random permutation
by means of a new parameter
q∈[0,1].
Take a pipe dream of order n obtained
by randomly placing the tiles in the same way as in (<ref>)
(depending on p∈ (0,1)). However, instead of reduction (<Ref>),
let us apply a modified (q-randomized) procedure:

	
	Assume that D is a pipe dream
	that is not necessarily reduced.
	For each crossing tile, consider the incoming pipes.
	There are two cases:
	
		
  * If the lower colored pipe enters from the left (and the higher colored one from the bottom),
			then the pipes cross through each other with probability 1.
		
  * Otherwise, if the lower colored pipe enters from the bottom (and the higher colored one from the left),
			then the pipes cross through each other with probability q.
			That is, with probability 1-q we replace the crossing with a bump.
	
	The random q-reduction steps are done independently at each crossing tile
	when
	reading the staircase diagonal by diagonal starting from the bottom left.
	See <Ref> below for a detailed definition
	of the Markov process for q=0, which has the same order of reduction steps.
	Denote the random permutation obtained by the q-reduction of a random pipe dream
	with the measure (<ref>) by 𝐰^(q)∈ S_n.
	See
	<Ref> for simulations.



	
	For q=0, the q-reduction procedure becomes the usual reduction from <Ref>.
	For q>0, the q-reduction no longer tracks whether the pipes
	have crossed before. Instead, it simply applies the local q-reduction rule at each
	crossing tile,
	based on the relative colors of the incoming pipes.


The asymptotic analysis of 𝐰^(q) for q∈ (0,1]
may be performed similarly to the case q=0.
The underlying particle system is a certain
q-deformation of TASEP which allows both left and right jumps (governed by different rules).
Setting q=1 removes the asymmetry, which changes the
normalization in the law of large numbers and fluctuations (as well as
the fluctuation distribution).
This behavior
parallels the different
scales of laws of large numbers
in the
KPZ and the
Edwards–Wilkinson (EW) universality classes.
The KPZ class asymptotics arise in TASEP and its asymmetric deformations,
while symmetric versions of TASEP fall into the EW class.
We refer to
<cit.> for a detailed discussion of these two universality classes.

In this paper, we mainly focus on the case q=0, and obtain
results outlined in <Ref>.  In
<Ref>, we also treat the case q=1.  In
the latter model, the q-reduction does not change the pipe
dream at all. Probabilistically, this is the most natural
way to associate a permutation to a random pipe dream with
distribution (<ref>). On the other
hand, the q=1 model apparently lacks a rich algebraic
underpinning.

Finding the asymptotic number of inversions of
𝐰^(1)=𝐰^(1)(n)
∈ S_n as n→∞ was posed as an open
problem by Colin
Defant at the Richard Stanley's 80th birthday
conference.[Conference “The Many Combinatorial Legacies of Richard P. Stanley: Immense Birthday Glory of the Epic Catalonian Rascal,” <https://www.math.harvard.edu/event/math-conference-honoring-richard-p-stanley/>. Accessed: 07/23/2024.]
We show
the following
asymptotics:

	
	For p∈[0,1) and any ε>0,
	the expected number of inversions satisfies for all sufficiently large n:
	
    2/3√(π) (1-ε)  n^3/2√(p/1-p)≤𝔼[(𝐰^(1)(n))] ≤4/3√(π) (1+ε)  n^3/2√(p/1-p).

	Moreover,
	the random permutations 𝐰^(1)(n) converge in distribution
	to the identity permuton supported
	on the diagonal of [0,1]^2.


Based on simulations (see <Ref> for examples)
and supporting numerics, we make
further conjectures about the behavior of 𝐰^(q).
(<ref>):

	
	For any p∈[0,1), we have convergence in probability:
	
    n^-3/2(𝐰^(1))→ϰ√(p/1-p),     n→∞,

	where ϰ is a constant independent of p whose value is near 0.53.

<Ref> implies that
ϰ (if it exists) must satisfy
2/(3√(π))≤ϰ≤ 4/(3√(π)), and the approximate value 0.53
is conjectured based on numerical data.

We also expect that as p=p(n)→1 with n, the random
permutations 𝐰^(1)(n) converge in
distribution to a nontrivial deterministic permuton (which
depends on the speed and the parameters in p(n)→1).



In the intermediate cases q∈(0,1), one has to make sure that pipes with larger numbers
do not get stuck close to the diagonal boundary. This leads to a
conjectural restriction pq≤1/2.
Let H^(q)(x,y) be the height function of 𝐰^(q),
defined in the same way as in <Ref> above.

	
	For any fixed p,q∈(0,1) with pq≤1/2, the random permutations
	𝐰^(q)(n)∈ S_n converge to a permuton
	determined by a height function
	𝗁^(q)(𝗑,𝗒) on [0,1]^2.
	That is, we have the limit in
	probability:
	
    lim_n→∞ n^-1 H^(q)(⌊ n𝗑⌋,
    		⌊ n𝗒⌋)=𝗁^(q)(𝗑,𝗒),
    		    
    		(𝗑,𝗒)∈[0,1]^2.

	When pq>1/2, the random
	permutations 𝐰^(q)(n) converge to the identity permuton.

When pq≤1/2, fluctuations of H^(q)(x,y) around 𝗁^(q) should
be governed by one of the laws appearing in the
KPZ universality class in the presence of a boundary
(Tracy–Widom GUE distribution,
its GOE/GSE counterparts and crossovers,
or Gaussian fluctuations on scale n^1/2).
We refer to <cit.>, <cit.> for further discussion of
fluctuations in half-space models from the KPZ universality class.






 §.§ Asymptotics and maxima of Grothendieck principal specializations



We are interested in the following principal specializations of the
Grothendieck polynomials:

    Υ_w(β) 𝔊_w^β (1,1,…,1_n),
    	     w∈ S_n.

In particular, Υ_w(0) is the principal specialization of the Schubert polynomial.
For β=1, these quantities are the (unnormalized) probability weights
of the Grothendieck random permutation 𝐰 with p=1/2,
see (<ref>).

Set

    v_n(β)∑_w ∈ S_nΥ_w(β),    
    	u_n(β)max_w∈ S_nΥ_w(β).

From <Ref>, we have:

    v_n(1)=2^n2,      u_n(1)=2^n2-o(n^2),      n→∞.

The first equality is exact, while the second one is asymptotic.
This asymptotic behavior
follows from
the cardinality
n!∼ e^nlog n+O(n)≪ 2^n2 of the set over which we take the maximum.

For u ∈ S_k, w ∈ S_m, we denote

    u × w  (u(1), …, u(k), w(1)+k, …, w(m) + k)∈ S_k+m.

For a composition b = (b_ℓ, …, b_1) of n = b_1 + ⋯ + b_ℓ, the layered permutation w(b) ∈ S_n is defined as follows:

    w(b) = w_0(b_ℓ) ×⋯× w_0(b_1),

where w_0(k) is the full reversal permutation of order k.
Denote by L_n⊂ S_n the subset of layered permutations,
and let

    u_n'(1)max_w∈ L_nΥ_w(1).





In <Ref>, we prove that
on layered permutations, the
β=1 Grothendieck polynomials
attain their asymptotic maximum:


	
There are sequences of layered permutations w(b^(n))∈ S_n so that

    lim_n →∞1/n^2log_2
    Υ_w(b^(n)) (1)
    = 1/2.


<Ref> implies that in leading order,
u_n'(1) asymptotically behaves in the same way as
v_n(1) and u_n(1) (<ref>).

Explicit constructions of such sequences of layered
permutations are given in <Ref>. In particular, we
can take the parts of the compositions b^(n) = (…, b_2, b_1)
to be
geometric b_i ∼
(1- α)α^i-1 n for any α∈ [1/√(2), 1).
See
<Ref> for an illustration.
Note, however that we do not know in the limit what
compositions b of size n yield the global maximum of
Υ_w(b) over all layered permutations. For
Schubert polynomials, the analogous question was settled in
<cit.>. We present numerics
for principal specializations of the β=1 Grothendieck polynomials in
<Ref>, and explain the difference between the (numerically)
optimal layered permutations and the ones constructed in the proof of <Ref>.

Regarding Stanley's problem on permutations achieving the
maximal Schubert specialization u_n(0) (the case
β=0), the Merzon–Smirnov conjecture
<cit.> states that the maximum
is attained on layered permutations. On one hand, our
<Ref>
establishes that an asymptotic analog of this conjecture holds for
the β=1
Grothendieck polynomials.
On the other hand, the typical shape of Grothendieck
random permutations
(<Ref>, center and right; see also
<Ref>)
suggests that the
global maximum u_n(1)
may be achieved on permutations whose shape is far from layered.
In <Ref>, we obtain new bounds on the quantities
v_n(β) and u_n(β) for general values of β,
but do not consider the relation
of maximal principal specializations to layered permutations.



 §.§ Outline


In the Introduction, we defined the model and formulated our main results.
In <Ref>, we explain how the Grothendieck random permutation
𝐰 from <Ref>
is sampled by running a Markov chain, more precisely, the colored stochastic six-vertex model.
In <Ref>, we connect the vertex model
to a discrete time TASEP with parallel update and geometrically distributed jumps.
This process is well-studied within Integrable Probability,
which leads to law of large numbers and Tracy–Widom fluctuation
results.
In <Ref>, we recast the asymptotic results about
TASEP into
limiting properties of
Grothendieck random permutations.
In <Ref>, we consider a variant of our
model coming from non-reduced pipe dreams (described in <Ref>).
In <Ref>, we
address the question of asymptotics of  principal specializations
of β=1 Grothendieck polynomials, and show that they are asymptotically the largest
on layered permutations.
We also obtain new bounds on the principal specializations for general β,
based on an interpretation of Grothendieck polynomials in terms of
Alternating Sign Matrices.
In <Ref>, we discuss further directions and a few
open problems.


In <Ref>, we provide tables of maximal  principal specializations of β=1 Grothendieck polynomials on layered permutations.
In <Ref>, we provide the steepest descent computations
leading to the asymptotics in TASEP. While these computations are standard by
now, we include them to justify the exact values of the constants
in TASEP limit shape and fluctuations.




 §.§ Acknowledgments


We are very grateful to Jehanne Dousse for her valuable insights in the development of this project and doing experiments in Sage. We thank Igor Pak for bringing us together and initiating our collaboration.
We thank Daoji Huang, Mark Shimozono and Tianyi Yu for sharing an early version of <cit.>. We would also like to thank Dave Anderson, Jacopo Borga, Filippo Colomo, Colin Defant, Philippe DiFrancesco, Rick Kenyon, Matthew Nicoletti, Mikhail Tikhonov, and Anna Weigandt for helpful discussions.

The project was begun at the SQuaRE “Young Tableaux Asymptotics”
at the American Institute of Mathematics. The authors thank
AIM for providing a supportive and mathematically rich
environment.
Part of this research was performed in Spring 2024 while
some of the authors were
visiting the
program
“Geometry, Statistical Mechanics, and Integrability”
at the Institute for Pure and Applied Mathematics
(IPAM), which is supported by the NSF grant DMS-1925919.

AM was partially supported by the NSF grant DMS-2154019, the NSERC Discovery Grant
RGPIN-2024-06246, and the FRQNT Team grant 341288. GP was partially supported by the NSF grant CCF-2302174. LP was partially supported by the NSF grant DMS-2153869 and the Simons Collaboration Grant for Mathematicians 709055. DY was partially supported by the  grant.



§ FROM PIPE DREAMS TO THE COLORED STOCHASTIC SIX-VERTEX MODEL





 §.§ Colored stochastic six-vertex model



Introduce the vertex weights

    w_p(a,b;c,d)
    =
    w_p
    (
    	[baseline=-3,scale=.7,very thick]
    			[fill] (0,0) circle [radius=0.025];
    			 (0.5,0) – (0.05,0);
    			 (-0.5,0) – (-0.05,0);
    			 (0,0.05) – (0, 0.5);
    			 (0,-0.05) – (0,-0.5);
    			 at (.25,-.4) a;
    			 at (-.7,0) b;
    			 at (-.25,.4) c;
    			 at (.7,0) d;
    ),
 where
a,b,c,d∈{0, 1,…,n  }.
Here 0 represents
the absence of a pipe, and
positive numbers indicate the
pipes' colors.
We view (a,b) and (c,d) as incoming and outgoing pipes, respectively.
The weights are
defined as follows:

    w_p(a,a;a,a)    =1;
    		
    
    		w_p(b,a;b,a)   =p,       w_p(b,a;a,b)=1-p;
    		
    
    		w_p(a,b;a,b)   =0,       w_p(a,b;b,a)=1,

where 0≤ a<b≤ n.
The weights of all other configurations not listed in (<ref>)
are zero. Let us indicate a few crucial properties of the weights:

	
  * The weights conserve the pipes:
		w_p(a,b;c,d)=0 unless {a,b}={c,d} as sets.
	
  * The weights are stochastic:
		
    w_p(a,b;c,d)≥0,    ∑_c,d=0^n w_p(a,b;c,d)=1   for all a,b.




	The weights w_p come from the
	fundamental stochastic R-matrix
	for the quantum group U_q(𝔰𝔩_n+1)
	<cit.>, see also <cit.>
	for a brief overview.
	For the application to random pipe dreams, we specialize
	the quantum parameter q to zero
	(equivalently, to infinity in the normalization of <cit.> — this choice
	depends only on
	the order in which colors correspond to basis vectors of
	the fundamental representation of 𝔰𝔩_n+1).
	The remaining parameter p is related to the spectral parameter
	in an integrable vertex model.




 §.§ Matching random pipe dreams to the stochastic vertex model



Attach coordinates
(i,j)∈ℤ_≥1 ^2
to boxes of the
staircase shape, with i increasing down,
and j increasing to the right.
The staircase shape is then
_n{ (i,j) i+j≤ n }.

Place a stochastic vertex with the weight w_p at each box (i,j)∈_n.
Let the initial condition along the left boundary of _n be
the rainbow one, with colors 1 to n from top to bottom.
Then we can sample a random configuration
of pipes as in <Ref>, right,
by running a discrete time Markov chain with time τ=j-i, where
-(n-1)≤τ≤ n-1. At each step τ→τ+1,
the configuration with j-i≤τ
is already sampled, which determines the incoming colors at all boxes
(i,j)∈_n with j-i=τ+1. The next
step τ→τ+1 consists in an independent update of the outgoing colors
at all boxes (i,j)∈_n with j-i=τ+1,
using the stochastic vertex weights
w_p(a,b;·,·)
(<ref>), 1≤ a,b≤ n.
Here we view each w_p(a,b;·,·) as a probability distribution on possible outputs,
(a,b) or (b,a).
Reading off the outgoing colors at the top boundary of _n,
we arrive at a random permutation 𝐰∈ S_n.


	
	The random permutation 𝐰∈ S_n obtained from the colored stochastic six-vertex model
	as described above has the same distribution as the Grothendieck random permutation
	defined in <Ref>.


	The Markov evolution of the stochastic vertex model
	is equivalent to a simultaneous
	exploration of a random pipe dream (that is, determining the state
	-2pt
    < g r a p h i c s >

	or
	-2pt
    < g r a p h i c s >

	at each box),
	and its reduction by following the pipes.
	Indeed, in
	the evolution of the stochastic vertex model,
	in each box (i,j), two strands of different colors meet as incoming pipes.
	There are two possibilities:
	
		
  * If the pipe of the lower numbered color
			is below, then these pipes have already met and crossed through each other
			(which is allowed only once in a reduced pipe dream).
			Thus, regardless of the state
			-2pt
    < g r a p h i c s >

			or
			-2pt
    < g r a p h i c s >

			at (i,j), the pipes must bump off each other.
			In
			(<ref>),
			this corresponds to the vertex weight w_p(a,b;b,a)=1, where 1≤ a<b≤ n.
		
  * If the pipe of the lower numbered color is to the left, then the pipes have not yet crossed
			through each other (but they may have bumped off at an elbow).
			Then, we place the tile
			-2pt
    < g r a p h i c s >

			or
			-2pt
    < g r a p h i c s >

			at (i,j) with probability p or 1-p, respectively.
			After placing the tile,
			the strands deterministically follow the paths drawn on it.
			The choice of the tile is equivalent
			to using the stochastic vertex weights
			w_p(b,a;b,a)=p or w_p(b,a;a,b)=1-p, where 1≤ a<b≤ n.
	
	We see by induction that the vertex model
	(<ref>)
	produces the same random permutation
	as reducing the random pipe dream.




 §.§ Height function and color forgetting



Let x,y∈{ 1,…,n  }.
For a fixed pipe dream D and the corresponding permutation w=w(D)
(see <Ref>),
define the permutation height function as

    H(x,y)
    		   #{pipes of colors ≥ x
    		which exit through positions j≥ y at the top}
    =   #(
    			{ w^-1(x),w^-1(x+1),…,w^-1(n)  }∩{ y,y+1,…,n  }).

Observe that H(x,y) depends only on the permutation
w and not on the structure of crossings and bumps in the pipe dream.
For a fixed w, the function H(x,y) decreases in x and y.
When the pipe dream and the permutation 𝐰
are random, the height function H(x,y) becomes a random variable.

In the permutation matrix of w∈ S_n,
H(x,y) is the number of entries in the rectangle
[y,n]×[x,n]. Recall that the pipe of color i exits at w^-1(i), so the
rectangle is transposed.
See <Ref> for an illustration.
This interpretation of H(x,y) implies that

    H(x,y)≤min(n-x,n-y).


In colored stochastic six-vertex models, quantities like H(x,y) are
referred to as colored height functions, cf. <cit.>.




	
	The color-position symmetry <cit.>
	of the colored stochastic six-vertex model
	implies that the distribution of H(x,y) corresponding to the
	Grothendieck random permutation 𝐰
	is symmetric in x,y.
	Indeed, by the color-position symmetry, 𝐰 and 𝐰^-1
	have the same distribution. Therefore,
	
    H(x,y)
    		=
    		∑_i=x^n∑_j=y^n1_𝐰^-1(i)=jd=∑_i=x^n∑_j=y^n1_𝐰(i)=j
    		= H(y,x).

	However, we can access the distribution of H(x,y) directly
	using color forgetting (<Ref> below), and do not
	rely on
	color-position symmetry.


Consider an uncolored (color-blind) stochastic
vertex model
with all pipes of the same color.
We indicate pipes and empty edges by 0 and 1, respectively.
The weights of the color-blind model are defined as

    w_p^∙(0,0;0,0)   =1,     w_p^∙(1,1;1,1)=1,
    		
    
    		w_p^∙(1,0;1,0)   =p,    
    		w_p^∙(1,0;0,1)=1-p,
    		
    
    		w_p^∙(0,1;0,1)   =0,    
    		w_p^∙(0,1;1,0)=1.



	
	Fix 1≤ x≤ n. In the colored stochastic vertex model of
	<Ref>,
	erase all pipes of colors <x,
	and identify all the remaining pipes for colors ≥ x.
	The resulting random configuration of
	uncolored pipes evolves
	according to
	the color-blind stochastic
	vertex model w_p^∙ (<ref>).



	This is a standard property of the colored stochastic six-vertex model,
	see <cit.> and references therein.
	In interacting particle systems,
	this property is known as the basic coupling, cf.
	<cit.>, <cit.>.
	Let us provide an idea of the proof.

	The crucial property is
	that the weights w_p (<ref>)
	depend only on the relative order of colors of the incoming pipes, and not the actual colors.
	Therefore, when we identify all colors ≥ x, the resolution of the pipe
	crossings does not affect the behavior of the color-blind system.
	In detail, one can consider four cases
	for the incoming pipes (a,b),
	1≤ a,b≤ n,
	depending on whether a≥ x and/or b≥ x.
	The resulting probability distribution on possible outputs,
	(a,b) or (b,a), depends only on these cases, and not on
	the actual values of a,b.
	These four cases correspond to the color-blind incoming pipes
	in w_p^∙ (<ref>).


By <Ref>,
the permutation height function H(x,y)
can be identified with an observable of the model
w_p^∙ with initial occupied configuration
{ x,x+1,…,n   } along the left boundary
(that is, the sites {1,…,x-1} are empty).
Indeed, in this color-blind model, H(x,y) is simply
the
number of pipes exiting through the top boundary at positions ≥ y.
We see that under the color forgetting, the color parameter x in H(x,y) became a parameter of the initial configuration.


	One can similarly forget the colors >x, and the resulting
	color-blind system evolves according to different
	weights (obtained from w_p^∙ by swapping 0↔ 1).
	It is more convenient for us to work with the model w_p^∙,
	which we directly relate
	to an interacting particle system in the next <Ref>.




§ FROM VERTEX MODELS TO TASEP





 §.§ TASEP with moving exit boundary




	
	Let k≥ 1.
	Let
	ξ(t)(ξ_1(t)>…>ξ_k(t) )⊂ℤ
	be the
	k-particle
	discrete time TASEP (Totally Asymmetric Simple Exclusion Process)
	having
	parallel updates
	and geometrically distributed jumps.
	In detail, ξ(t), t∈ℤ_≥0, is a discrete time Markov chain
	on particle configurations in ℤ which at each time step t→ t+1 evolves as follows:
	
    ξ_i(t+1)=ξ_i(t)+min( G_i(t+1), ξ_i-1(t)-ξ_i(t)-1 ),    
    		1≤ i≤ k,

	where G_i(t+1) are independent geometric random variables with parameter p∈(0,1),
	that is,
	
    ( G_i(t+1)=m )=(1-p) p^m,     m∈ℤ_≥0.

	The update (<ref>) occurs in parallel for all
	particles 1≤ i≤ k, that is,
	the new positions ξ_i(t+1) depend only on the configuration
	ξ(t) at the previous time step, and new independent random variables.
	By agreement, we have ξ_0(t)≡ +∞,
	so that the first particle ξ_1(t) performs an independent random walk
	with geometrically distributed jumps.
	See <Ref> for an illustration.




Start the TASEP from
the densely packed (also called step)
initial configuration
{ 1,2,…,k  }, that is,
ξ_i(0)=k+1-i, 1≤ i≤ k.
Fix n≥ k, and introduce a moving exit boundary
which starts at n+1/2, and deterministically
moves to the left with speed 1.
The exit time of a particle ξ_i(t) is defined by

    T_exit(i)min{ tξ_i(t)≥ n+1-t }.

For example, if n=k, then the first particle starting at
n exits at time t=1 (but have not exited at the initial
time t=0). The exit times are almost surely ordered:

    1≤ T_exit(1)<T_exit(2)<…<T_exit(k)≤ n.

Note that tor any n≥ k,
all particles have exited the system by the time t=n.
See <Ref> for an illustration and simulations.







Integrability of TASEP with parallel updates, geometric jumps,
and densely packed initial configuration
can be traced back to
<cit.>, see also
<cit.>,
<cit.>,
and <cit.>.
We recall the necessary
integrability
results in <Ref> below
after connecting TASEP to vertex models and
Grothendieck random permutations
in the next <Ref>.



 §.§ Identification with vertex models



Fix n and 1≤ x,y≤ n. Recall the
observables H(x,y)
defined by (<ref>).
That is, H(x,y) are random variables which are functions of the Grothendieck
random permutation (equivalently, of the colored stochastic six-vertex model).
In <Ref>,
we identified H(x,y) with observables of a color-blind vertex model with weights
w_p^∙ (<ref>).


	
	For any 1≤ x,y≤ n and 0≤ h≤ n-x+1, we have
	
    _𝐰(H(x,y)≤ h)
    			   =
    			_TASEP(T_exit(n-x+1-h)≤ y-1)
    			
       =
    			_TASEP(
    				ξ_n-x+1-h(y-1)≥ n-y+2
    			).

	Here
	_𝐰
	corresponds to the Grothendieck random permutation of order n,
	and
	_TASEP
	is the probability distribution of the TASEP
	with k=n-x+1 particles and moving exit boundary.
	In (<ref>)
	we have, by agreement,
	T_exit(m)= 0 and
	ξ_m(t)= +∞ for all m≤ 0.


	The second identity in (<ref>)
	immediately follows from the definition of the exit times
	(<ref>). Let us focus on the first identity.

	By <Ref>, let us consider the color-blind vertex model
	with k=n-x+1 identical pipes entering the left boundary
	of the staircase shape _n
	at locations { x,x+1,…,n  }.
	The event { H(x,y)≤ h } means that
	at most h of these pipes exit the top boundary at positions ≥ y.
	Note that the color-blind model has indistinguishable pipes, and thus the “resolution”
	of their crossings
	does not change the behavior of the system (see the proof of <Ref> for more detail).

	View the horizontal coordinate j
	as time t=0,1,…,n,
	and record the pipes' coordinates as
	
    η_1(t)<…<η_k(t),     t=0,1,…,n .

	The initial condition is η_k-m+1(0)=n-m+1, 1≤ m≤ k.

	At each time step, according to the w_p^∙ (<ref>),
	each pipe first deterministically turns up.
 	When a pipe faces up, it can travel up a further
	distance distributed as a geometric random variable with parameter p,
	see (<ref>).
	However, during a time step t→ t+1, each pipe η_m cannot
	travel further than the previous pipe' location η_m-1(t)
	(this is the exclusion mechanism).
	When a pipe stops moving up, it turns right and waits till the next time step.
	Once a pipe reaches the top boundary, it exits the system.
	See <Ref>, left, for an illustration.



	

	Setting ξ_m(t) n+1-t-η_m(t),
	one readily verifies that the evolution of ξ_1(t)> … > ξ_k(t)
	is the same as that of the TASEP
	with k=n-x+1 particles
	defined in
	<Ref>.
	In particular, subtracting t from n+1-η_m(t)
	eliminates the deterministic up movement of the pipes by one at each time step.

	The top boundary for pipes (as in <Ref>, left)
	becomes the moving exit boundary for TASEP. The event
	{H(x,y)≤ h} for pipes is equivalent to the event
	that at least k-h=n-x+1-h of the TASEP particles
	have exited before time y-1. Because the
	exit times are ordered as in (<ref>),
	we get the desired first identity in (<ref>).




 §.§ Integrability of TASEP



Fix m≥ 0, and let λ=(λ_1≥…≥λ_m≥ 0 )
be a partition with at most m parts.
The Schur symmetric polynomial in m variables indexed by λ is defined as

    s_λ(a_1,…,a_m)[a_i^λ_j+m-j]_i,j=1^m/[a_i^m-j]_i,j=1^m,

where a_i are variables. The denominator is the Vandermonde determinant
∏_1≤ i<j≤ m(a_i-a_j).

Let m,t≥1.
The Schur measure <cit.>
is a probability measure on partitions with at most min(m,t) parts
depending on parameters a_1,…,a_m,b_1,…,b_t. Its probability weights
are defined as

    (λ)∏_i=1^m∏_j=1^t
    	(1-a_ib_j)
    	·
    	s_λ(a_1,…,a_m)
    	s_λ(b_1,…,b_t).

When m=t=1, the weights (<ref>) define the geometric distribution
on λ=(λ_1≥0) with parameter a_1b_1.
In general, the form of the normalizing constant in (<ref>)
follows from the Cauchy summation identity for Schur polynomials.
The infinite sum
for the normalizing constant runs
over all λ of length ≤min(m,t) and converges if |a_ib_j|<1 for all i,j.

The next statement connects TASEP to Schur measure,
which provides the necessary integrability structure for the former.


	
	For any t,m≥1, the displacement of the m-th particle
	in the TASEP with parallel updates, geometric jumps with parameter p,
	and densely packed initial configuration (as in <Ref>)
	has the same distribution as λ_m, the last part of a partition
	under the Schur measure
	(<ref>)
	with a_1=…=a_m=1 and b_1=…=b_t=p:
	
    ξ_m(t)-ξ_m(0)d=λ_m,
    		    λ∼
    		(1-p)^m t s_λ(1,…,1 _m)
    		s_λ(p,…,p _t).



<Ref> is a well-known
result in Integrable Probability.
It follows either from the
Robinson–Schensted–Knuth (RSK) correspondence with column insertion,
or by representing the TASEP as a
marginally Markovian evolution of a process on the space of semistandard Young tableaux
(equivalently, interlacing arrays / Gelfand–Tsetlin patterns).

For the simpler continuous time TASEP (when each particle waits an exponentially
distributed time and jumps to the right by 1 if the destination
is unoccupied), the connection to Schur measures is
exploited in the celebrated work <cit.>
via last-passage percolation, which is related to the row RSK.
The RSK approach can be traced back to <cit.>.
Dynamics on interlacing arrays are constructed later in
<cit.> using a non-RSK approach.

For our discrete time TASEP with geometric jumps,
the RSK was explicitly used in
<cit.>
to establish <Ref>.
A systematic treatment of discrete time
dynamics on interlacing arrays connected to Schur measures
leading to various TASEPs and PushTASEPs may be found in
<cit.>.
It includes
RSK-type and other dynamics on interlacing arrays under one roof,
as well as a generalization of the Schur structure to the level of q-Whittaker
symmetric polynomials.



 §.§ Determinantal structure



The connection
to Schur measures (<Ref>)
allows to extract law of large numbers and
fluctuation results for the TASEP particles.
The key property of Schur measures which makes this possible is
the fact that they form a determinantal point process.

Fix m,t≥ 1, and define a random point configuration
X(λ){λ_j+m-j}_j=1^m on ℤ_≥0,
where λ is distributed as (<ref>).
If λ has less parts than m, append it by zeros.
These zeros translate into a part of X(λ) of the form
{0,1,…,l } for some l.


	
	The random point configuration X(λ) on ℤ_≥0 is a
	determinantal point process. This means that for
	any r≥1 and pairwise distinct points u_1,…,u_r ∈ℤ_≥0,
	the correlation functions have the form
	
    (X(λ) contains all of the points u_1,…,u_r )=
    		[ K(u_i,u_j) ]_i,j=1^r.

	The correlation kernel K is given by a double contour integral:
	
    K(u_1,u_2)=1/(2π𝐢)^2dz  dw/z-ww^u_2-m/z^u_1-m+1(1-p/z)^m/(1-z)^t(1-w)^t/(1-p/w)^m,
    		    
    		u_1,u_2∈ℤ_≥0,

	where the contours are positively oriented simple closed curves satisfying
	p<|w|<|z|<1.


<Ref> is due to
<cit.>,
see also
<cit.>.
General discussions and many examples of determinantal point processes
may be found
in <cit.>, <cit.>,
or
<cit.>.




 §.§ Asymptotics of TASEP



Determinantal structure powers the asymptotic behavior of
the TASEP particles.
We are interested in
probabilities of the form

    _TASEP(ξ̅_m(t)≥ u),
    	    where  ξ̅_m(t)ξ_m(t)-ξ_m(0).

Indeed, since
<Ref>
reduces the distribution of H(x,y) to
probabilities of the form (<ref>),
we can forget about the moving exit boundary in TASEP.
Indeed, the parameter n of the moving boundary is absorbed into m and u.



Let the parameters
t,m,u grow to infinity proportionally to each other:

    m=⌊ L𝗆⌋ ,     
    	t=⌊ L𝗍⌋ ,     
    	u=⌊ L𝗎⌋ ,
    	      L→∞.

First, let us heuristically discuss the law of large numbers and fluctuations
of
_TASEP(ξ̅_m(t)≥ u),
without providing exact constants.
The latter are given in <Ref> below,
and we formulate the asymptotic results in <Ref>.
The proof of <Ref> using the determinantal structure
is a standard application of the steepest descent method for double contour integrals.
We provide the computations (leading to the constants in <Ref>)
in
<Ref>.



At the “hydrodynamic” scale (<ref>),
TASEP has a limit shape
(first observed in the continuous time TASEP in <cit.>, see also
<cit.>). That is, there exists
a function 𝖼(𝗆,𝗍) such that
for any ε>0, we have

    lim_L→∞_TASEP(ξ̅_m(t)≥ L𝖼(𝗆,𝗍)+ε)
    	=0,
    	    lim_L→∞_TASEP(ξ̅_m(t)≥ L𝖼(𝗆,𝗍)-ε)
    	=1.

In other words, we have convergence in probability:

    lim_L→∞
    	L^-1ξ̅_⌊ L𝗆⌋(⌊ L𝗍⌋ )
    	=𝖼(𝗆,𝗍).

Let us note a certain a priori property of the limit shape:

	
	If 𝗍<𝗆/p, we have 𝖼(𝗆,𝗍)=0.


	This follows from the fact that the leftmost hole in the TASEP configuration
	(which starts immediately to the right of ξ_1(0) at time t=0)
	travels left by at most 1 at each time step.
	The probability that the hole moves is p=(G>0),
	where G is the geometric random variable (<ref>).
	Therefore, if 𝗍<𝗆/p, then the hole has not reached the
	m-th particle by time t with probability 1-e^-C L for some C>0.
	This implies the claim.


Next, since TASEP belongs to the Kardar–Parisi–Zhang (KPZ) universality class
<cit.>, we expect
fluctuations of order L^1/3 provided that 𝗍>𝗆/p.
More precisely,
modifying m and u on the scale
L^1/3 probes the scaling window in the law of large numbers
(<ref>). That is,
there exist constants
𝗏_1(𝗆,𝗍),
𝗏_2(𝗆,𝗍)>0, such that

    lim_L→∞_TASEP( ξ̅_⌊ L𝗆- L^1/3α𝗏_1(𝗆,𝗍)⌋
    		(⌊ L𝗍⌋ )≥
    	L𝖼(𝗆,𝗍)-
    	L^1/3β𝗏_2(𝗆,𝗍))
    	=F_2( α +β)

for all α,β∈ℝ.
Here, F_2 denotes the cumulative distribution function of
the Tracy–Widom GUE distribution. The plus signs
by α and β in the right-hand side of (<ref>)
are straightforward from the monotonicity of the pre-limit probabilities.

The distribution function F_2
was first discovered
in connection with the fluctuations of the largest
eigenvalue in large complex Hermitian random matrices
exhibiting unitary symmetry <cit.>.
Subsequently,
F_2 was put into a larger class of limiting distributions arising in random
growth models and interacting particle systems, known as the KPZ universality class.
We refer to the surveys <cit.>,
<cit.>,
<cit.> for a detailed exposition.
The appearance of F_2 in the fluctuations (out of several
candidates in the KPZ universality class)
is a feature of the step initial
configuration in our TASEP.


	
	Let 𝗆,𝗍>0. Define
	
    𝖼(𝗆,𝗍)
    		
    			0,
    			   if 𝗍≤𝗆/p;
    			
    (√(p𝗍)-√(𝗆))^2/1-p,
    			   if 𝗍≥𝗆/p.

	If 𝗍>𝗆/p, set
	
    𝗏_1(𝗆,𝗍)√(p)𝗆^1/3/𝗍^1/6(√(𝗍/p)-√(𝗆))^2/3/(√(p𝗍)-√(𝗆))^1/3,
    		    𝗏_2(𝗆,𝗍)√(p)
    		(√(p𝗍)-√(𝗆))^2/3
    		(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6(1-p)
    		.




	
	Let the constants 𝖼(𝗆,𝗍) and 𝗏_1,2(𝗆,𝗍) be
	given in <Ref>. Then
	
		
  * The law of large numbers (<ref>) holds for all
			𝗆,𝗍>0.
		
  * The Tracy–Widom fluctuation result (<ref>) holds
			under the condition 𝗍>𝗆/p>0.
	

The proof of <Ref> is given in <Ref>.



§ LIMIT BEHAVIOR OF GROTHENDIECK RANDOM PERMUTATIONS





 §.§ Law of large numbers for the permutation height function



In this section, we obtain the asymptotic behavior
of Grothendieck random permutations of growing order
via the
the pre-limit identity in distribution (<Ref>)
and
TASEP asymptotics (<Ref>).
As a result, we complete the proof of <Ref>
formulated in the Introduction.
First, we focus on
computations involving the law of large numbers, and
then add the fluctuation terms.
By <Ref>, we have

    _𝐰(H(x,y)≤ h)
    	=
    	_TASEP(
    		ξ_n-x+1-h(y-1)≥ n-y+2
    	),

where H is the height function of the Grothendieck random
permutation 𝐰, and
ξ_i(t) are the TASEP particles
with
the step initial configuration
ξ_i(0)=n-x-i+2, 1≤ i≤ n-x+1.
Therefore,

    _𝐰(H(x,y)≤ h)
    	=
    	_TASEP(
    		ξ̅_n-x+1-h(y-1)≥ n-y+1-h
    	),

where ξ̅_i(t)=ξ_i(t)-ξ_i(0) is the displacement.

Let h=⌊ n𝗁⌋, where 𝗁 is the
scaled permutation height function.
Let also
x=⌊ n𝗑⌋, y=⌊ n𝗒⌋,
where 𝗑,𝗒∈[0,1] are the scaled coordinates in the permutation matrix.
Note that
by (<ref>), we have
0≤𝗁≤min(1-𝗑,1-𝗒).

For each 𝗑,𝗒,𝗁, the
asymptotic location
of n^-1ξ̅_n-x+1-h(y-1) is determined
from the TASEP law of large numbers (<ref>), (<ref>).
Namely, this location is equal to

    𝖼(1-𝗑-𝗁,𝗒)=
    	(√(p𝗒)-√(1-𝗑-𝗁))^2/1-p1_p𝗒≥ 1-𝗑-𝗁
    	.

There are two cases.
If

    𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁,

then the probability (<ref>) goes to 1.
If the inequality is reversed, then this probability goes to 0.
As 𝗁 increases,
the left- and the right-hand sides of (<ref>)
increase and decrease, respectively.
Therefore, we should define

    𝗁^∘(𝗑,𝗒)inf{𝗁∈[0,min(1-𝗑,1-𝗒)]𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁}.

If the set in (<ref>) is empty, we
take the maximal possible value,
𝗁^∘(𝗑,𝗒)=min(1-𝗑,1-𝗒).

To describe 𝗁^∘ more explicitly,
introduce the northeast part of the ellipse

    ℰ_p{
    		(𝗑,𝗒)
    		
    		(𝗒-𝗑)^2/p+(𝗒+𝗑-1)^2/(1-p)=1,		1-p≤𝗑≤ 1
    	}⊂ [0,1]^2.





	
	The function
	𝗁^∘
	(<ref>) is given by
	
    𝗁^∘(𝗑,𝗒)
    		=
    		
    			1-𝗒,   (𝗑,𝗒)∈𝒜{ 0<𝗑<1-p, 𝗑/(1-p)<𝗒<1 }
    			;
    
    			1-𝗑,   
    			(𝗑,𝗒)∈ℬ{ 0<𝗑<1, 0<𝗒<(1-p)𝗑}
    			;
    
    			1+
    			2/p√((1-p) 𝗑𝗒)
    			-𝗑+𝗒/p
    			,   (𝗑,𝗒)∈𝒞{.380<𝗑<1, (1-p)𝗑<𝗒<𝗑/(1-p),
    			
    (𝗑,𝗒)belowℰ_p}
    			;
    
    			0,   
    			(𝗑,𝗒)∈𝒟{(𝗑,𝗒) above ℰ_p}
    			.

	It
	is continuous on [0,1]^2.

See <Ref>, left, for an illustration
of the subsets 𝒜,ℬ,𝒞,𝒟
in (<ref>). The graph of the function
𝗁^∘ is given in
<Ref>, right.

	In 𝒜 and 𝒞,
	we have
	𝖼(1-𝗑-𝗁^∘,𝗒)= 1-𝗒-𝗁^∘.
	Solving this for 𝗁^∘ leads to the result.
	In 𝒟, we have
	𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁
	for all 𝗁≥0, so 𝗁^∘(𝗑,𝗒)=0.
	In ℬ,
	we have
	𝖼(1-𝗑-𝗁,𝗒)< 1-𝗒-𝗁
	for all 0≤𝗁≤ 1-𝗑.
	This means that
	the set over which we take the infimum in
	(<ref>) is empty.
	This implies that
	𝗁^∘(𝗑,𝗒)=
	1-𝗑 in ℬ, which
	completes the proof.


<Ref>
and identity (<ref>)
immediately imply the law of large numbers for the
height function:

    lim_n→∞n^-1
    	H
    	( ⌊ n𝗑⌋,
    	⌊ n𝗒⌋)
    	=𝗁^∘(𝗑,𝗒),
    	    𝗑,𝗒∈[0,1],

with convergence in probability.
This completes the proof of the first part of
<Ref>
from the Introduction.



 §.§ Properties of the limiting permuton



The law of large numbers
(<ref>)
implies that the Grothendieck random permutations 𝐰∈ S_n
converge, as n→∞, to a (deterministic) permuton
prescribed by
𝗁^∘(𝗑,𝗒).
Recall that a permuton is a probability measure
on [0,1]^2 with uniform marginals.
We refer to
<cit.>,
<cit.>,
<cit.> for detailed treatment of permutons.

In detail, the permuton is connected to the height function as

    𝗁^∘(𝗑,𝗒) =
    	( X>𝗑, Y>𝗒),
    	    𝗑,𝗒∈[0,1],

where (X,Y)∈[0,1]^2 is a random point distributed according to the permuton.
Note that both X and Y are uniformly distributed on [0,1],
while their joint distribution is nontrivial.
Indeed, from (<ref>)
if follows that
𝗁^∘(𝗑,0) = 1-𝗑,
𝗁^∘(0,𝗒) = 1-𝗒,
as it should be for uniform marginals of a permuton.

Let is find the part of the permuton
that is concentrated on
the curve
ℰ_p (<ref>)
(this concentration is visible in simulations, see <Ref>).
For small ε>0, we have

    𝗁^∘(𝗑,𝗒-ε)-𝗁^∘(𝗑,𝗒)
    	=( X>𝗑, 𝗒-ε<Y≤𝗒).

Dividing this by ε=(𝗒-ε<Y≤𝗒),
taking the limit as ε→0,
and passing to the complement event,
we obtain the conditional cumulative distribution function (cdf) of X given Y=𝗒:

    (X≤𝗑| Y = 𝗒)=
    	1+
    	∂/∂𝗒𝗁^∘(𝗑,𝗒)=
    	
    		0,    (𝗑,𝗒)∈𝒜;
    
    		1,    (𝗑,𝗒)∈ℬ∪𝒟;
    p-1p+√((1-p)𝗑𝗒)p𝗒,    (𝗑,𝗒)∈𝒞.

This function has a jump discontinuity along the ellipse ℰ_p (<ref>),
that is, at the point

    𝗑_p(𝗒)
    	
    	p+𝗒
    	-2 p 𝗒+
    	2 √((1-p)  (1-𝗒) p𝗒),
    	    𝗒>1-p.

We conclude:

	
	Let 𝗒>1-p.
	The unique discontinuity of the conditional distribution of X given Y=𝗒 is
	an atom
	at 𝗑=𝗑_p(𝗒), and the atom's mass is
	
    ( X = 𝗑_p(𝗒)| Y = 𝗒)=
    			1/p
    			-[
    				(1-p) (p+𝗒
    			-2 p 𝗒+
    			2 √((1-p)  (1-𝗒) p𝗒))
    	]^1/2/p√(𝗒),

	where
	1-p<𝗒≤ 1.


In particular, from (<ref>) we have
( X = 1-p| Y = 1)
=1, as it should be.


	
	While conditionally on Y=𝗒 (where 𝗒>1-p),
	the distribution of X has an atom,
	the permuton (that is, the joint distribution of X and Y)
	does not have atoms. This follows from the fact that
	𝗁^∘(𝗑,𝗒)
	(<ref>) is continuous on [0,1]^2.


As an application, let us consider the law of large numbers of the number of inversions
(𝐰)#{i<j𝐰(i)>𝐰(j)}, where
𝐰∈ S_n is the Grothendieck random permutation.
By a result of
<cit.> (see also Theorem 1 in the survey <cit.> or
<cit.>),
the convergence of 𝐰∈ S_n to a permuton
is equivalent to the convergence (in probability) of all pattern counting statistics
of 𝐰.
The limiting pattern counts are determined by the permuton.
In particular,
for inversions, we have

    lim_n→∞(𝐰)/n2=
    	γ_p
    	[ (X_1>X_2,  Y_1<Y_2)  or  (X_1<X_2,  Y_1>Y_2) ],
    	     p∈(0,1),

where (X_1,Y_1) and (X_2,Y_2) are independent
points of [0,1]^2
sampled from the limiting permuton.


	
	We have γ_1/2=1-π/4.

The constant γ_p for general p can be expressed
as a sum of a double and a single
integral of algebraic functions,
and we were not able to evaluate it in closed form except for p=1/2.
The integrals for general p are given in the proof below.

	Fix (𝗑,𝗒)∈[0,1]^2, and let (X,Y)∈[0,1]^2
	be a random point sampled from the permuton.
	We have
	
    [ (X>𝗑,  Y<𝗒)  or  (X<𝗑,  Y>𝗒) ]
    		=1-𝗁^∘(𝗑,𝗒)-
    		𝗁^∙(𝗑,𝗒),

	where 𝗁^∘(𝗑,𝗒)=(X>𝗑,  Y>𝗒) is given by
	(<ref>), and
	
    𝗁^∙(𝗑,𝗒)(X≤𝗑,  Y≤𝗒).

	In (<ref>),
	we used the absence of atoms
	(<Ref>) and
	ignored the difference between strict and weak inequalities.
	The function 𝗁^∙ can be computed using the
	conditional distribution (<ref>):
	
    𝗁^∙(𝗑,𝗒)=
    		∫_0^𝗒(X≤𝗑| Y = w) dw=
    		𝗑,    (𝗑,𝗒)∈𝒜;
    𝗒,    (𝗑,𝗒)∈ℬ;
    p-1/p(𝗑+𝗒)+
    			2/p√((1-p)𝗑𝗒),
    			    (𝗑,𝗒)∈𝒞;
    𝗑+𝗒-1,    (𝗑,𝗒)∈𝒟.


	By (<ref>),
	the constant γ_p (for general p) is given by
	
    γ_p=
    		𝔼[ 1 - 𝗁^∘(X,Y) - 𝗁^∙(X,Y) ],

	where the expectation is taken with respect to the permuton.
	To evaluate this expectation, we use the conditional density
	of X given Y=y which is obtained from
	(<ref>):
	
    f_X| Y=𝗒(𝗑)=
    		1/2p√(1-p/𝗑𝗒)1_(𝗑,𝗒)∈𝒞.

	Recalling that for Y=𝗒>1-p the conditional distribution of X has an atom,
	we can express the expectation as follows:
	
    γ_p=
    			1   -
    			∬_𝒞(
    				𝗁^∘(𝗑,𝗒)+𝗁^∙(𝗑,𝗒)
    			)
    			f_X| Y=𝗒(𝗑) d𝗑 d𝗒
       -
    			∫_1-p^1
    			( 𝗁^∘(𝗑_p(𝗒),𝗒)+
    				𝗁^∙(𝗑_p(𝗒),𝗒)_𝗑_p(𝗒)+
    				𝗒-1)
    			( X=𝗑_p(𝗒)| Y=𝗒) d𝗒.

	For p=1/2, a
	computer algebra system
	like 
	is able to find explicit anti-derivatives, and
	evaluate these integrals.
	We have:
	
    ∬_𝒞4  √(2𝗑𝗒) - 3 𝗑 - 3 𝗒 + 1/√(2𝗑𝗒)  d𝗑 d𝗒=π+2/8,

	and
	
    ∫_1/2^1
    		(
    			𝗒 + √((1 - 𝗒) 𝗒) - 12)
    		(
    		2 - 𝗒^-1√(𝗒+2 𝗒√((1-𝗒 ) 𝗒)))  d𝗒=π-2/8.

	Combining the last two integrals, we obtain γ_1/2=1-π/4.
	This completes the proof.




 §.§ Fluctuations of the permutation height function



Let us now apply the second part of
<Ref> to the Grothendieck random permutations, and
obtain fluctuations of the height function
around its limit shape 𝗁^∘(𝗑,𝗒).
Throughout this subsection, we assume that the point (𝗑,𝗒)
belongs to the curved triangle 𝒞
where the height function 𝗁^∘ is not linear.

Recall that by H(x,y) we denote the random height function of the
Grothendieck random permutation 𝐰∈ S_n.
By (<ref>), we have
for all r∈ℝ:

    _𝐰(
    			H(⌊ n𝗑⌋,⌊ n𝗒⌋)
    			≤
    			n𝗁^∘+ r n^1/3)
    		
       =
    		_TASEP(
    			ξ̅_⌊ n(1-𝗑-𝗁^∘-r n^1/3) ⌋
    				(⌊ n𝗒⌋ )
    				≥
    				n(1-𝗒-𝗁^∘-r n^1/3)
    		)+o(1),

where we used the shorthand notation

    𝗁^∘
    	=
    	𝗁^∘(𝗑,𝗒)
    	=
    	1+
    	2/p√((1-p) x y)
    	-x+y/p
    	.

The term o(1) in (<ref>)
is due to the fact that we removed
shifts by (+1), and combined the integer parts. These
modifications are negligible because fluctuations live on the scale n^1/3.

By the fluctuation result for TASEP,
the probability in the right-hand side of
(<ref>)
has the following behavior:

    lim_n→∞_TASEP(
    			ξ̅_⌊ n(1-𝗑-𝗁^∘-r n^1/3) ⌋
    				(⌊ n𝗒⌋ )
    				≥
    				n(1-𝗒-𝗁^∘-r n^1/3)
    			)
       =
    		F_2( r( 1/𝗏_1(1-𝗑-𝗁^∘,𝗒)
    		+
    		1/𝗏_2(1-𝗑-𝗁^∘,𝗒)) ),

where F_2 is the Tracy–Widom GUE cumulative distribution function,
and the constants 𝗏_1,𝗏_2 are given by
(<ref>).
We have

    1/𝗏_1(𝗆,𝗍)
    	+
    	1/𝗏_2(𝗆,𝗍)=
    	(√(𝗍)-√(𝗆 p)) 𝗍^1/6/𝗆^1/3
    	(√(𝗍/p)-√(𝗆))^2/3
    	(√(𝗍 p)-√(𝗆))^2/3.

Let us denote the above expression with
𝗆=1-𝗑-𝗁^∘(𝗑,𝗒)
and 𝗍=𝗒 by

    𝗏(𝗑,𝗒)(√(𝗒)-√(p(1-𝗑-𝗁^∘(𝗑,𝗒))))
    	𝗒^1/6/(1-𝗑-𝗁^∘(𝗑,𝗒))^1/3
    	(√(𝗒/p)-√(1-𝗑-𝗁^∘(𝗑,𝗒)))^2/3
    	(√(𝗒 p)-√(1-𝗑-𝗁^∘(𝗑,𝗒)))^2/3,

where 𝗁^∘(𝗑,𝗒) is given in (<ref>).
While it is not evident from (<ref>),
inside 𝒞
the function 𝗏(𝗑,𝗒)
is symmetric in 𝗑,𝗒.

With the constant 𝗏(𝗑,𝗒), the convergence
in (<ref>)
implies the desired
Tracy–Widom GUE fluctuations of the height function
H(x,y).
This completes the proof of
<Ref>.



§ RANDOM PERMUTATIONS FROM NON-REDUCED PIPE DREAMS



Here we consider a different family of random permutations
which is
obtained by sampling a non-reduced pipe dream and not resolving any double crossings.
This corresponds to setting q=1 in <Ref>
from the Introduction.
For shorter notation, throughout this section we denote these random permutations
𝐰^(1) by 𝐰.



 §.§ Exact formula



Consider the staircase shape
_n= { (i,j) i+j≤ n }
(where i and j are the row and column indices, respectively)
and
place tiles
-2pt
    < g r a p h i c s >

or
-2pt
    < g r a p h i c s >

in each box of _n independently with probabilities p and 1-p, respectively.
Let the random permutation 𝐰∈ S_n
is obtained by following the pipes starting at (i,0), and ending at (0,𝐰_i^-1),
i=1,…,n.
For example, for the pipe dream in <Ref>, this procedure gives
w=241635.

Observe that each pipe is a random walk in the bulk of
_n, equipped with the mandatory turns by
90^∘ at the elbows at the diagonal boundary i+j=n.
We begin by deriving an exact formula for the probability
distribution of 𝐰_i^-1, the outgoing position
of the i-th pipe. For general p, this distribution
involves two Gauss hypergeometric functions _2F_1, which
for p=1/2 simplifies to a sum of binomial
coefficients. We do not use these exact formulas for
asymptotic analysis, but instead, in
<Ref> below, we obtain bounds on
the expected number of inversions using the random walk
approach.






Recall the Gauss hypergeometric function

    _2F_1(a,b;c;z) = ∑_k≥ 0 (a)_k (b)_k/(c_k)z^k/k!,

where (a)_k = a(a+1)⋯ (a+k-1). Set

    F_1(i,j)   
    		(1-p)p^i+j-2_2F_1(-i+1,-j+1;1; (1/p-1)^2)
    		
       = ∑_k=0^∞i-1kj-1k (1-p)^2k+1 p^i+j-2-2k;
    
    		F_2(s,n)     n (1-p)^2p^n+s-3_2F_1(-s+2,-n+1,2, (1/p-1)^2 )
    		
       = ∑_k=0^∞s-2knk+1 (1-p)^2k+2 p^n+s-3-2k.

Note that both hypergeometric series terminate after finitely many terms.
For p=1/2, these functions simplify to binomial coefficients:

    F_1(i,j)|_p=1/2
    	=2^-i-j+1i+j-2j-1,
    	    
    	F_2(s,n)|_p=1/2
    	=2^-s-n+1n+s-2n-1.



	
	We have for all i,j=1,…,n:
	
    (𝐰^-1_i=j) = F_1(i,j) +
    			p^n 1_i+j=n+1
    			+
    			F_2(i+j-n,n)1_i+j>n+1.



    We proceed by induction on i.
		For i=1, we have
		
    (𝐰^-1_1=j) =
        
        (1-p) p^j-1,     if  1≤ j<n;
    
    		p^n-1,     if  j=n,

    which coincides with the proposed formula. Now let i>1.
		Let X_i^(n) denote the exit position of the pipe started at (i,0),
		assuming that all squares on the diagonal i+j=n+1 are elbows.
		By considering where this pipe crosses the first row,
		we can represent
		X_i^(n)=X_i-1^(n-1) + Y,
		where Y depends on X_i-1^(n-1) and has the following
		conditional distribution
		given X_i-1^(n-1):
		
    (Y=m| X_i-1^(n-1)) =
    			
    				p,    m=0;
    
    				(1-p)^2  p^m-1,    1≤ m< n-X_i-1^(n-1);
    
    				(1-p)  p^n-X_i-1^(n-1)-1,    m=n-X_i-1^(n-1).

		Since 𝐰^-1_i=X_i^(n),
		we can write
    
    (X_i-1^(n-1)=j)· p+∑_r< j(X_i-1^(n-1)=r) ·(𝐰^-1_i =j)
    					
       
    					= 
    					(1-p)^2 p^j-r-1,     j<n;
    ∑_r< n(X_i-1^(n-1)=r)· (1-p)  p^n-r-1,      j=n.

		Using the induction hypothesis,
		we know the distribution of X_i-1^(n-1)
		in terms of F_1 and F_2.
		Therefore, it remains to verify the recursion
		(<ref>)
		for the answer (<ref>).
		This is easily done by grouping binomial coefficients,
		summing over r, and using the hockey-stick identity.





 §.§ Estimates from random walks



We are interested in the asymptotics of the (expected) number of inversions
(𝐰) in the random permutation 𝐰 obtained from the non-reduced pipe dream model.
First, we use the following known bound in terms of a
displacement (disarray)
dis(w) ∑_i=1^n |i-w_i|
=
∑_i=1^n|i-w_i^-1|:


	For any permutation w, we have
	 
    1/2dis(w) ≤(w) ≤dis(w).


Taking the expectation of (<ref>) and using linearity,
we see that it suffices to understand the asymptotic behavior
of 𝔼[|i-𝐰_i^-1|] for all i.
In principle, the exact distribution of 𝐰_i^-1
from <Ref>
should allow us to compute this expectation.
However, this is not straightforward.
Instead, we use the random walk interpretation of each individual
pipe:


	Fix ε>0 and let i<(1-ε)n. Then
	
    𝔼[|i-𝐰_i^-1|]
    		=
    		√(4i/πp/1-p) + O(1),     n→∞.



	Fix i. Let J_i,J_i-1,…,J_1,J_0
	be the (random) column coordinates of the positions of the i-th pipe
	in row i,i-1,…,1,0.
	In particular, J_i=0 and J_0=𝐰^-1_i.
	For example, in the pipe dream in <Ref>, left, we have
	J_3=0, J_2=3, J_1=3, and J_0=5 for i=3.

	We aim to upper bound the probability that
	the pipe reaches the diagonal i+j=n first time at row k_0, that is,
	J_k_0=n-k_0. Denote this event by R_k_0,
	and observe that (R_a)≥(R_b) for a<b.
	Before the pipe reaches the diagonal,
	the horizontal displacements J_k-J_k+1, k_0<k<i-1, are iid (independent
	identically distributed), and are distributed as
	
    ( {J_k-J_k+1=m}∩ R_k_0)=p1_m=0
    		+
    		(1-p)^2p^m-11_m≥ 1,     m∈ℤ_≥ 0,
    		   k_0<k<i-1.

	One readily checks that the
	expectation and variance of the random variable in the right-hand side are equal to 1
	and 2p/1-p, respectively.
	We can represent
	
    J_k_0=
    		J_k_0-J_k_0+1+J_i-1+∑_k=k_0+1^i-2( J_k-J_k+1).

	For the event R_k_0 to occur, the above sum must be
	equal to n-k_0=i-k_0+(n-i)>i-k_0+n ε.
	Since the expectations of the iid summands are equal to 1,
	by a standard large deviation estimate, this probability
	is upper bounded by e^-c_ε n for suitable
	c_ε(k_0)>0.[Note that if i-k_0≪ n, the actual bound is even stronger than this, but we do not need this precision.]
	By monotonicity, we can choose these constants such that c_ε(k_0)≥ C_ε c_ε(0)>0
	for all k_0.
	Taking the union over all k_0, we see that the probability
	that a pipe started at i ever reaches the diagonal
	is exponentially small.

	Therefore, J_0=𝐰^-1_i
	is close (with exponentially small error in probability)
	to a sum of i iid random variables
	J_k-J_k+1.
	By the Central Limit Theorem,
	this sum is approximately normal with mean i and variance
	2ip/(1-p). Subtracting i from 𝐰^-1_i, taking the absolute value,
	and using the fact that 𝔼( |Z| )=√(2/π)
	for standard normal Z, we obtain the result.


We can now bound the expected number of inversions in 𝐰 on the scale
n^3/2:

	
	Fix p∈[0,1).
	For every ε
		>0 and sufficiently large n, we have
		
    2/3√(π) (1-ε)  n^3/2√(p/1-p)≤𝔼[(𝐰)] ≤4/3√(π) (1+ε)  n^3/2√(p/1-p).



		For p=0, the permutation is identity with probability 1,
		so the bounds
		(<ref>) hold trivially.

    Let p>0 and let us calculate 𝔼[dis(𝐰)] to apply <Ref>.
		With <Ref> for i< (1-ε)^2/3n, we obtain
		for large n:
    
    𝔼[dis(𝐰)] =∑_i=1^n 𝔼[|𝐰_i^-1-i|] ≥∑_i=1^(1-ε)^2/3n2/√(π)√(p/1-p)√(i)≥2/√(π)√(p/1-p)2/3 n√(n) (1-ε),

    where the last inequality is a simple Riemann sum approximation. The upper bound follows similarly, this time extending the Riemann sum to n.


Numerical simulations suggest the following behavior of the
number of inversions:

	
	For any p∈[0,1), we have convergence in probability:
	
    lim_n→∞(𝐰)/n^3/2 = ϰ√(p/1-p).


Note that the bounds
2/(3√(π))≈ 0.376 and 4/(3√(π))≈ 0.752
in <Ref>
are at the same time bounds on ϰ.
Simulations show that ϰ is close to 0.5, but is not exactly equal to it.

The fact that the number of inversions lives on scale
n^3/2
implies a trivial permuton limit of 𝐰.
Denote
by id the
deterministic identity permuton
supported on the diagonal of [0,1]^2.


	
	For any fixed p∈ [0,1), the random permutations 𝐰∈ S_n converge in distribution
	to id as n→∞.


	For any τ∈ S_k and w∈ S_n, k< n, denote by
	t(τ,w)
	<cit.>
	the relative frequency of
	the pattern τ in w.
	This is simply the number of times the pattern τ appears in w,
	divided by nk.
	In particular, t( 21, w)=(w)/ n2.
	We have for any 2<ℓ < n
	<cit.>:
	
    t(21, w) = ∑_τ∈ S_ℓt(21, τ) t(τ, w).

	The term in the sum with τ=id vanishes since t(21, id)=0.
	For our random permutations
	𝐰∈ S_n, the left-hand side of (<ref>)
	converges to zero in probability by <Ref>
	and Markov inequality:
	
    (
    			(𝐰) ≥δ n^2)≤ C_δ n^-1/2→ 0,     n→∞

	for all δ>0.
	For a fixed ℓ, this implies that
	t(τ,𝐰)→0,
	n→∞,
	in probability
	for all idτ∈ S_ℓ, ℓ≥3. We see that all relative frequencies
	converge to zero except for t(id, 𝐰)→ 1.
	Therefore,
	by
	<cit.>,
	we get the result.


This completes the proof of <Ref> from the Introduction.



§ MAXIMAL PRINCIPAL SPECIALIZATIONS OF GROTHENDIECK POLYNOMIALS





 §.§ Grothendieck polynomials via bumpless pipe dreams



In this subsection, we outline another
combinatorial model for Schubert and Grothendieck polynomials
based on bumpless pipe dreams. Some of these definitions and results
go back to
<cit.>, and are described in detail in
<cit.> and <cit.>.

A bumpless pipe dream is a
tiling D of the n× n square
with six types of tiles:

    SE bump  -2pt
        < g r a p h i c s >
    ,   NW bump  -2pt
        < g r a p h i c s >
    ,   cross  -2pt
        < g r a p h i c s >
    ,   empty  -2pt
        < g r a p h i c s >
    ,   horizontal  -2pt
        < g r a p h i c s >
    ,   vertical  -2pt
        < g r a p h i c s >
    .

These tiles form a set of pipes labeled 1,…,n going
from the bottom boundary (left to right) to the right
boundary. See <Ref>, left, for an illustration.
We denote the set of all bumpless pipe dreams of size n by (n).


	
	Bumpless pipe dreams are the same as
	configurations
	of the six-vertex model in the n× n square
	with domain wall boundary conditions (see <cit.> or <cit.>).
	We refer to <cit.>
	or <cit.> for background and many properties
	on the six-vertex model. This object is also sometimes called
	the oscullating lattice paths model <cit.>.

	The same configurations can be identified with
	alternating sign matrices (ASMs) of size n,
	see <cit.> for a detailed exposition,
	and <cit.>, <cit.> for applications
	of the six-vertex model to the enumeration of ASMs.






	
	A permutation w(D) is associated to a bumpless pipe dream D
	by tracing the pipes from left to right, starting from the bottom boundary,
	and ignoring the second and subsequent crossings of each pair of pipes.
	See <Ref>, right, for a reading of the
	bumpless pipe dream in the left part of the figure
	which leads to w(D). The local rules for labeling the
	tiles in a bumpless pipe dream
	(which produce the permutation w(D))
	are given in
	<Ref>.
	Denote by (w)⊂(n) the set of bumpless pipe dreams
	whose associated permutation is w.





	A bumpless pipe dream is called reduced if no two pipes
	cross twice.
	Denote the set of reduced bumpless pipe dreams of size n by (n),
	and by (w)⊂(n) the subset
	whose associated permutation is w.


For a bumpless pipe dream D (reduced or not), denote by
(D) and (D) the set of NW bumps and empty tiles in D, respectively
(see (<ref>) for the notation).
For a tile in a bumpless pipe dream, we denote its
horizontal and vertical coordinates by (k,l), 1≤ k,l≤ n.
Here k increases from left to right, and l increases from top to bottom.

 
	For any permutation w∈ S_n, we have
	
    𝔊^β_w(x_1,…,x_n ) = β^-ℓ(w)∑_D ∈(w)∏_(k,l)∈(D)β x_k ∏_(k,l)∈(D) (1+β x_k).

	In particular, for β=0, we have
	for the Schubert polynomials:
	
    𝔖_w(x_1,…,x_n ) = ∑_D∈(w)∏_(k,l)∈(D) x_k.




 
	Recall the notation (<ref>).
	We have
	
    v_n(β)    =   ∑_D ∈(n)β^#(D)-ℓ(w(D)), 
       =∑_P ∈(n)β^#(P)-ℓ(w(P))(1+β)^#(P),

	where w(D) and w(P) are the
	permutations associated to the pipe dream D and bumpless pipe dream P, respectively.


	These identities follow from the formulas for Grothendieck polynomials in <Ref>.
For (<ref>), we immediately get

    v_n(β) = ∑_w 𝔊_w^β(1^n) = ∑_D ∈ PD(n)β^#(D)-ℓ(w(D)).

For (<ref>), we have

    v_n(β)= ∑_w 𝔊_w^β(1^n)    = ∑_P ∈ BPD(n)β^-ℓ(w(P))∑_D ∈(w)∏_(k,l)∈(D)β∏_(k,l)∈(D) (1+β )
       = ∑_P ∈ BPD(n)β^#(P)-ℓ(w(P))(1+β)^#(P).

This completes the proof.



Gao and Huang  <cit.> gave a bijection between reduced pipe dreams and reduced bumpless pipe dreams that preserves the corresponding weight of the diagrams in the corresponding expansions (<ref>) and (<ref>) of Schubert polynomials. This bijection was recently generalized to the case of pipe dreams and bumpless pipe dreams with marked NW-bumps and Grothendieck polynomials by Huang–Shimozono–Yu <cit.>.



 
By the product
formula for the number of alternating sign matrices of size n
<cit.>, <cit.>, we have
#(n) = ∏_k=0^n-1(3k+1)!/(n+k)!. Moreover,
since (n)⊂(n), we have

    u_n(0) ≤ v_n(0) ≤∏_k=0^n-1(3k+1)!/(n+k)!.

From the asymptotics of the number of AMSs of order n
(which is relatively straightforward using the product formula),
we conclude that

    lim sup_n→∞1/n^2log_2 u_n(0)   ≤ 0.37.

The asymptotic behavior of the number of bumpless pipe dreams
(six-vertex model configurations)
with periodic boundary is originally due to
Lieb <cit.>. The domain wall case
with general six-vertex parameters is treated
by Korepin–Zinn-Justin <cit.>.




 §.§ Asymptotics of Grothendieck random permutations via domino tilings



Consider a random bumpless pipe dream 𝐃∈(n)
with the distribution

    _2-ASM(𝐃=D) = 2^-n2 2^#(D),    
    	D∈(n).

Via the bijection with Alternating Sign Matrices
(<Ref>),
this distribution is equivalent to choosing each alternating
sign matrix of size n with probability
proportional to 2^#(number of -1's).
We call this distribution the 2-enumerated ASM model,
and it is equivalent (via a two-to-one bijection) to
the uniform probability distribution on
domino tilings of the Aztec diamond of size n-1.
This connection, as well as
the asymptotics of uniformly random domino tilings,
has been a subject of extensive research in
statistical mechanics and integrable probability,
see, for example,
<cit.>, <cit.>, and
<cit.>, <cit.>, <cit.>.

The results from the previous
<Ref> immediately imply:

	
	The Grothendieck random permutation 𝐰∈ S_n
	with the parameter p=1/2
	(see <Ref> for the definition)
	coincides with the permutation w(𝐃)
	associated
	(as in <Ref>)
	to the
	random bumpless pipe dream 𝐃∈(n)
	with the distribution (<ref>).



	
	For general p, an analog of <Ref>
	would involve a more complicated weighting of bumpless pipe dreams
	(six-vertex configurations). This weighting is still in the
	free-fermion family of six-vertex weights.
	At the level of domino tilings, the p-weighting corresponds to a
	deformation of the uniform model
	in which vertical and horizontal dominoes have different weights.
	This model is also well-studied,
	for example, see <cit.>.
	For simplicity, here we focus only on the case p=1/2.


Our asymptotic results for
Grothendieck random permutations (<Ref>)
readily
translate into results about the behavior of the permutations
w(𝐃) coming from the 2-enumerated ASM model.
On the other hand, the asymptotic behavior of the bumpless pipe dream 𝐃
in various regimes
has also been extensively studied,
and some of these results can be
brought back to Grothendieck random permutations.
Let us present just one example leading to asymptotically Gaussian fluctuations:


	
	Let 𝐰∈ S_n be the Grothendieck random permutation with p=1/2.
	Let 𝐰_n be the image of n under 𝐰.
	We have
	
    𝐰_n-n/2/√(n/4)d⟶𝒩(0,1),    
    		n→∞.

	Here 𝒩(0,1) is the standard normal distribution.


	Let
	𝐥_n be the position of the single NW bump in the
	rightmost column of 𝐃.  In
	<Ref>, we have
	𝐥_n=5.
	By <Ref>, 𝐰_n has the
	same distribution as 𝐥_n.
	In the language of domino tilings
	(see the map in, e.g., <cit.>), 𝐥_n is the position of the
	first particle in the particle process associated to the domino tiling.
	By
	<cit.>,
	𝐥_n satisfies a central limit theorem as in
	(<ref>). Note that our standard deviation
	√(n/4) matches the normalization in <cit.>
	as the latter work deals with GUE random matrices
	whose diagonal elements have variance 2.
	This completes the proof.



	
	<Ref> can also be proven directly by looking at the trajectory of the
	pipe of color n in the random pipe inside the staircase shape
	(<Ref>), as this trajectory is a random walk.




 §.§ Exact formulas for elementary layered permutations



We are now in a position to discuss principal specializations
of Grothendieck polynomials on layered permutations.
We begin with a few exact formulas, for which we need some notation.

Let 𝖢_n1/n+12nn denote the
nth Catalan number which counts the number of
Dyck paths of size n. Let _n denote the n-th
little Schröder number
<cit.> and
_n the n-th large Schröder number
<cit.>. It is
known that _n=2_n for n≥ 1.

The Narayana polynomial is defined as

    ℒ_n(x) ∑_d ∈Dyck(n) x^v(d),

where the sum is over Dyck paths of size n and v(d) is the number of valleys of the path d. The coefficients of ℒ_n(x) are the Narayana numbers N(n,k)=1/knknk-1 <cit.>. Note that 𝖢_n=ℒ_n(1) and s_n=ℒ_n(2). The polynomials ℒ_n(x) have the following generating function:

    ∑_n≥ 0 y^n ℒ_n(x) = 1-y(1+x)-√((1-y(1+x))^2-4xy^2)/2y.



Consider the following
elementary layered permutation
(see also (<ref>) for notation):

    w_0(k;n)id_k × w_0(n) = (1,…,k, k+n,k+n-1,…,k+1)
    	 ∈ S_k+n.

Recall the notation Υ_w(β)
(<ref>).
A
formula due to Fomin and Kirilov <cit.> relates the
Schubert principal specialization Υ_w_0(a;b)(0)
to Proctor's formula <cit.> counting bounded
plane partitions of staircase shape:


	
	For nonnegative integers k and n, we have
    
    Υ_w_0(k;n)(0) = 𝔖_w_0(k;n)(1,…,1)  = [𝖢_n-2+i+j]_i,j=1^k  = ∏_1≤ i<j≤ n2k+i+j-1/i+j-1.



In <cit.>, an analogous formula for
the β=1
Grothendieck polynomials is proven:


	
	For nonnegative integers k and n, we have
	
    Υ_w_0(k;n)(1) = 𝔊^β=1_w_0(k;n)(1,…,1) = 2^-k2[_n-2+i+j]_i,j=1^k
    		= 2^-k+12[_n-2+i+j]_i,j=1^k.



A general determinantal formula for all β may be derived
from <cit.>:
  
	For nonnegative integers k and n, we have

    Υ_w_0(k;n)(β)  =  (1+β)^-k2[ℒ_n+i+j-2(1+β)]_i,j=1^k.



The determinants
in
<Ref>
can be efficiently computed via the Dodgson condensation (the Desnanot–Jacobi identity).
We use it to get the numerical data for β=1
presented in
<Ref>.



 §.§ Asymptotically maximal specializations for layered permutations



Here we prove that principal
specializations of β=1
Grothendieck polynomials on layered permutations
(see (<ref>) for the definition)
attain the asymptotically largest
value. That is, we prove <Ref> from the Introduction.


Consider the compositions b^(n) = (…, b_2, b_1) of n
such that

    b_1 + … + b_i-1 + (2 + √(2))b_i ≤ n  for  i = 1,2,…,
     while b_1 + … + b_i-1≤ n - 4,

and b^(n) has o(n) parts.
Then for the layered permutations w(b^(n)) we have

    lim_n →∞1/n^2log_2
    Υ_w(b^(n)) (1)
    = 1/2,

which is asymptotically maximal.



	In particular, one can take compositions to be geometric
	b_i ∼ (1 - α) α^i - 1 n, for any α∈ [1/√(2), 1).


In the rest of this <Ref>,
we prove <Ref>. First, observe that the Grothendieck
specializations enjoy the following basic factorization property:

    Let u ∈ S_k, w ∈ S_n. We have
    
    Υ_u × w(β) = Υ_u(β) ·Υ_id_k × w(β).




	This follows from properties of pipe dreams
	corresponding to permutations
	of the form u × w.
	See <cit.> or <cit.>.


Let us denote (recall the notation (<ref>))

    F(k, n) Υ_w_0(k;n)(1),      G(k,n)   2^k2· F(k,n)=
    	[_n-2+i+j]_i,j=1^k,

where the last equality follows from <Ref>.



An important step in obtaining the asymptotics of Υ_w(1) for layered permutations is to understand the behavior of the function F(k,n). A similar analysis <cit.> was possible in the Schubert case (β = 0) thanks to an explicit product formula (<Ref>). For other β's, we have determinantal (but not product) formulas (<Ref> and <Ref>). However, in the case β=1 we are able to asymptotically analyze these determinants via the correspondence with 2-enumerated ASMs.


Let
k ∈ (n/√(2), n]. We have

    log_2 F(k, n - k) = n^2/2 - k^2/2 - O(n),       n →∞.


In (<ref>) and throughout the proofs below in this subsection,
we assume that the constants in the O(·) notation are nonnegative.

	Let 𝐃∈ PD(n) be the
	random bumpless pipe dream with the
	distribution (<ref>).
	Then we have
	
    F(k,n-k)=2^ n2(w(𝐃)=w_0(k;n-k)).

	Let us first consider a different event,
	
    R_k,n-k{restriction of w(𝐃) to the last n-k entries equals
    				(n,n-1,…,n-k+1)}.

	That is, under R_k,n-k, the first k entries can be an arbitrary
	permutation of {1,…,k }, while the last n-k entries are in the fully reversed order.
	Geometrically, event (<ref>)
	means that the bumpless pipe dream 𝐃
	develops a right triangle of side n-k in the lower right corner
	whose diagonal boundary is made
	of the SE bumps -2pt
    < g r a p h i c s >
, and
	the rest of the triangle
	entirely consists of crosses -2pt
    < g r a p h i c s >
.
	See <Ref> for an illustration.



	The 2-enumerated ASM model is equivalent to uniform domino tilings of the Aztec diamond,
	which develops an arctic circle phenomenon <cit.>. In terms of bumpless pipe dreams, this means that the pipe configuration outside of the circle inscribed in the n× n square is nonrandom (frozen). The four frozen phases adjacent to the four vertices of the square are different, but the one in the lower right corner consists entirely of crosses.
	By the variational principle for domino tilings
	<cit.>,
	as long as our triangle of side n-k is outside the arctic circle
	(which is equivalent to n/√(2)<k≤ n), the condition
	in R_k,n-k does not lower the asymptotics of the probability on the
	scale e^-cn^2.
	Moreover,
	
    (R_k,n-k)=1-e^-O(n),      n→∞.

	Indeed, the complementary event to R_k,n-k
	is that there is at least one non-cross in the lower right triangle of side n-k.
	By <cit.>,
	this event has probability e^-O(n).

	Returning to F(k,n-k),
	using the factorization property
	(<Ref>) and the fact that v_k(1)=2^k2, we have
	
    2^ n2-O(n)∼∑_w∈ S_kΥ_w × w_0(n-k)(1)
    		=
    		∑_w∈ S_kΥ_w(1) · F(k,n-k)=
    		2^ k2· F(k,n-k).

	This implies the desired asymptotic equivalence.


	
	The recent work <cit.>
	provides asymptotics of
	log_2 F(k, n - k) for all k, not necessarily in the range k>n/√(2).
	For k<n/√(2), the leading term
	is strictly smaller than n^2/2,
	which agrees with the variational principle
	<cit.>.
	We do not need the regime k<n/√(2) for <Ref>.




Let us denote k_0  n and

    k_i  n - b_1 - … - b_i,
       i ≥ 1,

so that b_i = k_i-1 - k_i.
We have by <Ref> and (<ref>) that

    Υ_w(…, b_2, b_1)(1)    = Υ_w(…, b_3, b_2)(1) · F(n-b_1,b_1)  
       = ∏_i ≥ 1 F(n - b_1 - … - b_i, b_i) = ∏_i ≥ 1 F(k_i, k_i-1 - k_i).

From the inequalities on the b_i's, we get

    k_i/k_i-1 = n - b_1 - … - b_i/n - b_1 - … - b_i-1∈ (1/√(2), 1).

Hence by <Ref>, we have

    log_2 F(k_i, k_i-1 - k_i) = k_i-1^2/2 - k_i^2/2 - O(k_i-1).

Therefore,

    1/n^2log_2 Υ_w(…, b_2, b_1)(1,…, 1)    = ∑_i ≥ 11/n^2log_2 F(k_i, k_i-1 - k_i) 
       = ∑_i ≥ 11/n^2(k_i-1^2/2 - k_i^2/2 - O(k_i-1) ) 
    
       = 1/2 - 1/n^2 O(∑_i ≥ 1 k_i-1).


Since k_i-1 = O(n) and there are o(n) elements in our composition, we have

    ∑_i ≥ 1 k_i = o(n^2),  and so 1/n^2 O(∑_i ≥ 1 k_i-1) = o(1),








which completes the proof of <Ref>.






	
	Recall the notation
	(<ref>),
	(<ref>).
	<Ref> establishes that on layered permutations,
	the Grothendieck specializations Υ_w(1)
	reach their asymptotic maximum: u_n'(1)∼ 2^ n2.
	However, it is not clear whether
	the absolute maximum, u_n(1),
	is attained on layered permutations.
	Dennin
	<cit.> verified that for n≤ 9, one has
	u_n(1)=u_n'(1).

	We present numerical data of
	optimal (i.e., achieving the exact, non-asymptotic maximum in u_n'(1))
	layered
	permutations for n≤ 500 in
	<Ref>.
	We also comment on why these optimal layered permutations
	have parts whose behavior does not fall under the assumptions of <Ref>.




 §.§ Bounds on the constant for Grothendieck specializations with general β



In <cit.>, the authors conjectured that for β fixed there exists a limit

    c(β)lim_n→∞1/n^2log_2 υ_n(β).

If the limit exists, it must satisfy
for β≥ 0:

    1/4log_2(2+β) ≤ c(β) ≤1/2log_2(2+β),

see <cit.>.
The limit
(<ref>)
exists when β=1, see (<ref>),
and we have c(1)=1/2.
Here we
improve the bounds (<ref>) in certain regimes:



 
	If the limit c(β) (<ref>) exists, then it must satisfy the following bounds.
	For 0 < β≤ 1, we have
	
    1/4max{log_2(2+β), 2log_2(1+β)}≤ c(β) ≤1/2min{log_2(2+β), log_2(1+1/β)}.

For β≥ 1, we have

    1/4max{log_2(2+β), 2log_2(1+1/β)}≤ c(β) ≤1/2log_2(1+β).

				See <Ref> for an illustration of the bounds.


For 0<β≤ 1 and
D∈(n), we can bound 1 ≤β^-ℓ(w(D))≤β^-n2.
Thus by (<ref>), we have

    ∑_D ∈(n)β^#(D)≤ v_n(β) ≤β^-n2∑_D ∈(n)β^#(D),

which leads to
(1+β)^n2≤ v_n(β) ≤(1+β/β)^n2.
Taking the logarithms and comparing with (<ref>) gives the result.

For β≥ 1 and D∈(n), let us estimate
β^-n2≤β^-ℓ(w(D))≤ 1. Thus by (<ref>), we have

    β^-n2∑_D ∈(n)β^#(D)≤ v_n(β) ≤∑_D ∈(n)β^#(D),

yielding
(1+β/β)^n2≤ v_n(β) ≤ (1+β)^n2,
which leads to the
desired bounds.








§ ADDITIONAL REMARKS AND OPEN PROBLEMS





 §.§ Optimal permutations achieving the exact maxima u_n(1) and u_n(β)


This project was motivated by Stanley's original question
from <cit.> on determining the family of
permutations w for which the Schubert specialization 𝔖_w(1^n) is
asymptotically maximal. This question as well as the
asymptotic value remain widely open (see <cit.> for data up to n≤ 10).
Dennin <cit.> extended this
question to Grothendieck specializations 𝔊_w^β=1(1^n),
but one can ask the same question for general β.
Presumably, answering this
question would give rise to a family of optimal permutations
w^β, which for β=0 would answer the original
Schubert specialization question.

Among the candidates for
such a family w^β are the layered permutations
<cit.>,
<cit.>
(see
<Ref> for numerical evidence for β=1).
When β=1,
Dennin verified that
layered permutations indeed achieve the maximum in u_n(1) for n≤ 9.
Our results of
<Ref> suggest that layered
permutations are indeed good candidates
as they at least attain the asymptotic maximum in u_n(1).

To obtain the asymptotics of Grothendieck specializations
on layered permutations for general β, one
has to study the asymptotics of the following Hankel
determinants:

    F(k,n,β)(1+β)^-k2[ℒ_n+i+j-2(1+β)]_i,j=1^k,

where ℒ_n(x) is the Narayana polynomial, see (<ref>)
and <Ref>.
For β=0, these determinants admit a product formula
and were analyzed in <cit.>.
We could access the special case β=1 due to the connection to
2-enumerated ASMs and domino tilings.
It would be interesting to find a deformation of the
domino tiling model related to the determinants (<ref>).



 §.§ Typical vs optimal permutations


One of our main results, <Ref>,
concerns “typical” Grothendieck
permutations.
By comparing the
permutation matrices in <Ref>, center,
and <Ref>, left, we see that
typical permutations are far from layered.
However, the
Shannon–McMillan–Breiman theorem (e.g., see
<cit.>)
suggests that probabilities of typical Grothendieck random permutations
also achieve the asymptotic maximum.
For our permutations, this is not yet a rigorous statement,
as one has to prove the asymptotic uniformity of the probability weights
of typical Grothendieck permutations.



 §.§ Local structure of Grothendieck permutations


What is the local structure of the Grothendieck random permutations?
In <Ref>, left and center, one
can observe a certain local layered structure.
Understanding this local structure goes beyond the permuton convergence.
Can the local structure be
understood in terms of descent statistics, major index, or
pattern statistics other than the overall pattern counts?
For instance, layered permutations are the
ones maximizing the packing density of the
pattern 132 <cit.>.
Potential tools
for this analysis may be found in
<cit.>. Alternatively,
one could translate the local behavior of random domino
tilings of the Aztec diamond <cit.>
to the local behavior of Grothendieck permutations.



 §.§ Random and maximal permutations for domain wall six-vertex model


The Bumpless Pipe Dreams provide a surprising connection
of Schubert calculus and K-theory to
the six-vertex model, one of the central objects in
statistical mechanics.
The analysis of typical and maximal permutations for Schubert
and Grothendieck specializations prompts the analogous questions
about the six-vertex model.

Namely, consider the six-vertex
model with
domain wall boundary conditions and
general Boltzmann weights a,b,c (for the 2-enumerated
ASMs, we have a=b=1 and c=√(2)).
Represent a random configuration of the six-vertex model
as a collection of pipes, and associate to it a random
permutation 𝐰 by resolving the intersections
as in
<Ref>.
What is the behavior of 𝔼[(𝐰)] and
how does it vary depending on the weights of the six-vertex
model? Do the permutations 𝐰 have a permuton limit?



 §.§ q-analogues and generalized principal specialization


Zhang <cit.> extended the analysis in <cit.> to the generalized principal Schubert specialization 𝔖_w(1,q,q^2,…), when q is a  root of unity. The analogue of layered permutations is the multi-layered ones <cit.>. It would be interesting to extend our results to the Grothendieck specializations
	𝔊^β_w(1,q,q^2,…) at a root of unity.
	However, having q complex makes connections to probabilistic models less clear.







§ DATA FOR MAXIMAL SPECIALIZATIONS ON LAYERED PERMUTATIONS





 §.§ Dodgson condensation and data


Principal specializations Υ_w(b)(β) of
Grothendieck polynomials with general β on layered permutations
(see <Ref> for notation)
can be maximized using dynamic programming, similarly to how this is done
for Schubert specializations in <cit.>.
To implement it, one needs an efficient way to compute the
determinants
D_β(n,k)[ℒ_n+i+j-2(1+β)]_i,j=1^k
from
<Ref>.
A recursion
follows from
the Dodgson condensation
(Desnanot–Jacobi identity)
<cit.>, <cit.>:


	For k,n∈ℤ_≥0,
	we have

    D_β(n,k)  = 1/D_β(n+2,k-2)(D_β(n+2,k-1)D_β(n,k-1) - D_β(n+1,k-1)^2),

with initial conditions D_β(n,0)=1, D_β(n,1)=ℒ_n(1+β), and D_β(0,k)=(1+β)^k2.


	Let M(n,k)=[ℒ_n+i+j-2(1+β)]_i,j=1^k, so
	D_β(n,k)= M(n,k). Clearly,
	
    D_β(n,0)=1,    
    		D_β(n,1)=ℒ_n(1+β).

	For n=0, the
	permutation w_0(0;k)=12⋯ k is the identity, and so
	𝔊^β_w_0(0;k)=1. By
	<Ref>, we have

    1 = Υ_w_0(n,k)(β) = (1+β)^-k2[ℒ_i+j-2(1+β)]_i,j=1^k.

Thus, D_β(0,k)= M(0,k)=(1+β)^k2.

For 1≤ i,j,p,q≤ k, let M_i,j^p,q denote the matrix obtained from M
by deleting rows i and j and columns p and q.
Similarly, let M_i^p be obtained from M by deleting row i and column p.
By the Dodgson condensation, we have

    (M(n,k))(M(n,k)^1,k_1,k)  = (M(n,k)^1_1)(M(n,k)^k_k) - (M(n,k)^k_1)(M(n,k)^1_k).

Observe that

    M(n,k)^1,k_1,k   = M(n+2,k-2),   M(n,k)^1_1 = M(n+2,k-1),
       M(n,k)^k_k   =M(n,k-1),   M(n,k)^k_1=M(n,k)^1_k=M(n+1,k-1).

This implies the result.


The implementation of the recursion for D_β(n,k) combined with dynamic programming
yields the data in <Ref>.
This data agrees with the asymptotic behavior f(n)→1/2
obtained in <Ref>.







 §.§ Optimal layered permutations and asymptotic maximum



Observe that the optimal layered permutations given in
<Ref> do not fall into the
description of sequences given in <Ref>.
For example, for n=700, the numerically optimal layered permutation
has parts b^(700)_opt=(1, 3, 5, 9, 14, 21, 33, 49, 73, 107, 157, 228),
whereas a sequence b^(700) from <Ref>
would satisfy the bound b_1=n-k ≤ 700/(2 + √(2)) ≈
205 on its largest part.
The parts of b^(700)_opt are suspiciously close to
the Fibonacci sequence (see <Ref>).
Let us explain this discrepancy.

Optimal
layered permutations can be constructed inductively via the
optimization

    max_b_1 + b_2 + ⋯ = nΥ_w(…, b_2, b_1)(1) =
    max_kmax_b_2 + b_3+⋯ = k{Υ_w(…,b_3, b_2)(1) · F(k, n-k)
    },

where b_1 = n-k.
Assume that max_b_2 +b_3+ ⋯ = kΥ_w(…,
b_3,b_2)(1) = 2^ k2· g(k), where g(k) =
2^-o(k^2). To extract the first part b_1, we thus need to
find

    max_k
    	{ 2^ k2· g(k) · F(k, n-k) }
    	=
    	max_k {g(k)· G(k,n-k)},

see (<ref>) for the notation.
On the other hand, in the proof of <Ref> (in particular, see
<Ref>),
we estimated the asymptotic behavior of
F(k,n-k), and
found that the coefficient
by n^2 in log_2 F(k,n-k) does not depend on k as
long as k>n/√(2).
These two problems lead to different choices of k.
See also
the
illustration in <Ref>.





The discrepancy between optimal layered permutations and the ones appearing in
<Ref> should be of order o(n) in the large n limit. Indeed,
assume that k/n<1/√(2). Then, as in the proof of
<Ref> (see also <Ref>),
one can show that log_2 F(k,n-k)∼α n^2-k^2/2-O(n) for some
α<1/2, so
2^ k2 g(k)  F(k, n-k)∼ 2^α n^2-o(n^2),
which is not maximal.
We believe that the discrepancy of the part sizes
in optimal layered permutations is of order
o(n) as n→∞.



§ LIMIT SHAPE AND FLUCTUATIONS IN TASEP FROM CONTOUR INTEGRALS



In this section we compute the constants
𝖼,𝗏_1,𝗏_2
from <Ref>, and
show how to establish
<Ref> by an asymptotic analysis of the correlation kernel.
We omit tedious but standard
estimates,
as they are
very similar to those in <cit.>
or <cit.>.



 §.§ Fredholm determinant



Recall that ξ̅_m(t)=ξ_m(t)-ξ_m(0) is the displacement of the m-th particle
at time t.
By <Ref>,
we have

    _TASEP(ξ̅_m(t)≥ u)=
    	(λ_m≥ u),

where λ_m is a point of the determinantal point process
X(λ)={λ_j+m-j }_j=1^m⊂ℤ_≥0 with the correlation kernel
(<ref>):

    K(u_1,u_2)=1/(2π𝐢)^2dz  dw/z-ww^u_2/z^u_1+1( z-p/w-p)^m( 1-w/1-z)^t,
    	    
    	u_1,u_2∈ℤ_≥0,

where the contours are positively oriented simple closed curves satisfying
p<|w|<|z|<1.

	
	The probability
	in the right-hand side of
	(<ref>) is equal to the Fredholm determinant
	of the kernel K on {0,1,…,u-1}:
	
    (λ_m≥ u)=
    		[ Id-K ]_≤ u-1
    		1+∑_ℓ=1^∞(-1)^ℓ/ℓ!∑_i_1,…,i_ℓ=0^u-1[ K(i_a,i_b) ]_a,b=1^ℓ,

	where Id is the identity operator.


	The event {λ_m≥ u} means that in the determinantal
	point process X(λ), no particles
	are located in {0,1,…,u-1 }.
	It is a well-known property of determinantal point processes that this
	probability is given by a Fredholm determinant, see e.g.
	<cit.>.
	This statement can be traced back to the inclusion–exclusion principle
	(e.g., see <cit.>).
	Note that the sum over ℓ in (<ref>) is finite, as
	for u>ℓ, the ℓ×ℓ determinant must have identical rows.
	Further details on Fredholm determinants may be found in
	<cit.> or <cit.>.


Let us assume that
m=⌊ L𝗆⌋,
t=⌊ L𝗍⌋,
u=⌊ L𝗎⌋,
where L→∞. By <Ref>,
it suffices to consider the case 𝗍≥𝗆/p,
which we assume throughout the present
<Ref>.



 §.§ Steepest descent and law of large numbers



The asymptotic analysis of K(u_1,u_2) is done by
steepest descent (see, e.g., <cit.> for an accessible introduction).
The integrand in (<ref>) has the form

    1/z-ww^u_2/z^u_1+1( z-p/w-p)^m( 1-w/1-z)^t
    	=exp{ L (
    	S(z;u_1/L)
    	-S(w;u_2/L)
    	)
    	+O(1)
    	}/z(z-w),

where

    S(z;𝗎)
    	-𝗎log z+
    	𝗆log(z-p)
    	-𝗍log(1-z).

The term O(1)
arises from dropping integer parts in m and t.
It is
negligible in the limit because the determinantal process
will be scaled from discrete to continuous space.

The point λ_m is at the left edge of the determinantal process.
Therefore, to catch its asymptotic location 𝗎=𝖼(𝗆,𝗍),
we need to find a double critical point of the function
z↦ S(z;𝗎).


	
	Let us explain the need for the double critical point
	in more detail.
	By (<ref>), we have
	K(u,u)=(u∈ X(λ)), which is the density
	of the random point configuration X(λ).
	The equation S'(z;𝗎)=0 for single critical points
	is quadratic, and its discriminant is
	a function of p and our parameters (𝗆,𝗍,𝗎).
	Fox fixed 𝗆 (viewed as a parameter of the determinantal
	process and not the scaled index on the last part λ_m),
	there are three possibilities depending on (𝗍,𝗎):
	
		
  * If the discriminant is negative,
			then
			K(u,u) converges to a value strictly between 0 and 1. Therefore,
			at the scaled time 𝗍 around the scaled location
			𝗎 there is a random configuration of points from X(λ)
			of density strictly between zero and one.
		
  * If the discriminant is positive, then
			K(u,u) converges to 0 or 1,
			and around 𝗎 there is either a densely packed, or an empty
			region of points from X(λ).
		
  * The case of zero discriminant (double critical points)
			is at the transition between the two cases, and
			thus
			corresponds to the edge of the random configuration X(λ).
	


Solving the double critical point equations S'(z;𝗎)=S”(z;𝗎)=0 in
z,𝗎, we find

    z_cr=
    	√(p𝗍)-√(𝗆)/√(𝗍/p)-√(𝗆)
    	,    𝗎_cr=
    	(√(p𝗍)-√(𝗆))^2/1-p.

Here out of two solutions to quadratic equations, we picked
the one with the smaller 𝗎_cr which corresponds to the left edge of X(λ).
Note that the solution 𝗎_cr in (<ref>) is the same as
the constant 𝖼(𝗆,𝗍)
from <Ref>, which is not a coincidence.

Let us deform the integration contours in
the kernel (<ref>) to pass near the double critical point
z_cr. This point lies inside the interval (0,p), while
the original contours satisfy p<|w|<|z|<1.
Due to high multiplicity poles in the integrand,
the w contour should not be deformed through p, and
the z contour should not be deformed through 0 and 1.

We deform the contours as follows. First,
deform w such that the new w contour goes around p and passes to the right of z_cr.
Then, deform z through w, such that
the new z contour goes around 0 and passes to the left of z_cr.
Dragging z through w picks up a residue at z=w, which needs to be integrated over the new w contour.
However, taking the residue at z=w eliminated the pole at w=p, so the
new single integral is zero.
This implies that the contour deformation does not produce extra terms in the kernel.
See <Ref>
for an illustration.



To arrive at the law of large numbers λ_m/L →𝖼(𝗆,𝗍)
(recall that this is the same as 𝗎_cr),
it remains to show that changing 𝖼
by ε or (-ε) makes the Fredholm determinant (<ref>) go to 0 or 1,
respectively. To show convergence to 1, it suffices to
estimate |K(u_1,u_2)|≤ e^-δ L for
u_1,u_2≤ L (𝖼-ε), which follows from the steepest descent analysis.
To show convergence to 0,
it suffices to look at the density function K(u,u) for u around L(𝖼+ε).
In the limit, this density
is strictly positive, and so the event
λ_m≥ L (𝖼+ε) has probability tending to zero.
We omit further details.



 §.§ Asymptotic fluctuations



Consider the probabilities of fluctuations as in <Ref>:

    _TASEP( ξ̅_⌊ L𝗆- L^1/3α𝗏_1⌋
    		(⌊ L𝗍⌋ )≥
    	L𝖼 -
    	L^1/3β𝗏_2
    )= [
    		Id
    		-
    		K^L𝗍,
    		L𝗆 - L^1/3α𝗏_1]_≤ L𝖼-
    	L^1/3β𝗏_2 +o(1).

Assume that 𝗆 and 𝗍 are fixed
(recall that 𝖼,𝗏_1,𝗏_2 depend on them).
In the right-hand side of
(<ref>),
we dropped integer parts as
they do not affect the limit to continuous space.

One can show that in the
series
(<ref>)
for the
Fredholm determinant (<ref>),
terms where
i_a< L𝖼-s L^1/3 for at least one a=1,…,ℓ
(and s sufficiently large but finite), are negligible in the limit.
Thus,
it suffices to consider
K̃(ũ_1,ũ_2)
K^L𝗍,
L𝗆 - L^1/3α𝗏_1
(L𝖼+ũ_1 L^1/3,L𝖼+ũ_2 L^1/3),
where ũ_1,ũ_2∈ℝ.
This computation will also produce
the normalization
constants 𝗏_1 and 𝗏_2
for the Tracy–Widom GUE fluctuations.

In K̃,
deform the contours to pass near the double critical point
z_cr=z_cr(𝗆,𝗍) given by (<ref>).
The contribution outside a small neighborhood of z_cr vanishes in the limit.
In the neighborhood,
make a change of variables

    z=z_cr+L^-1/3z̃/ϱ,
    	    
    	w=z_cr+L^-1/3w̃/ϱ,

where ϱ>0 is to be determined.
The new contour z̃ goes from
e^-2π𝐢/3∞ to -1 to
e^2π𝐢/3∞,
and w̃ goes from
e^-π𝐢/3∞ to 0 to
e^π𝐢/3∞.
The function
(<ref>)
is Taylor expanded in L as

    L( S(z;𝖼+L^-2/3ũ_1 )-S(z_cr;𝖼+L^-2/3ũ_1 ) )
    		
       =
    		-z̃^3/3ϱ^3p^3/2/√(𝗆𝗍)(√(𝗍/p)-√(𝗆))^4/(1-p)^3 z_cr
    		+
    		z̃/ϱ z_cr( -ũ_1
    			+
    			α𝗏_1(√(p𝗍/𝗆)-1)/1-p).

Let us pick ϱ>0 such that the coefficient
by z̃^3 is (-1/3),
that is,

    ϱ=
    	√(p)/(1-p) z_cr(√(p𝗍)-√(𝗆))^2/3
    	(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6.

We also have
in the integrand:

    dz  dw/z(z-w)∼ -L^-1/3/ϱ z_crdz̃ dw̃/w̃- z̃.

Adding the difference
S(z_cr;𝖼+L^-2/3ũ_2 )-S(z_cr;𝖼+L^-2/3ũ_1 )
inside the exponent in the integrand (<ref>)
does not change the correlation kernel, since it is a
gauge transformation of the form
K̃(ũ_1,ũ_2)↦K̃(ũ_1,ũ_2)f(ũ_2)/f(ũ_1),
where f(·) is a nonvanishing function.
Thus, we have

    K̃(ũ_1,ũ_2)f(ũ_2)/f(ũ_1)∼
    	-L^-1/3/ϱ z_cr∫_e^-2π𝐢/3∞^e^2π𝐢/3∞
    	dz̃∫_e^π𝐢/3∞^e^-π𝐢/3∞
    	dw̃ exp[ w̃^3/3
    			-
    			z̃^3/3
    			+
    			z̃ U_1
    			-
    			w̃ U_2
    	]/w̃-z̃,

where

    U_1,21/ϱ z_cr(
    	-ũ_1,2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p)
    	.

The integral in (<ref>) together with the minus sign from the prefactor
is equal to the celebrated Airy_2 kernel
𝖠(U_1,U_2)
(e.g., see <cit.>).
The Tracy–Widom GUE cumulative distribution function
is the Fredholm determinant of the Airy_2 kernel, that is,
F_2(r)=[Id-𝖠]_≥ r, r∈ℝ.

The additional prefactor (L^1/3ϱ z_cr)^-1
in (<ref>)
combined with the sums in
the pre-limit Fredholm determinant (<ref>)
turns them into Riemann integrals of the Airy_2 kernel.
The normalization by
ϱ z_cr
is absorbed by
a change of variables:

    1/L^1/3ϱ z_cr∑_i=⌊ L^1/3ũ⌋≤ -L^1/3β𝗏_2
    	g(ũ)
    	∼1/ϱ z_cr∫_-∞^-β𝗏_2
    	g(ũ) dũ=
    	∫_1/ϱ z_cr(
    		β𝗏_2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p)
    	^+∞𝗀(U) dU.

Here we incorporated the bound
L𝖼- L^1/3β𝗏_2 from (<ref>)
into the bound for i, and shifted
all indices by L𝖼.
The function g(ũ) represents
terms like K(·, i_a)K(i_a,·) in the
Fredholm determinant (<ref>)
arising when expanding the individual ℓ×ℓ
determinants as sums over permutations.
Here ũ and i_a are related as
i_a=⌊ L^1/3ũ⌋.
We also set 𝗀(U)=𝗀(ũ).

We see that
the following
convergence of Fredholm determinants holds:

    lim_L→∞[
    		Id
    		-
    		K^L𝗍,
    		L𝗆 - L^1/3α𝗏]_≤ L𝖼
    	=
    	[
    		Id
    		-
    		𝖠]_≥ r,

where

    r=
    	1/ϱ z_cr(
    		β𝗏_2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p).

We see that setting

    𝗏_1=
    		𝗏_1
    		(𝗆,𝗍)
    		   ϱ z_cr(1-p)/√(p𝗍/𝗆)-1
    		=
    		√(p)𝗆^1/3/𝗍^1/6(√(𝗍/p)-√(𝗆))^2/3/(√(p𝗍)-√(𝗆))^1/3
    		;
    𝗏_2=
    		𝗏_2
    		(𝗆,𝗍)
    		   ϱ z_cr
    		=
    		√(p)/(1-p)(√(p𝗍)-√(𝗆))^2/3
    		(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6

turns the right-hand side of (<ref>) into the Tracy–Widom GUE distribution F_2(α+β).
This completes the proof of <Ref>.





alpha





A. H. Morales, Université du Québec à Montréal, Montreal, QC, Canada and University of Massachusetts Amherst, Amherst, MA, USA

E-mail: 



G. Panova, University of Southern California, Los Angeles, CA, USA

E-mail: 



L. Petrov, University of Virginia, Charlottesville, VA, USA

E-mail: 



D. Yeliussizov, Kazakh–British Technical University, Almaty, Kazakhstan

E-mail: 













 Paper Title: A new unit-bimodal distribution based on correlated Birnbaum-Saunders random variables.


	Dynamical friction in rotating ultralight dark matter galactic cores
    V.M. Gorkavenko^1, O.V. Barabash^1,
  T.V. Gorkavenko^1,  
O.M. Teslyk^1,

 A.O. Zaporozhchenko^1, Junji Jia^3,  A.I. Yakimenko^1,4,5, E.V. Gorbar^1,2

^1 Faculty of Physics, Taras Shevchenko National University of Kyiv,

64, Volodymyrs'ka str., Kyiv 01601, Ukraine

^2 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,

14-b Metrolohichna str., Kyiv 03143, Ukraine

^3 School of Physics and Technology, Wuhan University, 299 Bayi Road, Wuhan, Hubei Prov., China

^4 Dipartimento di Fisica e Astronomia ’Galileo Galilei’, Universit'a di Padova,
 via Marzolo 8, 35131 Padova, Italy

^5 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy

    
===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
		
			In this paper, we propose a new distribution over the unit interval which can be characterized as a ratio of the type Zd=Y/(X+Y) where X and Y are two correlated Birnbaum-Saunders random variables. The stress-strength probability between X and Y is calculated explicitly when the respective scale parameters are equal. Two applications of the ratio distribution are discussed.
			
			
			
		
		
		
		
	
	Keywords. Unit distribution · Type I ratio ·  Maximum likelihood method · Monte Carlo simulation ·  software.
	

	Mathematics Subject Classification (2010). MSC 60E05 · MSC 62Exx · MSC 62Fxx.
	
	
		
		
		
	
	

§ INTRODUCTION

	
	Traditionally, the activities in statistical distribution theory have been concentrated on distributions with unbounded support. Only in recent years, attention has been devoted to the construction of distributions with bounded support. Bounded data have applications in economics, finance, biology and medicine among others, see e.g., the introductory remarks in  <cit.> and <cit.>. Such data arise naturally in the form of rates and proportions. Some of the most prominent bounded distributions are the beta distribution and their generalizations and the distributions of Kumaraswamy and Topp-Leone. 
	
	A natural way to create a model for bounded data is to transform one or two unbounded random variables in a bounded variable Z. For example, Z=X/(X+1), Z=exp(-X) and Z=X/(X+Y) have the support (0, 1), when X and Y are positive variables. In applications of the last model it is usually assumed that X and Y are independent. We consider some specific approaches using these transformations or more complicated ones. The first four techniques transform a single variable: 
	<cit.> derive a bimodal beta distribution, applying a quadratic transformation technique due to Elal-Olivero to the beta distribution.
	<cit.> apply an approach of Marshall and Olkin to the reduced Kies distribution.
	<cit.> propose a general framework in which a monotone function is applied to a single variable. 
	<cit.> applies a procedure due to  <cit.> to a uniformly distributed variable, resulting in a three-parameter family of distributions over the unit interval.
	<cit.> (unit log) use the approach Z=X/(X+Y) where X and Y are the components of a two-dimensional random vector, following a bivariate log-symmetric distribution. In this model, X and Y may be correlated.
	
	The distribution of ratios of the latter type is of great importance in practice, for example in modeling COVID-19-related death rates, see <cit.>. 
	






















	
	
		On the other hand, the Birnbaum–Saunders (BS) fatigue-life model, introduced by <cit.>, is based on renewal theory principles. This model focuses on the number of cycles needed to exceed a critical threshold and cause fatigue crack propagation. In the same year, <cit.> provided maximum likelihood estimates for the BS model parameters. Later, <cit.>, in his work on stochastic models of failure in random environments, offered an alternative derivation of the distribution using a biological model and relaxed several of the assumptions made by <cit.>. The BS distribution has been extensively studied and applied across various fields, including business, industry, economics, finance, management, engineering, and medical sciences. For a review of recent work on the BS distribution, see <cit.> and the references therein.
		
		Bivariate (and multivariate) versions of the BS distribution have also been explored. <cit.> introduced the bivariate BS distribution with five parameters using the bivariate normal distribution function. <cit.> revisited the bivariate BS distribution and discussed several new properties. <cit.> introduced a mean-based bivariate BS distribution, demonstrating that the mean is one of its parameters.
	
	
	
	
	In the present paper we propose the distribution of a random variable of the type Z=Y/(X+Y), where X and Y are two (possibly correlated) Birnbaum-Saunders random variables. The article is structured as follows. 
	In Section <ref>, we introduce the unit Birnbaum-Saunders (UBS) model and some of its forms are outlined in graphs. Note that bimodality in the model is observed as a function of its shape and correlation parameters.
	In Section <ref>, we show one of the main properties that the UBS model has, that is, the corresponding random variable to the UBS distribution originally comes from a type I ratio <cit.> of two variables following a bivariate BS distribution.
	Also in this section, we derive the cumulative distribution function (CDF), which is used to explicitly calculate the stress-strength probability in the case of the presence of correlation and equality of scale parameters. Finally, the maximum product of spacings method for parameter estimation is presented. In Section <ref>, we carry out a Monte Carlo simulation study to evaluate the performance of the maximum product of spacings estimators by means of their bias and root mean square error. In Section <ref>, we present two applications to income-consumption and body mass data. We demonstrate that these data can also be fitted well by the UBS distribution. Finally, in Section <ref>, some concluding remarks are made.




	
	
	
	
	
	

§ THE UNIT BIRNBAUM-SAUNDERS DISTRIBUTION

	
	
	We say that a continuous random variable Z, with support (0, 1), follows a unit BS (UBS) distribution with parameter vector θ_ρ=(α_1,α_2,β_1,β_2,ρ)^⊤, α_1> 0,α_2> 0 ,β_1> 0, β_2> 0 and ρ∈(-1,1),
	denoted by Z∼ UBS(θ_ρ), if its probability density function (PDF) is given by (for 0<z<1 and s=1/z-1)
	
		
    f_Z(z;θ_ρ)
    			   =
    			exp{1 1-ρ^2(
    				1α_1^2+1α_2^2)} 4πα_1α_2 √(1-ρ^2) (s+1)^2 s exp{
    			-
    			ρα_1α_2(1-ρ^2)(√(β_2 sβ_1)
    			+
    			√(β_1β_2 s))
    			}
       ×{(
    			√(β_2 sβ_1)
    			+
    			√(β_1β_2 s)) 
    			K_0(√(u_ρ v_ρ) 1-ρ^2)
    			+
    			[
    			√((v_ρ/u_ρ)sβ_1β_2)
    			+
    			√(β_1β_2 (v_ρ/u_ρ)s) ] 
    			K_1(√(u_ρ v_ρ) 1-ρ^2)
    			},

	where 
	
    u_ρ≡sα_1^2β_1
    		+
    		1α_2^2β_2
    		-
    		2ρα_1α_2√(sβ_1β_2),
    		   
    		v_ρ≡β_1α_1^2 s
    		+
    		β_2α_2^2
    		-
    		2ρα_1α_2√(β_1β_2 s).

	Furthermore, in formula (<ref>),
	
    K_λ(x)
    	   =
    	1 2∫_0^∞ w^λ-1exp{-1 2  x(w+1 w)} dw,    x>0, λ∈ℝ,
    	
       =
    	π 2sin(λπ)[I_-λ(x)-I_λ(x)]

	is the modified Bessel function of the third kind (also known as modified Bessel function of the second kind) <cit.>,
	with
	
    I_λ(x)
    	=
    	(x 2)^λ∑_k=0^∞1 k!Γ(λ+k+1)(x 2)^2k

	being the modified Bessel function of the first kind. When λ∈ℤ the following integral representation of I_λ is valid <cit.>:
I_λ(x)
	=
	(1/π) ∫_0^πexp{xcos(θ)}cos(λθ) dθ, 
	 λ∈ℤ.
	
	When ρ=0, that is, X and Y are independent, the PDF (<ref>) of Z is simplified as (for 0<z<1 and s=1/z-1)
		
    f_Z(z;θ_0)
    			   =
    			exp(
    				1α_1^2+1α_2^2) 4πα_1α_2 (s+1)^2 s{(
    			√(β_2 sβ_1)
    			+
    			√(β_1β_2 s)) 
    			K_0(√(uv))
    			+
    			[
    			√((v/u)sβ_1β_2)
    			+
    			√(β_1β_2 (v/u)s) ] 
    			K_1(√(uv))
    			},

	where u=u_0 and v=v_0, and θ_0
	=θ_ρ=0=(α_1,α_2,β_1,β_2)^⊤.
	
		Let S=X/Y be the type II ratio, where X∼ BS(α_1,β_1) and Y∼ BS(α_2,β_2) are independent random variables (that is, ρ=0), and let f_S(s;θ_0) denote its respective PDF.
	By using 
	the identity f_S(s;θ_0)=∫_0^∞ y f_X(sy)f_Y(y)  dy it is simple to check that the PDF of S is (for s>0)
	
    f_S(s;θ_0)
    	   =
    				exp(
    		1α_1^2+1α_2^2) 4πα_1α_2 1 s{(
    	√(β_2 sβ_1)
    	+
    	√(β_1β_2 s)) 
    	K_0(√(uv))
    	+
    	[
    	√((v/u)sβ_1β_2)
    	+
    	√(β_1β_2 (v/u)s) ] 
    	K_1(√(uv))
    	}
       =
    	1 (s+1)^2  f_Z(1 s+1;θ_0),

	where f_Z(z;θ_0) is as given in (<ref>). The PDF (<ref>) of S with β_1=β_2=1 has appeared in reference  <cit.>.
	
	
	Figures <ref>, <ref> and <ref> show the behavior of f_Z for some parameter choices.
	Note that the density modality changes according to the value of α_1 and α_2 parameters.
	
	
	
	
	
	


§ CHARACTERIZATION AND ESTIMATION


In this section, we establish some mathematical results of the UBS distribution. In particular, we define the UBS model as a ratio, present the cumulative distribution function, and calculate the stress-strength probability. We also derive parameter estimators for the UBS model based on the maximum product of spacings method.





 §.§ UBS model arising as a ratio

	In this section, we establish one of the most important properties of the UBS model. This property characterizes a UBS random variable Z in (<ref>) as a ratio of the form Y/(X+Y), where the random vector (X,Y)^⊤ follows a bivariate Birnbaum-Saunders distribution as discussed in <cit.>.
	
	Indeed, let X, Y and Z be continuous and positive random variables
	such that
	
    Zd=YX+Y,

	being d= equality in distribution, and the ratio in (<ref>) has support in the unit interval (0, 1). The random variable Z = Y/(X+Y) is known in the literature as type I ratio <cit.>.
	
	From (<ref>), it is simple to verify that the cumulative distribution function (CDF) of Z is given by
	
    F_Z(z)
    		=
    		
    		
    		
    		
    		
    		
    		
    		ℙ(X Y⩾s)
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		=
    		1-
    		∫_0^∞ℙ(X⩽ sy| Y=y) f_Y(y)  dy,
    		   s=1 z-1.

	
	
	
	
	
	
	
	
	
	
	
	
	Hence, the	 PDF of Z can be expressed as
	
    f_Z(z)
    		=
    		(s+1)^2
    		∫_0^∞ y f_X,Y(sy,y)  dy.

	
	
		Equation in (<ref>) can also be obtained using the Jacobian method by determining the marginal with respect to Z of the random vector (Z,U)^⊤, where Z=Y/(X+Y) and U=Y.
	
	
	Now, let's suppose that (X,Y)^⊤ has a bivariate Birnbaum-Saunders distribution with parameter vector θ_ρ=(α_1,α_2,β_1,β_2,ρ)^⊤, denoted by (X,Y)^⊤∼ BS(θ), that is, its joint density is given by
	
    f_X,Y(x,y)
    		=
    		ϕ_2(a(x;α_1,β_1),a(y;α_2,β_2);ρ)
    		a'(x;α_1,β_1)a'(y;α_2,β_2),

	where ϕ_2(u, v;ρ) denotes the standard bivariate normal density function;
	
    ϕ_2(u, v;ρ)
    		=
    		1 2π√(1-ρ^2) exp{-1 2(1-ρ^2)  (u^2+v^2-2ρ uv)},
    		   (u,v)^⊤∈ℝ^2

	and
	
    a(t;α,β)
    		=
    		1α(√(tβ)-√(β t)),
    		   
    		a'(t;α,β)
    		=
    		1 2α t(√(tβ)+√(β t)),
    		   t>0.

	
	Hence, the density (<ref>) of Z is written as
	
    f_Z(z)
    		=
    		(s+1)^2
    		∫_0^∞ y 
    		ϕ_2(a(sy;α_1,β_1),a(y;α_2,β_2);ρ)
    		a'(sy,α_1,β_1)a'(y,α_2,β_2)
    		 dy.

	
	Since
	
    a^2(sy;α_1,β_1)+a^2(y;α_2,β_2)-2ρ a(sy;α_1,β_1)a(y;α_2,β_2)
    		
       =
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		
    		(
    		u_ρ y
    		+
    		v_ρ y)
    		-
    		2
    		[
    		1α_1^2
    		+
    		1α_2^2
    		-ρα_1α_2(√(β_2 sβ_1)
    		+
    		√(β_1β_2 s))
    		],

	with u_ρ and v_ρ as given in (<ref>), note that the function ϕ_2(a(sy;α_1,β_1),a(y;α_2,β_2);ρ) in (<ref>) can be expressed as
	
    ϕ_2(a(sy;α_1,β_1),     a(y;α_2,β_2);ρ)
    		=
    		1 2 exp{-1 2(1-ρ^2)(
    		u_ρ y
    		+
    		v_ρ y)}
       ×1π√(1-ρ^2) exp{1 1-ρ^2[
    		1α_1^2
    		+
    		1α_2^2
    		-ρα_1α_2(√(β_2 sβ_1)
    		+
    		√(β_1β_2 s))
    		]}.

	
	Furthermore, notice that
	
    a'(sy;α_1,β_1)a'(y;α_2,β_2)
    		   =
    		
    		
    		
    		
    		
    		1 4α_1α_2 sy[
    		1 y^2√(β_1β_2 s)
    		+
    		1 y(
    		√(β_2 sβ_1)
    		+
    		√(β_1β_2 s))
    		+
    		√(sβ_1β_2)].

	
	By replacing (<ref>) and (<ref>) in formula (<ref>), we have
	
    f_Z(z)
    		   =
    		1π√(1-ρ^2) (s+1)^2 4α_1α_2 s exp{1 1-ρ^2[
    		1α_1^2
    		+
    		1α_2^2
    		-ρα_1α_2(√(β_2 sβ_1)
    		+
    		√(β_1β_2 s))
    		]}
       ×[
    		√(β_1β_2 s) 𝕀_1
    		+
    		(
    		√(β_2 sβ_1)
    		+
    		√(β_1β_2 s)) 𝕀_2
    		+
    		√(sβ_1β_2) 𝕀_3
    		],

	where
	
    𝕀_1
    		=
    		1 2 ∫_0^∞1 y^2 exp{-1 2(1-ρ^2)(
    		u_ρ y
    		+
    		v_ρ y)} dy
    		=
    				√(u_ρ v_ρ)  
    		K_-1(√(u_ρ v_ρ) 1-ρ^2),
    		
       𝕀_2
    		=
    		1 2 ∫_0^∞1 y exp{-1 2(1-ρ^2)(
    		u_ρ y
    		+
    		v_ρ y)} dy
    		=
    		K_0(√(u_ρ v_ρ) 1-ρ^2)

	and
	
    𝕀_3
    		=
    		1 2 ∫_0^∞exp{-1 2(1-ρ^2)(
    		u_ρ y
    		+
    		v_ρ y)} dy
    		=
    		√(v_ρ u_ρ)  
    		K_1(√(u_ρ v_ρ) 1-ρ^2).

	In the above, K_λ(x), x>0, denotes the modified Bessel function of the third kind defined in (<ref>).
	
	Finally, by using that the Bessel function is symmetric with respect to the index λ: K_λ(x)=K_-λ(x), and by employing the above formulas of 𝕀_1, 𝕀_2 and 𝕀_3, from (<ref>) the PDF of Z can be expressed as in (<ref>).
	
	

 §.§ The cumulative distribution function

	
	For (X,Y)^⊤∼ BS(θ_ρ), it holds that X∼ BS(α_1,β_1) and Y∼ BS(α_2,β_2). Furthermore, the  conditional CDF of X, given Y=y, is given by <cit.>
	
    ℙ(X⩽ x| Y=y) 
    		=
    		Φ{
    			a(x;α_1,β_1)
    			-
    			ρ a(y;α_2,β_2)	
    		√(1-ρ^2)},

	where a(t;α,β) is as given in (<ref>) and Φ is the CDF of the standard normal distribution.
	Hence, from (<ref>) it follows that
	
    F_Z(z;θ_ρ)
    		   =
    		1-
    		∫_0^∞ℙ(X⩽ sy| Y=y) 
    		f_Y(y)  dy,
    		   s=1 z-1,
    		
       =
    		1-
    		∫_0^∞Φ{
    			a(sy;α_1,β_1)
    			-
    			ρa(y;α_2,β_2)
    		√(1-ρ^2)}ϕ(a(y;α_2,β_2))a'(y;α_2,β_2)
    		 dy,

	where ϕ denotes the PDF of the standard normal distribution.
	By using the change of variable w=a(y;α_2,β_2), we write the above expression as
	
	
	
    1-
    		∫_-∞^∞Φ{
    			a(sa^-1(w;α_2,β_2);α_1,β_1)
    			-
    			ρw
    		√(1-ρ^2)}ϕ(w)
    		 dw
    		
       =
    		1 2
    		-
    		1 2∫_-∞^∞ erf{
    			a(sa^-1(w;α_2,β_2);α_1,β_1)
    			-
    			ρw
    		√(2(1-ρ^2))}ϕ(w)
    		 dw,

	where we have used the well-known identity 
	Φ(x)=(1/2)[1+(x/√(2))]. In the above, a^-1(t;α,β) denotes the inverse function of a(t;α,β).
	As 
	a^-1(t;α,β)
	=
	(β/4)[α t+√((α t)^2+4)  ]^2,
	the integral above can be written as
	
    1 2
    		-
    		1 2∫_-∞^∞ erf{(α_2α_1√(sβ_2β_1)-ρ)√(2(1-ρ^2)) 
    			w
    			+
    			√(2)(sβ_2-β_1)α_1√(sβ_1β_2(1-ρ^2)) 1α_2w+√((α_2 w)^2+4)}ϕ(w)
    		 dw.

	
	Therefore, the CDF of the UBS random variable Z is given by (for s=1/z-1 and 0<z<1)
	
    F_Z(z;θ_ρ)
    		=
    		1 2
    		-
    		1 2∫_-∞^∞ erf{(α_2α_1√(sβ_2β_1)-ρ)√(2(1-ρ^2)) 
    			w
    			+
    			√(2)(sβ_2-β_1)α_1√(sβ_1β_2(1-ρ^2)) 1α_2w+√((α_2 w)^2+4)}ϕ(w)
    		 dw.

	
	
	
	
	
	
	
	

 §.§ The stress-strength probability

	
	
	In what follows we explicitly calculate the stress-strength probability R≡ℙ(X<Y)  in the case where (X,Y)^⊤∼ BS(θ_ρ), that is,
	X and Y can admit correlation. 
	
	Indeed, by (<ref>), we have
	
    R
    		=
    		ℙ(X Y< 1)
    		=
    		1-ℙ(X Y⩾ 1)
    		=
    		1-F_Z(z;θ_ρ)|_z=1/2.

	Hence, as z=1/2 if and only if s=1, by using (<ref>),
	
	we get
	
    R
    		=
    		1 2
    		+
    		1 2∫_-∞^∞ erf{(α_2α_1√(β_2β_1)-ρ)√(2(1-ρ^2)) 
    			w
    			+
    			√(2)(β_2-β_1)α_1√(β_1β_2(1-ρ^2)) 1α_2w+√((α_2 w)^2+4)}ϕ(w)
    		 dw.

	
	In the particular case β_1=β_2, from (<ref>), the stress-strength probability R is simply written as
	
    R
    		=
    		1 2
    		+
    		1 2∫_-∞^∞ erf{(
    			α_2α_1
    			-
    			ρ)
    		√(2(1-ρ^2)) 
    		w
    		}ϕ(w)  dw
    		=
    		1 2,
    		  ∀ρ∈(-1,1),

	where in the last equality we have used the fact that erf(x) and ϕ are odd and even functions, respectively.
	
	To our knowledge, in this setting (β_1=β_2), identity R=1/2 was known only for the special case in which X and Y are independent (that is, ρ=0); see, e.g. <cit.> and <cit.>.
	
	



































































































































































































































	
	





 §.§ Maximum product of spacings method


The maximum product of spacings (MPS) method was introduced by <cit.> and <cit.> as a different approach to estimating parameters of continuous univariate distributions compared to maximum likelihood estimation. Let {Z_i:i = 1,…,n} be a univariate random sample of size n from the UBS(θ_ρ)
distribution with PDF as in (<ref>), and let z_i be the sample observation of Z_i. Then, the uniform spacings of  {z_i:i = 1,…,n} is given by

    Δ_i(θ_ρ)=F_Z(z_i:n;θ_ρ)-
     F_Z(z_i-1:n;θ_ρ),     1,…,n,

with F_Z(z_0:n;θ_ρ)=0, F_Z(z_n+1:n;θ_ρ)=1,
and Δ_0(θ_ρ)+Δ_1(θ_ρ)+⋯+Δ_n+1(θ_ρ)=1.

The maximum product of spacings estimator, θ_ρ, is obtained by maximizing the geometric mean of the spacings

    G(θ_ρ)=( ∏_i=1^n+1Δ_i(θ_ρ) )^1/n+1,

or, equivalently, maximizing the function

    H(θ_ρ)= 1/1+n∑_i=1^n+1log( Δ_i(θ_ρ)).











§ SIMULATION STUDY



In this section, we carry out a Monte Carlo simulation study for evaluating the performance of the above described maximum product of spacings method for the UBS distribution (<ref>). The simulation scenario considers the following setup: 300 Monte Carlo replications, sample size n ∈{100,200,400,800}, and vector of true parameters (α_1,α_2,β_1,β_2)^⊤= (0.5,0,5,1.0,1.0)^⊤, ρ∈{0.10,0.25,0.5,0.75}.


The performance and recovery of the maximum product of spacings estimators are evaluated by means of relative bias (RB), and root mean square error (RMSE), which are calculated from the Monte Carlo replicates, as

    RE(γ)    =   1/N∑_i = 1^N| (γ^(i) - γ)/γ| ,  RMSE(γ) = √(1/N∑_i = 1^N (γ^(i) - γ)^2),

where γ and γ^(i) are the true parameter value and its i-th estimate, and N is the number of Monte Carlo replications. The steps for the Monte Carlo simulation study are as:
i. set the values of the parameters of the UBS distribution; ii. generate 300 samples of size n from the chosen model; iii. estimate the model parameters using the maximum product of spacings method method for each sample; and iv. compute the empirical RB and RMSE.

The maximum product of spacings estimation results so obtained are shown in Figures <ref> and <ref> wherein the empirical bias and RMSE are both plotted. We observe that the results obtained for the UBS distribution are as expected in that as the sample size increases, both RE and RMSE decrease. In terms of the impact of the parameter ρ, we observe that both RB and RMSE of α_1, α_2, β_1 tend to increase as ρ increases. In the case of ρ, we observe the opposite behavior.


















§ APPLICATION

	

    We now give two applications using real data previously analyzed in the literature. We study the validity of the new model for the data sets and show that UBS distribution presents good fits on both cases.




 §.§ Income-consumption data



In order to evaluate the new probabilistic model, a public data set of income and consumption data was used.
The data (available at <https://www.bancaditalia.it/statistiche/tematiche/indagini-famiglie-imprese/bilanci-famiglie/documentazione/ricerca/ricerca.html?min_anno_pubblicazione=2008   max_anno_pubblicazione=2008>, accessed on July 22, 2024) comes from the Bank of Italy's Survey on Household Income and Wealth from the year 2008.


The income comprises payroll income, pensions and net transfers, net self-employment income, and property income. 






Consumption consists of expenditures on both durable and non-durable goods. Durable expenditure represents the net balance of purchased and sold goods. Non-durable expenditures include monthly expenses such as rent, food, and other essentials, as well as annual expenses.

We used the data from the survey, available as the data sets RFAM08 (income as Y variable) and RISFAM08 (expenditures as C variable), according to the Survey on Household Income and Wealth 2008 data description file.
We removed from the data set lines whose income or consumption was negative or unavailable.
Thus, the analyzed data set included information from 7,957 families, after removing 20 lines of the data.

The same data set was also used in <cit.> and <cit.>, however, they used a copula approach to model the data and compared income and consumption through stress-strength measures of the type ℙ(X<Y).
Our framework involves constructing a new variable U=X/(X+Y).
In summary, when X and Y represent expenditures and income, respectively,  U < 1/2 indicates that the family ended the year with more expenditures than income. If U > 1/2, the opposite is true. The case U = 1/2 means expenditures and income are equal.


Table <ref> presents descriptive statistics for U=X/(X+Y).
The estimated parameters of UBS model was θ_ρ = (0.275, 0.274, 1.041, 1.331, 0.149).
Figure <ref> shows the fit of the UBS distribution to the density and ECDF of the data. The histogram of U shows that with high (81.5%) probability U>1/2. That means, in general, the families have income greater than expenditures.

















 
 
 
 
 
 



Given that the U data has unit support, we can make a comparison between the utilization of the UBS model and other families of unit distributions, such as Beta and the unit bimodal BS (UBBS[The UBBS model was proposed in <cit.>.]) models, as illustrated in Table <ref>.
To obtain the estimation results for Beta and UBBS models in Table <ref>, we used the package AdequacyModel in the R software.
Deciding the better distribution to model the data was oriented by AIC (Akaike information criterion) and BIC (Bayesian information criterion) criteria.
In Table <ref>, these criteria suggest UBS fits the data better than Beta and UBBS models.









 §.§ Body mass data



Biomedical measurements of 202 athletes from different sports are presented in the Australian Institute of Sport (AIS) data set. <cit.> proposed the UBBS model and showed its fit on the distribution of the percentage of the body fat complement of the athletes. In this subsection, we show that the new UBS model is an alternative distribution to fit that data.
The data were imported directly through the software R by the command:


Descriptive statistics for 𝐮 (the athlete's body mass in the data set) are presented in Table <ref>.
As the data set has unit support, the UBS distribution is a candidate to model such data. 
Figure <ref> shows the fit of UBS to the data.
Maximum likelihood estimates and maximum product of spacings estimates can be compared graphically. 







In Table <ref>, when comparing the UBS model (with parameter estimates via product of spacings) with the Beta and UBBS models using information criteria, the BIC indicates the new model is better to fit the athlete’s body mass data. However, when considering the AIC, the UBBS model should be chosen.
	
	
	



	


§ CONCLUDING REMARKS

	

	
	

In this paper, we have introduced a new unit distribution. Its generation as a ratio of possibly dependent Birnbaum-Saunders random variables has been discussed in detail. As a more theoretical part, we have derived the cumulative distribution function and the stress-strength probability. As practical issues, we presented maximum product of spacings estimators and applications to income-consumption and body mass data sets.

The research could be continued in several directions. One could investigate the moments and the moment generating function and try to express shape characteristics like skewness and kurtosis. The analysis of the entropy of the distribution could also be of interest. Instead of using the ratio Y/(X+Y) one could explore other construction principles like those mentioned in the introduction to obtain a unit distribution. Finally, one could use a bivariate distribution other than the Birnbaum-Saunders to construct a unit distribution.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

  
Acknowledgements
	This study was financed in part by the
	Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
	
	

  
Disclosure statement
	There are no conflicts of interest to disclose.
	
	
	
	
	
	
	
	
	
	
	
	
	10
		
		[Abramowitz and Stegun, 1972]Abramowitz1972	
		M. Abramowitz and I. A. Stegun, 
		(eds.): Handbook of Mathematical Functions,
		9th edn. Dover, New York (1972).
		
		[Afifi et al., 2022]Afifi2022		
		A. Z. Afifi, M. Nassar, D. Kumar, and G.M. Cordeiro, 
		A New Unit Distribution: Properties, Inference, and Applications,
		Electronic Journal of Applied Statistical 
		Analysis (2022), 
		15(02), pp. 460–484, 
		
		
		[Bekker et al., 2009]Bekker2009
		 A. Bekker, J. Roux, and T. Pham-Gia,
		The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables,
		Statistical Papers (2009), 50, pp. 323–338.	
		
		[Birnbaum and Saunders, 1969a]Birnbaum1969a	
		Z. W. Birnbaum and S. C. Saunders,  
		A new family of life distributions,
		Journal of Applied Probability (1969), 
		6(2), pp. 319-327.
		
		[Birnbaum and Saunders, 1969b]Birnbaum1969b
		Z. W. Birnbaum and S. C. Saunders, 
		Estimation for a family of life distributions with applications to fatigue,
		Journal of Applied Probability (1969), 
		6(2), pp. 328–347.
		
		[Bourguignon et al., 2024]Bourguignon2024	
		M. Bourguignon, H. Saulo, and D. Gallardo. 
		Parametric quantile beta regression model, 
		International Statistical Review (2024), 92, pp. 106–129.
		
		[Cheng and Amin, 1979]ChengAmin1979
		R. C. H. Cheng and N. A. K. Amin,
		Maximum product-of-spacings estimation with applications to the lognormal distribution, Technical Report, Department of Mathematics, University of Wales (1979).

		[Cheng and Amin, 1983]ChengAmin1983
		R. C. H. Cheng and N. A. K. Amin,
		Estimating parameters in continuous univariate distributions with a shifted origin, Journal of the Royal Statistical Society. Series B (1983), 3, pp. 394–403.

		[Condino and Domma, 2023]Condino2023	
		F. Condino and F. Domma,
		Unit Distributions: A General Framework, Some Special 
			Cases, and the Regression Unit-Dagum Models, 
		Mathematics (2023), 11, 2888. 
		
		
		[Cortés et al., 2018]Cortes2018	
		M. A. Cortés, D.  Elal-Olivero, and J.,F. Olivares-Pacheco, 
		A New Class of Distributions Generated by the Extended Bimodal-Normal Distribution, 
		Journal of Probability and Statistics, (2018), 1-10.

	[Desmond, 1985]Desmond1985		
		A. Desmond,  
		Stochastic models of failure in random environments,
		Canadian Journal of Statistics (1985), 
		13(3), pp. 171–183. 
		


	[Domma and Giordano, 2012]domma2012stress	
		F. Domma and S. Giordano,
		A stress–strength model with dependent variables to measure household financial fragility, 
		Statistical Methods & Applications (2012), 21, pp. 375–389. 
  
		[Johnson  et al.(1995)]Johnson95
		N. L. Johnson, S. Kotz, and N. Balakrishnan,   
		Continuous Univariate Distributions - Vol. 2. 
		John Wiley & Sons, New York (1995).	
		
		[Jørgensen, 1982]Jorgensen1982	
		B. Jørgensen, 
		Statistical Properties of the Generalized Inverse Gaussian Distribution,
		Lecture Notes in Statistics, vol. 9. 
		Springer, New York (1982).
		
		[Kundu and Gupta, 2017]Kundu2017
		D. Kundu and R. C. Gupta,  
		On Bivariate Birnbaum-Saunders Distribution,
		American Journal of Mathematical and Management Sciences (2017),  36(1), pp. 21–33.
		
		[Kundu et. al, 2010]Kundu2010
		D. Kundu, N. Balakrishnan and A. Jamalizadeh,
		Bivariate Birnbaum-Saunders distribution and associated inference,
		Journal of Multivariate Analysis (2010), 101, pp. 113–125.

	[Lima et al., 2024]Lima2024assessing
		R. Lima,  F. Quintino, T. da Fonseca, L. Ozelim, P. Rathie and H. Saulo,
		Assessing the Impact of Copula Selection on Reliability Measures of Type P(X<Y) with Generalized Extreme Value Marginals,
		Modelling (2024), 5(1), pp. 180-200.

[Martínez-Flórez et al., 2024]Martinez2024unit
G. Martínez-Flórez, N. Olmos and O. Venegas,  Unit-bimodal Birnbaum-Saunders distribution with applications, Communications in Statistics-Simulation and Computation (2024), 53(5), pp. 2173-2192.
  
		[Teimouri et. al, 2013]Teimouri2013
		M. Teimouri, S. M. Hosseini and S. Nadarajah,
		Ratios of Birnbaum-Saunders Random Variables,
		Quality Technology & Quantitative Management (2013),
		10, pp. 457–481.
		
		[Quintino et. al, 2024]Quintino2024
		F. S. Quintino, L. C. d. S. M. Ozelim, T. A. d. Fonseca and P. N. Rathie,  
		Stress-Strength Reliability of the Type ℙ(X<Y) for Birnbaum-Saunders Components: A General Result, Simulations and Real Data Set Applications,
		Modelling (2024), 5, pp. 223–237.
		
		[Ranneby, 1984]Ranneby1984
		B. Ranneby,
		The maximum spacing method. An estimation method related to the maximum likelihood method,
		Scandinavian Journal of Statistics (1984),
		11, pp. 93–112.

		[Saulo et. al, 2019]Saulo2019
		H. Saulo, J. Leão, R. Vila, V. Leiva and V. Tomazella,
		On mean-based bivariate Birnbaum-Saunders distributions: Properties, inference and
			application, 
		Communications in Statistics - Theory and Methods (2019), 49(24), pp. 6032–6056.
		
		[Vila et al., 2024a]Vila2024a	
		R. Vila,  L. Alfaia, A.F.B. Menezes, M.N. Çankaya, and  M. Bourguignon,  
		A Model for Bimodal Rates and Proportions, 
		Journal of Applied Statistics (2024), 51(4), pp. 664–681.
		
		[Vila et al., 2024b]Vila2024b	
		R. Vila, N. Balakrishnan, and Bourguignon, M,
		On the distribution of a random variable               
			involved in an independent ratio, 
		Communications in Statistics-Theory and 
		Methods (2024), 
		<https://doi.org/10.1080/03610926.2024.2340608>.
		
		[Vila et al., 2024c]Vila2024c	
		R. Vila, N. Balakrishnan, H. Saulo, and P. Zörnig, Unit-log-symmetric models: characterization, statistical properties and their applications to analyzing an internet access data, 
		Quality and Quantity (2024),
		<https://doi.org/10.1007/s11135-024-01879-w>.
		
		[Zörnig, 2023]Zornig2023	
		P. Zörnig, 
		A new type of skew uniform distribution, 
		Asian Journal of Statistical Sciences (2023), 
		3, pp. 27–39.	
		
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	


 Paper Title: Comparative Analyses of the Type D ASEP: Stochastic Fusion and Crystal Bases.



Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================



Accessibility Statement: a WCAG2.1AA compliant version of this PDF will be posted at https://www.math.tamu.edu/undergraduate/research/REU/the Texas A&M University Mathematics REU Website. We thank the National Science
Foundation’s Office for Strategic Initiatives on ‘STEM Access for Persons with Disabilities’ for funding the creation of the accessible version.



§ ABSTRACT
 
The Type D asymmetric simple exclusion process (ASEP) is a particle system involving two classes of particles that can be viewed from both a probabilistic and an algebraic perspective <cit.>. From a probabilistic perspective, we perform stochastic fusion on the Type D ASEP and analyze the outcome on generator matrices, limits of drift speed, stationary distributions, and Markov self-duality. From an algebraic perspective, we construct a fused Type D ASEP system from a Casimir element of _q(𝔰𝔬_6), using crystal bases to analyze and manipulate various representations of 𝒰_q(𝔰𝔬_6). We conclude that both approaches produce different processes and therefore the previous method of <cit.>, which analyzed the usual ASEP, does not generalize to all finite-dimensional simple Lie algebras.



§ INTRODUCTION
 
<cit.> is the first paper considering two-class ASEP. Later research by <cit.> shows that generators of multi-class interacting particle systems can be constructed using a central element from various Drinfeld-Jimbo quantum subgroups of _q(𝔤𝔩_n) <cit.>. In <cit.>, the probabilistic procedure of stochastic fusion is developed and it is shown to produce the same process as central elements on normal ASEP. We focus on an interacting particle system, labeled the Type D ASEP <cit.>, that is a generalization of Spitzer's ASEP <cit.>. The Type D ASEP involves two classes of particles jumping on a lattice, where two particles of different classes can exist at a site but two particles of the same class cannot. The jump rate of an individual particle depends on its class and configuration in relation to other particles.

We question whether constructing a generator according to the probabilistic method of stochastic fusion defined in <cit.> will result in the same generator as the algebraic method, as studied in <cit.>, of applying a ground state transformation on a Casimir element in the second tensor power of an irreducible representation of _q(_6). We find that, unlike in Type A Lie algebras, this is not the case for all Type D Lie algebras; in fact, the probabilistic generator allows for nine states at each lattice site while the algebraic generator allows for fourteen. We therefore explore various characteristics of the different processes generated using these separate approaches. The probabilistic characteristics include limits of component generator matrices, stationary distributions, spectral gaps, and Markov self-duality. In terms of algebraic characteristics, we study the algebraic structure of various representations needed to construct the generator, creating crystal bases and Young tableaux to decompose _q(_6)-modules into irreducible representations as well as weight spaces, then reflecting on the impacts these decompositions have on the eigenvalues and overall structure of the constructed Markov generator.

The paper proceeds by the following outline: algebraic and probabilistic notation is introduced in the remainder of [section–Intro]Section 2, probabilistic and algebraic results are stated in [section–ProbResults]Section 3 and [section–AlgResults]Section 4, respectively, and the respective proofs are discussed in [section–ProbProofs]Section 5 and [section–AlgProofs]Section 6. We list a variety of matrices and methods of calculating them in the [section–Appendix]Appendix.

Acknowledgements. This research was conducted during a mathematics REU at Texas A&M University, funded by the National Science Foundation under REU Site Grant DMS-2150094 and the National Science Foundation’s Office for Strategic Initiatives on ‘STEM Access for Persons with Disabilities’. The authors express their gratitude to their mentor, Dr. Jeffrey Kuan, and teaching assistant, Kennedi Hatcher, for their involvement in the project. 



 §.§ Probability Notation
 


  §.§.§ Type D ASEP


The Type D ASEP is a Markov process in which there are two classes of particles jumping on a lattice. Two particles of the same class cannot share a lattice site, but two particles of different classes can. The Type D ASEP is determined by parameters (q,n,δ = 0). The variable n measures drift speed, with larger values of n indicating more rapid drifts in the Type D ASEP. The variable q represents drift direction, with rightwards drift for 0<q<1 and leftwards drift for q > 1. Note that q cannot equal 1. The variable δ represents the interaction between the two particle classes. The jump rate of a particle in the Type D ASEP is determined by the particle's class and configuration on the lattice.

We use Γ to denote lattice sites of the Type D ASEP. For each Γ site, we use 0 to denote an empty site (contains no particles), 1 to denote a particle of class 1 at a site, 2 to denote a particle of class 2, and 3 to denote a particle of class 1 and a particle of class 2.  Furthermore, we use parentheses to denote a Γ state in the Type D ASEP. For example, we let (3,1,0,2) denote the four Γ state particle configuration in the top line of Figure <ref>.

We now consider the Type D ASEP over two Γ lattice sites with δ = 0. See <cit.> for a rigorous definition and <cit.> for a picture of the generator of Type D ASEP on two Γ lattice sites. Note that there are sixteen (a,b) possible states since 0 ≤ a,b ≤ 3. We use L_p^(2) to denote the generator for Type D ASEP on two Γ sites, which is a direct sum of 

    L_1 = [                               *     (- q^1 - n + q^n - 1)^2/q^2 - q^2 n - 4 + q^2 n - 2 + 2/q^2;           q^2 n - q^2 n - 2 + 2                               *       - q^2 - 2 n + 2 + q^- 2 n         (- q^1 - n + q^n - 1)^2;      q^2(- q^1 - n + q^n - 1)^2   2 q^2 + q^2 - 2 n - q^4 - 2 n                               *   2 q^2 + q^2 - 2 n - q^4 - 2 n;           q^2 n - q^2 n - 2 + 2         (- q^1 - n + q^n - 1)^2       - q^2 - 2 n + 2 + q^- 2 n                               * ]

corresponding to the communicating classes {(3,0),(2,1),(0,3),(1,2)}, four 2 by 2 blocks

    L_2 = [                         *   q^1 - 2 n + q^2 n - 1/q; q (q^1 - 2 n + q^2 n - 1)                         * ]

corresponding to the communicating classes {(1,0),(0,1)},{(2,0),(0,2)},{(3,1),(1,3)},{(3,2),(2,3)}, and four 1 by 1 blocks of zeros corresponding to the communicating classes {(0,0)},{(1,1)},{(2,2)},{(3,3)}. Finally we express L_p^(2),

    L_p^(2) = L_1 ⊕⊕_i=1^4 L_2 ⊕⊕_i=1^4 [0]

with respect to the ordered basis

    Ω = ( (3,0),(2,1),(0,3),(1,2),(1,0),(0,1),(2,0),(0,2),(3,1),(1,3),(3,2),(2,3),(0,0),(1,1),(2,2),(3,3) ).

Futhermore, we expand the definition of Type D ASEP to K lattice sites where the generator matrix is given by

    L = L^1,2+L^2,3+...+L^K-1,K

We let L^x,x+1 denote the matrix the matrix on lattice sites x and x+1. For this paper, we almost always restrict to the case of Type D ASEP on 4 lattice sites. Therefore we define,

    L_p  L^1,2+L^2,3+L^3,4

to represent the generator of Type D ASEP on four Γ sites. The generator L_p is a 256 by 256 matrix, 256 = 4^4. Furthermore, we define L_m to be the generator where only the two middle particles interact, ie. L_m  L^2,3.



  §.§.§ Stochastic Fusion

Taking a lattice with an even number of sites, we define stochastic fusion as the merging of each set of two consecutive lattice sites, such that the particles in each original site of the set are stacked at the new site. Note that two particles of the same species can exist at one new site, and that the total number of sites involved in the process is decreased by one half through stochastic fusion. See <cit.> for a thorough treatment of stochastic fusion.
We proceed to define notation for the fused Type D ASEP, using γ to denote lattice sites and states that have undergone the process of fusion, and Γ to denote lattice sites and states that have not. For γ lattice  sites, we use 11 to denote two particles of class 1 at a site, 22 to denote two particles of class 2, 3 to denote a particle of class 1 and a particle of class 2, 33 to denote two particles of class 1 and two particles of class 2, 31 to denote two particles of class 1 and one particle of class 2, and 32 to denote one particle of class 1 and two particles of class 2.  We will use angle brackets to denote γ states. See Figure <ref> for a visualization of this notation. As another example of γ-state notation, note that ⟨ 31,2 ⟩ is the γ state of (3,1,0,2) and is depicted in the bottom line of Figure <ref>.





Let S be the state space for the four-site Type D ASEP, in which there are 256 = 4^4 possible states. Rigorously, 
    S = {(a, b, c, d) | a, b, c, d ∈{0,1,2,3}}
 Fusing our lattice, we reach the state space Ŝ of two γ lattice  sites, in which there are 81 = 9^2 possible states. Rigorously, 
    Ŝ = {⟨ x, y ⟩| x, y ∈{0,1,2,3,11,22,31,32,33}}


Our generator, fusion, and fission matrices depend on the ordering of states, so we order states in the Type D ASEP over two Γ lattice  sites as Ω and we order states in the Type D ASEP over one γ lattice site as 


    χ = (  0 , 1 ,  2 ,  11 ,  12 ,  22 , 31 , 32 ,  33 )
 
 A visualization of this ordering and the possible states over one γ lattice site can be seen in Figure <ref>. See section <ref> for the ordering of Γ and γ states over four Γ lattice sites and two γ lattice sites, respectively.

We now formally define the stochastic fusion map.



We define ϕ:Ω→χ as the stochastic fusion map for the Type D ASEP on two Γ lattice  sites, where x_i ∈{0,1,2,3} describes the particles at Γ lattice site i:

    ϕ(x_1,x_2)  0     x_1,x_2 = 0 
     x_1     x_1 ≠ 0, x_2 = 0 
     x_2     x_1 = 0, x_2 ≠ 0  
     x_1x_2     x_1,x_2 ≠ 0, x_1 ≥ x_2 
     x_2x_1     x_1,x_2 ≠ 0, x_1 ≤ x_2



This definition is implied as a specific case in <cit.> when there are only class 1 and class 2 particles. Notice that this map corresponds to stacking particles where order does not matter. To perform stochastic fusion on four Γ lattice sites of the Type D ASEP the stochastic fusion map is ϕ⊗ϕ, and so on for additional sites of stochastic fusion.



  §.§.§ Transition and Generator Matrices


We now turn to the transition matrices of the Type D ASEP. Let P_t be the transition matrix and L_p be the generator matrix for the Type D ASEP on four Γ sites defined previously. Note that L_m, the generator matrix for the Type D ASEP on four Γ lattice sites where only the two middle Γ lattice sites interact, is defined in <ref>. Fusing down to two γ lattice sites, Q_t is the fused transition matrix and L_Q is the generator matrix for the Type D ASEP on two γ lattice  sites. Summarizing this process, we let Λ: Ŝ→ S be the matrix of the fission map from the Type D ASEP on two γ lattice  sites to Type D ASEP on four Γ sites, and Φ: S →Ŝ be the matrix of the stochastic fusion map from the Type D ASEP on four Γ lattice sites to two γ sites. See Figure <ref> for an example of Λ and Φ mappings.

We also define notation for the two-Γ site Type D ASEP. Let Λ^(2) be the matrix of the fission map from the Type D ASEP on two Γ lattice  sites to one γ site, and Φ^(2) be the corresponding matrix of the stochastic fusion map.




 §.§ Algebraic Notation

We must first define _q(_6) as well as its fundamental representation in order to construct more complex _q(_6)-modules. We introduce crystal base theory as needed in Section <ref>.

    The special orthogonal Lie group SO_6 is the multiplicative group with elements of the form
    
    SO_6 = { X ∈ M_6 × 6 (): XX^T = I,  X = 1}




    The special orthogonal Lie algebra _6 is the Lie algebra with elements of the form
    
    𝔰𝔬_6 = {[    A    C; -C^T    B;      ]: A,B,C ∈ M_3 × 3 (),  A = - A^T, B=-B^T}





    The Universal Enveloping Algebra of _6, denoted (_6), is generated by 

    {E_1, E_2, E_3, F_1, F_2, F_3, H_1, H_2, H_3} and the following relations:
    
    [E_i, F_i] = H_i,  1 ≤ i ≤ 3

    and
    
    E_l^2E_j + E_jE_l^2 = 2E_lE_jE_l;  F_l^2F_j + F_jF_l^2 = 2F_lF_jF_l

    for (l,j) ∈{ (1,2), (2,1), (1,3), (3,1) }.



    The q-deformed quantum group _q(_6), as described by <cit.> and <cit.>, is generated by {E_1, E_2, E_3, F_1, F_2, F_3, q^H_1, q^H_2, q^H_3} and the relations:
    
    [E_i, F_i]    = q^H_i - q^-H_i/q-q^-1
    
        q^H_iE_j    = q^α_i ·α_jE_j q^H_i
    
        q^H_iF_j    = q^-α_i ·α_jF_j q^H_i

    for 1 ≤ i,j ≤ 3 and 
    
    E_l^2E_k + E_kE_l^2 = (q+q^-1)E_lE_kE_l;  F_l^2F_k + F_kF_l^2 = (q+q^-1)F_lF_kF_l

    for (l,k) ∈{ (1,2), (2,1), (1,3), (3,1) }.
    
    For a matrix H in the Cartan subalgebra 𝔥 of _6, let L_i ∈𝔥^* map H to H_i,i. Then we define the simple roots of _q(_6) to be
        
    Π = {α_1 = L_1 - L_2, α_2 = L_2 - L_2, α_3 = L_2 + L_3}

    and the fundamental weights to be
        
    {ω_1 = L_1, ω_2 = 1/2(L_1+L_2-L_3), ω_3 = 1/2(L_1+L_2+L_3) }.



 
    We define the coproduct Δ, counit ϵ, and antipode S of _q(_6) to be
    
    Δ (E_i) = E_i ⊗ 1 + q^H_i⊗ E_i, ϵ(   E_i) = 0, S(E_i) = -E_iq^H_i
    Δ(F_i) = 1 ⊗ F_i + F_i ⊗ q^-H_i, ϵ(   F_i) = 0, S(F_i) = -q^-H_iF_i 
    Δ(q^H_i) = q^H_i⊗ q^H_i,ϵ(   q^H_i) = 1,
            S(q^H_i) = q^-H_i.

    Equipping _q(_6) with Δ, ϵ, and S makes the quantum group a Hopf algebra.




 
    A fundamental representation of 𝒰_q(𝔰𝔬_6) is defined to be the subset of M_6 × 6([q,q^-1]) as follows: 

    
    
    E_i,j indicates the 6× 6 zero matrix with a 1 in the (i,j)^th entry, and q^-H_i is defined as the multiplicative inverse of q^H_i. We denote V to be ^6 ⊗^6.





§ PROBABILISTIC RESULTS



We begin by defining the generator for stochastic fusion on the Type D ASEP, noting the block diagonal form of this matrix. We then examine the unique characteristics of the generator's component blocks (communicating classes), particularly their limits and spectral gaps as drift speeds are taken to infinity and their stationary distributions in relation to drift speed. Fixing drift speed to two, we explore two approaches to finding the matrix of Markov self-duality for our system. The matrix of Markov self-duality will be expressed in terms of the eigenvectors of the block-diagonalized fusion generator, allowing for the dual and spectral gap to be found through the same eigenvalue-eigenvector calculation. 



 §.§ Producing Stochastic Fusion Matrix

In this section we perform stochastic fusion on Type D ASEP for four Γ lattice sites following the procedure developed in Section 3 of <cit.>. To begin, we define 

    Q_t Λ P_t Φ

Taking the derivative on both sides at zero, we express the same equation in terms of generators.

    L_Q = Λ L_p Φ

The goal of this section is to explicitly calculate L_Q, the generator matrix for two γ sites. This process reduces the 256 × 256 matrix L_p to the fused 81 × 81 matrix L_Q.

We construct Λ^(2) based on the reversibility measures given in Proposition 1.3 of <cit.> and we define ΛΛ^(2)⊗Λ^(2) in <ref>. Similarly, we construct Φ^(2) based on the map ϕ from two Γ lattice sites to a single γ lattice site, and we define ΦΦ^(2)⊗Φ^(2) in <ref>.

The generator matrix L_m corresponds to only the two middle of the four Γ lattice sites interacting, and can be expressed as a permutation of Id_16 × 16⊗ L_p^(2) as shown in <ref>, this stands in contrast to L_p where all four Γ sites can interact. We show that only considering when the two middle of the four Γ sites interact will lead to the same stochastically fused process L_Q.

[]theoremMiddleSwap

The generator for Type D ASEP on two γ sites denoted L_Q is equal to Λ L_m Φ, where L_m is the generator for Type D ASEP on four Γ sites where only the middle two Γ sites interact.


See a proof of this theorem in section <ref>.

Next, we want to understand the communicating classes of L_Q. We show that L_Q can be broken up into twenty-five communicating classes, where several of the generators on these twenty-five communicating classes are the same and every communicating class with the same amount of states has the same generator. We define L^Q_D as a permutation of L_Q that is a block diagonal matrix, where the ordering is based on the communicating classes.

[]propositionDiagonalBlockOrdering

Let L^D_Q = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕⊕_i=1^4 [0], where ℒ_9 is the generator for the 9-state communicating class, ℒ_6 is the generator for each 6-state communicating class, and so on for ℒ_4,ℒ_3 and ℒ_2. There exists a permutation matrix C such that L_Q = CL^D_QC^-1.

The permutation of states from L_Q to the block diagonal L^D_Q which defines the permutation matrix C is listed in <ref>. Below are the four three-state communicating classes of L_Q: 
    { 1,1 ,  0,11 ,  11,0 }, { 2,2 ,  0,22 ,  22,0 }, { 31,31 ,  11,33 ,  33,11 }, { 32,32 ,  22,33 ,  33,22 }
.

    ℒ_3 =  [                           * q^2+q^4n/(q^6+2q^4+q^2)q^2n q^6+q^4n+4/(q^4+2q^2+1)q^2n;               q^2+q^4n/q^2n                           *                           0;             q^2+q^4n/q^2n+2                           0                           * ]

The remaining ℒ_9,ℒ_6,ℒ_4,ℒ_2 matrices are outlined in section <ref>.



 §.§ Taking n to Infinity and Spectral Gaps


Recall that n affects drift speed, with larger values of n causing faster drifting in the Type D ASEP. Also recall that Type D ASEP particles drift to the right for 0<q<1 and to the left for q > 1. We are interested in how the drift speed of the Type D ASEP affects the drift speed of the stochastically fused process and its relaxation times.

Faster drift speeds in the Type D ASEP correspond to faster drift speeds in the fused Type D ASEP process. Recall that L_m is the generator for the Type D ASEP on four lattice sites where only the two middle Γ sites interact, and note that L_m depends on n. Thus, increasing values of n will indicate faster drift speeds for the Type D ASEP on four Γ lattice sites where only the two middle Γ sites interact. We see that Λ and Φ do not depend on n, so faster drift speeds in L_m will lead to faster drift speeds in L_Q, the fusion matrix. This statement is proved rigorously below.

[]lemmacommutativityProposition

We have the following limits: lim_n →∞ L_Q = lim_n →∞ (Λ L_m Φ) = Λ (lim_n →∞ L_m) Φ. In other words, the limit as n →∞ is commutative on L_Q.


[]propositionfiniteProposition

The following are finite: lim_n →∞ (q^-2n L_m) for q >1 and lim_n →∞ (q^2n L_m) for 0 < q < 1.



[]theoremfiniteLimitTheorem

The following are finite: lim_n →∞ (q^-2n L_Q) for q > 1, and lim_n →∞ (q^2n L_Q) for 0 < q < 1.


See a proof of this theorem in section <ref>.

The previous theorem proves the intuitive statement that, as the drift speed of the Type D ASEP on four Γ lattice sites increases, so does the drift speed of the stochastically fused process.

Continuing our study of limits as n approaches infinity, we now consider the limits of spectral gaps. The spectral gap of a Markov process is the absolute value of the second-largest (ie. least negative) eigenvalue of the generator matrix. Note that eigenvalues of generator matrices are less than or equal to zero. From our calculations, the spectral gaps corresponding to generators ℒ_2 and ℒ_3 simplify to reasonable expressions, while the spectral gaps for ℒ_4,ℒ_6, and ℒ_9 do not simplify to expressions of a reasonable length. 


    
    The spectral gaps for ℒ_2 and ℒ_3 communicating classes are denoted in the form |λ_ℒ_i| and are:
    
    |λ_ℒ_2| = (q^4n-2+1)(q^4+1)/q^2n(q^2+1)


    |λ_ℒ_3|   =  | -(q^4n-2+1)(2q^4+q^2+2)/2q^2n(q^2+1) +(q^2 + q^4n)/q^2n(q^2+1) | = (q^4+1)(q^2+q^4n)/q^2n+2(q^2+1).



For q > 1 and 0 < q < 1, notice that the spectral gaps |λ_ℒ_2| and |λ_ℒ_3| go to infinity as n goes to infinity, meaning the relaxation times 1/λ_ℒ_2 and 1/λ_ℒ_3 go to zero. This implies that as the drift speed of the Type D ASEP goes to infinity, the time to convergence to the stationary distribution approaches zero for the 2 and 3-state communicating classes. The authors suspect the same is true for the 4, 6, and 9-state communicating classes.

Note that, for i ∈{ 2, 3 }, the eigenvalues of q^± 2nℒ_i are those of ℒ_i multiplied by q^± 2n. Thus, the spectral gaps of  q^± 2nℒ_2 and q^± 2nℒ_2 are |q^± 2nλ_ℒ_2| and |q^± 2nλ_ℒ_3|, respectively. Note that the following spectral gaps and relaxation times are the same for the 2 and 3-state communicating classes. The limit as n goes to infinity of the spectral gap of q^-2nℒ_i for q > 1 is 1. The limit as n goes to infinity of the spectral gap of q^2nℒ_i for 0 < q < 1 is q^4 + 1/q^2 + 1. Thus, the relaxation times for both generators for q > 1 are 1, and the relaxation times for 0 < q < 1 are q^2 + 1/q^4 + 1. This result suggests that the time to convergence to the stationary distribution increases (relative to the time to convergence in the previous paragraph) as n goes to infinity for the 2 and 3-state communicating classes. The authors suspect the same is true for the 4, 6, and 9-state communicating classes.


As n goes to infinity, some of the jump rates in the (time-rescaled) fused Type D ASEP converge to 0, which can be seen by taking the limit as n →∞ of entries in the fusion generator. Thus, as n approaches infinity, the fused Type D ASEP (where δ = 0 and time is rescaled by a factor of q^± 2n for respective values of q) degenerates to the usual ASEP. In this degeneration, we would expect the 2-state communicating class and the 3-state communicating class to have the same relaxation time, which is consistent with the calculations in the previous paragraph.     


Due to the difficulty of displaying the eigenvalues for general q and n, we note that for n=2, the eigenvalues simplify nicely and are used to find a dual for L_Q when n=2 in <ref>.

[]lemmaLeigenvalues

The eigenvalues for ℒ_9, ℒ_6, ℒ_4, ℒ_3, and ℒ_2 when n = 2 are the eigenvalues of L_Q^D for n = 2.





 §.§ Stationary Distributions
 

Take Markov process X_t with generator matrix L and stationary distribution π. Then π L = 0. In other words, π is a left eigenvector of L. 
We use this result to obtain Proposition <ref>, which depends on the communicating classes listed in Section <ref>.

[]propositionallStationary 
For the 9-state communicating class, the stationary distribution is the normalization of 

    [q^4(q^2+1)^4,1,q^16,q^2(q^2+1)^2,q^10(q^2+1)^2,q^2(q^2+1)^2,q^10(q^2+1)^2,q^8,q^8].


For the 6-state communicating class, the stationary distribution is the normalization of 

    [q^2(q^2+1)^2,q^6(q^2+1)^2,1,q^12,q^4,q^8].


For the 4-state communicating class, the stationary distribution is the normalization of 

    [q^8,q^4,q^4,1].


For the 3-state communicating class, the stationary distribution is the normalization of 

    [q^2(q^2+1)^2,1,q^8].


For the 2-state communicating class, the stationary distribution is the normalization of 

    [1, q^4].


for all q > 0 where q ≠ 1. 



Our advisor, Dr. Jeffrey Kuan, noticed that the stationary distributions we found resemble q-exchangeable measures <cit.>, but we do not delve into the details of this relationship in our paper.


In order to find the stationary distributions, we found the left eigenvectors corresponding to zero for n = 2 (drift speed 2), and then verified that the same eigenvectors work for general n. We then normalized the vectors to obtain our final stationary distributions. This emphasizes that, although the jump rates of particles in the Type D ASEP do depend on n, the stationary distributions do not.



 §.§ Markov Duality



Let X_t be a time-homogeneous Markov process on a discrete state space 𝒳. Label the generator matrix of the Markov process as L_X. Let D be a matrix with rows and columns indexed by 𝒳. Then X_t is self-dual with respect to the matrix D if L_X D = D L_X^T
<cit.>.




  §.§.§ Duality with Quantum q-Krawtchouk Polynomials
 

Our mentor, Dr. Jeffrey Kuan, previously dropped the term “quantum” from “quantum q-Krawtchouk polynomials” <cit.>. We have used the correct term in our paper. 


There are several papers on quantum q-Krawtchouk polynomials, see <cit.> for example. Furthermore, in Theorem 3.1 of <cit.>, the authors propose a self-duality function for the unfused Type D ASEP with parameters (q, n = 2, δ = 0) and (q, n = 3, δ = 0). We question whether their self-duality function works for L_Q^D, the generator of the Type D ASEP over two γ lattice sites with ordered state space 𝒳_2 and parameters (q, n = 2, δ = 0), when approached from a probability perspective.

Using the notation of <cit.> with minimal changes for clarity, we denote the self-duality function given by <cit.> as D_α_1, α_2^(L) (η, ξ) = D_α_1^(L) (η_1, ξ_1) · D_α_2^(L) (η_2, ξ_2) where D_α_i^(L) (η_i, ξ_i) is taken over particle class i ∈{1,2}. We take variables α_1, α_2 ∈ (0, q^4). Using the parameters L = 2 and  n = 2, we find a potential self-duality function (given by Theorem 3.1 in <cit.>) for this 2-site Type D ASEP, where D is the matrix of this proposed self-duality function. For each 𝒳_2[i] = η and 𝒳_2[j] = ξ, the (i,j)th entry of D is given by  D_α_1, α_2^(2) (η, ξ).

However, using the matrix D that we computed, L_Q^D D[1,2] ≠ D(L_Q^D)^T[1,2], so L_Q^D D ≠ D(L_Q^D)^T and thus D is not the matrix of the self-duality function of the Type D ASEP with two γ lattice sites. In our case, 
constructing our particle system from a probabilistic rather than an algebraic process <cit.>, resulted in our particle systems having different duality functions.



  §.§.§ Other Avenues to Duality


Since the aforementioned duality function does not generate a valid dual for the stochastically fused Type D ASEP, another logical approach is to find an algebraic connection between the dual of the Type D ASEP generator on two Γ lattice sites and the dual of the stochastically fused process on two γ lattice sites. Unfortunately, we were not able to find a direct algebraic relation between a dual of L_p^(2) and a dual of L_Q.

Due to the eigenvalues of L_Q not simplifying to reasonable expressions, we now restrict to the case where n = 2 and find a dual based on the eigenvectors. We construct a non-trivial dual for L^D_Q, a permutation of L_Q in block diagonal form defined in <ref>, for n = 2. Note that the eigenvalues for the communicating classes of L_Q^D when n=2 were already given in <ref>.

Recall the structure of the block diagonal generator in <ref>:

    L^D_Q = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕⊕_i=1^4 [0].

Since each communicating class is diagonalizable, we diagonalize L^D_Q block by block and express L_Q^D in a diagonalized form.

[]lemmaLdiagonalizable


    
    
  a. The block generators ℒ_9, ℒ_6, ℒ_4, ℒ_3, ℒ_2 are diagonalizable for n = 2. Let 𝒫_i denote the matrix of the right eigenvectors for the diagonalization of ℒ_i, and let 𝒜_i be the diagonal matrix of eigenvalues listed in <ref> such that ℒ_i = 𝒫_i 𝒜_i 𝒫^-1_i for i ∈{2,3,4,6,9}.
    
  b. Define 𝒫 = 𝒫_9 ⊕⊕_i=1^4 𝒫_6 ⊕⊕_i=1^4 𝒫_4 ⊕⊕_i=1^4 𝒫_3 ⊕⊕_i=1^8 𝒫_2, 𝒜 = 𝒜_9 ⊕⊕_i=1^4 𝒜_6 ⊕⊕_i=1^4 𝒜_4 ⊕⊕_i=1^4 𝒜_3 ⊕⊕_i=1^8 𝒜_2 and 𝒵 = ⊕_i=1^4 [0].
    
  c. The block diagonal matrix L_Q^D = 𝒫𝒜𝒫^-1⊕𝒵.




[]theoremDiagonalStrategy

The matrix 𝒟 = 𝒫𝒫^T ⊕𝒵 is a non-trivial dual for L_Q^D.

See a proof of this theorem in <ref>.

Since 𝒫 has a block diagonal form, it follows that 𝒫𝒫^T has a block diagonal form,

    𝒟 = 𝒫(𝒫)^T ⊕𝒵 = 𝒫_9(𝒫_9)^T ⊕⊕_i=1^4 𝒫_6(𝒫_6)^T ⊕⊕_i=1^4 𝒫_4(𝒫_4)^T ⊕⊕_i=1^4 𝒫_3(𝒫_3)^T ⊕⊕_i=1^8 𝒫_2(𝒫_2)^T ⊕𝒵.

Note that 𝒟 is a non-trivial dual because 𝒟 is not a constant matrix. For example

    𝒫_4(𝒫_4)^T =
    [          1 + 1/q^8 + 1/q^16    1 - q^4 - 1/q^8 - 1/q^12                  1 - 1/q^12           1 - 1/q^4 + 1/q^8;    1 - q^4 - 1/q^8 - 1/q^12 2 + (q^4 - 1)^2/q^8 + 1/q^8                       1/q^8     1 + q^4 - 1/q^4 - 1/q^4;                  1 - 1/q^12                       1/q^8                   2 + 1/q^8                   1 - 1/q^4;           1 - 1/q^4 + 1/q^8     1 + q^4 - 1/q^4 - 1/q^4                   1 - 1/q^4                           3 ].







§ ALGEBRAIC RESULTS
 
Previous research used the symmetries of Type A, C, and D quantum groups to construct asymmetric interacting particle systems on various lattice sites <cit.>. The main algebraic focus of this project was to investigate how this construction differs for the Type D Lie algebra _6. We are also interested in whether this method produced the same Markov generator as the probabilistic approach did. We found the following key result, with the second research question answered in Proposition <ref>. Our definition of a ground state transformation can be found in Definition <ref>.
 
    A Markov generator is produced by performing a ground state transformation on the representation W ⊗ W of <cit.>'s Casimir element; the explicit generator can be found in Section <ref>.

To prove this, we need to first construct the aforementioned representation of _q(_6). We therefore prove the following two propositions.
[]propositionpropositionWtwentydim

    Define W to be the 20-dimensional subspace of ^6 ⊗^6 satisfying 
    Sym_q^2(^6) = W ⊕span{q^-2e_1 ⊗ e_4 + q^2 e_4 ⊗ e_1 +q^-1e_2⊗ e_5 + qe_5⊗ e_2+ e_3 ⊗ e_6 + e_6 ⊗ e_3 }
  with {e_i| i = 1,...,6} standard basis vectors of ^6. Then, W is an irreducible representation of _q(_6).


    As a representation of _q(_6), W ⊗ W decomposes into a direct sum of irreducible representations and thus a direct sum of weight spaces, listed respectively as follows:
         
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0)

         Let ⟨ i,j,k ⟩ denote the weight space W ⊗ W [i,j,k]. Then
         
    W ⊗ W ≅   ⟨0,0,0⟩⊕⟨1,1,0⟩⊕⟨1,0,1⟩⊕⟨0,1,1⟩⊕⟨-1,1,0⟩⊕⟨-1,0,1⟩⊕⟨0,-1,1⟩⊕⟨1,-1,0⟩
       ⊕⟨1,0,-1⟩⊕⟨0,1,-1⟩⊕⟨-1,-1,0⟩⊕⟨-1,0,-1⟩⊕⟨0,-1,-1⟩⊕⟨2,0,0⟩⊕⟨0,2,0⟩
       ⊕⟨0,0,2⟩⊕⟨-2,0,0⟩⊕⟨0,-2,0⟩⊕⟨0,0,-2⟩⊕⟨2,1,1⟩⊕⟨1,2,1⟩⊕⟨1,1,2⟩⊕⟨2,1,-1⟩
       ⊕⟨2,-1,1⟩⊕⟨1,2,-1⟩⊕⟨-1,2,1⟩⊕⟨1,-1,2⟩⊕⟨-1,1,2⟩⊕⟨2,-1,-1⟩⊕⟨-1,2,-1⟩
       ⊕⟨-1,-1,2⟩⊕⟨-2,1,1⟩⊕⟨1,-2,1⟩⊕⟨1,-2,1⟩⊕⟨-2,1,-1⟩⊕⟨-2,-1,1⟩⊕⟨1,-2,-1⟩
       ⊕⟨-1,-2,1⟩⊕⟨1,-1,-2⟩⊕⟨-1,1,-2⟩⊕⟨-2,-1,-1⟩⊕⟨-1,-2,-1⟩⊕⟨-1,-1,-2⟩
       ⊕⟨2,2,0⟩⊕⟨2,0,2⟩⊕⟨0,2,2⟩⊕⟨-2,2,0⟩⊕⟨-2,0,2⟩⊕⟨0,-2,2⟩⊕⟨2,-2,0⟩⊕⟨2,0,-2⟩
       ⊕⟨0,2,-2⟩⊕⟨-2,-2,0⟩⊕⟨-2,0,-2⟩⊕⟨0,-2,-2⟩⊕⟨3,1,0⟩⊕⟨3,0,1⟩⊕⟨0,3,1⟩
       ⊕⟨1,3,0⟩⊕⟨1,0,3⟩⊕⟨0,1,3⟩⊕⟨3,-1,0⟩⊕⟨3,0,-1⟩⊕⟨0,3,-1⟩⊕⟨-1,3,0⟩⊕⟨-1,0,3⟩
       ⊕⟨0,-1,3⟩⊕⟨-3,1,0⟩⊕⟨-3,0,1⟩⊕⟨0,-3,1⟩⊕⟨1,-3,0⟩⊕⟨1,0,-3⟩⊕⟨0,1,-3⟩
       ⊕⟨-3,-1,0⟩⊕⟨-3,0,-1⟩⊕⟨0,-3,-1⟩⊕⟨-1,-3,0⟩⊕⟨-1,0,-3⟩⊕⟨0,-1,-3⟩
       ⊕⟨4,0,0⟩⊕⟨0,4,0⟩⊕⟨0,0,4⟩⊕⟨-4,0,0⟩⊕⟨0,-4,0⟩⊕⟨0,0,-4⟩.


Once we characterized W ⊗ W, we were able to extract enough information to construct a 400 × 400 representation of <cit.>'s Casimir element without significant computation, which brings us to our next proposition. 
 
     We can write π_W⊗ W(C) as a block matrix with one 22× 22 block, twelve 12× 12 blocks, six 8 × 8 blocks, twenty-four 4 × 4 blocks, twelve 3 × 3 blocks, twenty-four 2 × 2 blocks, and six 1 × 1 blocks. Denoting v_i to be the i^th basis vector of W, the blocked matrix is with respect to the ordered basis in Section <ref>.

Finally, we must perform a ground state transformation so that each row of π_W⊗ W(C) sums to 0. We introduce and tweak the method in Section <ref> to compute this transformation. 

[]propositionpropositiongroundstate

    Applying variations of
    
    ⟨ u_i ⊗ u_j| Δ(F_1)^k_1Δ(F_2)^k_2Δ(F_3)^k_3| e_1 ⊗ e_1 ⟩

    will generate zero values for the lowest weight vector of the _q(_6)-modules V, W, and W ⊗ W.

Using the previous four propositions, we compute a 196 × 196 fused Type D ASEP system generator from <cit.>'s central element represented in W ⊗ W, which can be found in Section <ref>. This proves Theorem <ref>, and also has probabilistic significance: as an interacting particle system, the process allows for 14 different states at each site, i.e., all states except four particles of the same class. This differs from the probabilistically-constructed generator, which does not allow for three particles of the same class on a lattice site.

Finally, we compare our result to the fused generator matrix L_Q defined in Section <ref> and written explicitly in Section <ref>.


    Regardless of which ground state transformation is applied to π_W ⊗ W(C), the resulting Markov generator will not match the probabilistically-generated matrix L_Q in Section <ref>. 





§ PROBABILISTIC PROOFS
 


 §.§ Producing Stochastic Fusion Matrix


To begin, consider the function ϕ. The left side of Figure <ref> depicts ϕ(3,1) =  31 and the right side depicts ϕ(0,2) =  2. Note that the function ϕ is surjective.

In order to construct Λ^(2), we recall from <cit.>, that Λ^(2) is dependent on the reversible measures of the process generated by L_p^(2), in other words Type D ASEP on two Γ sites. We define the function G^2 below which is expressed fully in Proposition 1.3 of <cit.> and gives the reversible measures for the Type D ASEP on two Γ lattice sites.

Any particle configuration on two lattice sites can be represented as a function η:{1,2}→{0,1,2,3}, a function from a lattice site to the particle configuration on that lattice site. Furthermore, A_1(η) ⊂{1,2} is the set of lattice sites that class 1 particles occupy and similarly for A_2(η). Finally, below is the function from Proposition 1.3 of <cit.>,

    G^2(η) = ∏_x ∈ A_1(η) q^-2x∏_x ∈ A_2(η) q^-2x

For simplicity, we represent the function η as an ordered pair (η_1,η_2), where η(1) = η_1 and η(2) = η_2. As an example for the reader to check their understanding with the new notation,

    G^2(2,1) = ∏_x ∈{2}q^-2x∏_x ∈{1}q^-2x = q^-6



    
    We construct and express Λ^(2), and define the fission map ΛΛ^(2)⊗Λ^(2).



Let the rows of Λ^(2) be indexed by x ∈χ and the columns be indexed ω∈Ω. We define

    (Λ^(2))_x, ω = 
           
        G^2(ω)    ϕ(ω) = x 
    
        0    ϕ(ω) ≠ x

with the additional step that the rows of Λ^(2) are normalized to sum to 1. Λ^(2) is below.

    [             0             0             0             0             0             0             0             0             0             0             0             0             0             0             0             1;             0             0             0             0     q^2/q^2+1       1/q^2+1             0             0             0             0             0             0             0             0             0             0;             0             0             0             0             0             0     q^2/q^2+1       1/q^2+1             0             0             0             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0             0             0             0             1             0             0; q^4/(q^2+1)^2 q^2/(q^2+1)^2   1/(q^2+1)^2 q^2/(q^2+1)^2             0             0             0             0             0             0             0             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0             0             0             0             0             1             0;             0             0             0             0             0             0             0             0     q^2/q^2+1             0             0             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0     q^2/q^2+1             0             0             0             0             0;             0             0             0             0             0             0             0             0             0             0             0             0             0             0             0             1 ]




    
    We construct and express Φ^(2), and define the fusion map ΦΦ^(2)⊗Φ^(2).



    
    Let the rows of Φ^(2) be indexed by ω∈Ω and the columns be indexed by x ∈χ. We define
    
    (Φ^(2))_ω , x = 
        
            1    ϕ(ω) = x 
    
            0    ϕ(ω) ≠ x

    Note that Φ^(2) is a deterministic map and Φ^(2) is the matrix representation of ϕ. Φ^(2) is below.
    

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



We now prove that L_m, the generator matrix for the Type D ASEP on four γ lattice sites where only the middle two Γ states interact, can be permuted into _16 × 16⊗ L^(2)_p. 


    
    There exists a permutation matrix J such that L_m = J(_16 × 16⊗ L_p^(2))J^-1. In other words, the generator matrix L_m can be reordered into _16 × 16⊗ L_p^(2).



    
    For a short proof by existence, consider the diagonal ordering of states of four Γ lattice sites listed in <ref>, we permute L_m according to these states and get _16 × 16⊗ L_p^(2).
    
    For a more intuitive proof, consider that any state in the Type D ASEP on four Γ lattice sites can be represented as (x_1,x_2,x_3,x_4) where x_1,x_2,x_3,x_4 ∈{0,1,2,3}. Letting only the middle Γ sites interact corresponds to x_1 and x_4 being held constant, while the interaction of x_2 and x_3 is governed by L_p^(2). Now consider the bijective function h:(x_1,x_2,x_3,x_4) ↦((x_1,x_4),(x_2,x_3)). Since (x_1,x_4) is held constant while (x_2,x_3) interact, it follows that L_m can be permuted into ⊕_i=1^16 L_p^(2), since there are 16 possibilities for (x_1,x_4). Finally, it follows that ⊕_i=1^16 L_p^(2) = _16 × 16⊗ L_p^(2). This proves that L_m can be permuted into _16 × 16⊗ L_p^(2).

Finally we prove <ref>, and therefore that the fused process only depends on the middle interacting particles.
*

    
    We can express L_p = L^1,2+L^2,3+L^3,4. Note that L_m  L^2,3, as stated in <ref>. Now we show that,
    
    L_Q Λ L_p Φ = Λ (L^1,2+L_m+L^3,4) Φ

    
    = Λ L^1,2Φ+Λ L_m Φ+Λ L^3,4Φ

    
    = (Λ^(2)⊗Λ^(2))L^1,2(Φ^(2)⊗Φ^(2)) + Λ L_m Φ + (Λ^(2)⊗Λ^(2))L^3,4(Φ^(2)⊗Φ^(2))

    
    = 0+Λ L_m Φ+0

    
    = Λ L_m Φ

    The second to last step follows because L^1,2 and L^3,4 are the generators acting on the sites that get fused down by Φ and fissioned by Λ. The fused state of the Type D ASEP on two Γ lattice sites which are being fused cannot change because the stochastically fused state is completely dependent on the number of type 1 and type 2 particles on those two Γ lattice sites. This cannot change when only those two sites interact.
    

*

    
    Permute L_Q according to the diagonal ordering for two γ given in <ref>. This is a permutation of L_Q according to its communicating classes.





 §.§ Taking n to Infinity
 

*
 
We remind the reader that Λ and Φ do not depend on n. Therefore,

    lim_n →∞ L_Q = lim_n →∞ (Λ L_m Φ) = Λ (lim_n →∞ L_m) Φ



*


Take q >1. Observing L^(2)_p, we see that multiplying it by q^-2n then taking the limit as n →∞ is finite. The same applies to multiplying L^(2)_p by q^2n for 0 < q< 1.  Finally, use <ref> and notice that

    q^± 2n L_m = q^± 2nJ(_16 × 16⊗ L^(2)_p)J^-1 = J(_16 × 16⊗ (q^± 2n L^(2)_p))J^-1
 for respective boundaries on q, by the properties of the Kronecker product and standard matrix multiplication. Noting that J and J^-1 do not depend on n, we take the limit as n →∞ of both sides and see that lim_n →∞ (q^± 2n L_m) = lim_n →∞ (J(_16 × 16⊗ (q^± 2n L^(2)_p))J^-1) = J(_16 × 16⊗ (lim_n →∞ q^± 2n L^(2)_p))J^-1, where the right (and therefore left) side of the equation is finite since lim_n →∞ (q^± 2n L^(2)_p) is finite for respective boundaries on q. This implies that the rate matrix grows on the order of q^2n irrespective of drift direction.


*


Note that Proposition <ref> suggests that multiplying q^± 2n L_m (for respective boundaries on q) by any constant preserves the finite limit, since the constant can be moved outside the limit. Note that 
    q^± 2n L_Q = q^± 2nΛ L_m Φ = Λ q^± 2n L_m Φ
 by the commutativity of scalar multiplication. Taking the limit of both sides of the equation, we have 
    lim_n →∞ (q^± 2n L_Q) = lim_n →∞ (Λ q^± 2n L_m Φ) = Λlim_n →∞ (q^± 2n L_m) Φ
 by Lemma <ref>. We know that lim_n →∞ (q^± 2n L_m) is finite for respective values of q, so lim_n →∞ q^± 2n L_Q must also be finite for such values of q.


*

We list the eigenvalues for the generator ℒ_9 of the 9-state communicating class:


    0, -q^8 - q^4 - 1/q^4, -q^8 + q^6 - 2q^4 + q^2 - 1/q^4, -q^12 - 4q^10 - 2q^8 - 6q^6 - 2q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4,


    -q^12 - 4q^10 + q^8 - 8q^6 + q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4

The eigenvalues for the fusion generator ℒ_6 of the 6-state communicating class:

    0, -q^8 - q^4 - 1/q^4,  -q^8 + q^6 - 2q^4 + q^2 - 1/q^4,


    -q^12 - 4q^10 - 2q^8 - 6q^6 - 2q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4,  -q^12 - 4q^10 + q^8 - 8q^6 + q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4

The eigenvalues for the fusion generator ℒ_4 of the 4-state communicating class:

    0, -q^8 + q^6 - 2q^4 + q^2 - 1/q^4,  -q^12 - 4q^10 + q^8 - 8q^6 + q^4 - 4q^2 - 1/q^8 + 2q^6 + q^4

The eigenvalues for the fusion generator ℒ_3 of the 3-state communicating class:

    0, -q^8 - q^4 - 1/q^4,  -q^8 + q^6 - 2q^4 + q^2 - 1/q^4

The eigenvalues for the fusion generator ℒ_2 of the 2-state communicating class:

    0, -q^8 + q^6 - 2q^4 + q^2 - 1/q^4





 §.§ Stationary Distribution

*
 
For communicating class i with generator matrix ℒ_i, the stationary distribution for the class is represented by the normalized left eigenvector π corresponding to the eigenvalue zero for the generator matrix, ie. π L_i = 0. Through standard computation, one can see that a left eigenvector exists for each communicating class and is equal to the one listed in the proposition, making the normalization of that vector the stationary distribution for the corresponding communicating class.




 §.§ Markov Duality



*

To prove part a, we show that for each communicating class ℒ_i, the matrix of right eigenvectors is invertible for q > 0 where q ≠ 1 (all possible values q can take), and therefore that all the eigenvectors are linearly independent, which implies diagonalizability for ℒ_i. Below we show 𝒫_2 is invertible. We list all the other cases in <ref>.

    𝒫_2 = [    1 -q^4;    1    1 ]

It follows that (𝒫_2) = q^4+1 ≠ 0 for q > 0. Therefore, 𝒫_2 is invertible for q > 0, and therefore ℒ_2 is diagonalizable for q > 0.

To prove part c, we show that L_Q^D is the direct sum of the diagonalization over all of the communicating classes.

    𝒫𝒜𝒫^-1⊕𝒵


    = (𝒫_9 ⊕⊕_i=1^4 𝒫_6 ⊕⊕_i=1^4 𝒫_4 ⊕⊕_i=1^4 𝒫_3 ⊕⊕_i=1^8 𝒫_2)(𝒜_9 ⊕⊕_i=1^4 𝒜_6 ⊕⊕_i=1^4 𝒜_4 ⊕⊕_i=1^4 𝒜_3 ⊕⊕_i=1^8 𝒜_2)


    (𝒫^-1_9 ⊕⊕_i=1^4 𝒫^-1_6 ⊕⊕_i=1^4 𝒫^-1_4 ⊕⊕_i=1^4 𝒫^-1_3 ⊕⊕_i=1^8 𝒫^-1_2) ⊕𝒵


    = 𝒫_9 𝒜_9 𝒫^-1_9 ⊕⊕_i=1^4 𝒫_6 𝒜_6 𝒫^-1_6 ⊕⊕_i=1^4 𝒫_4 𝒜_4 𝒫^-1_4 ⊕⊕_i=1^4 𝒫_3 𝒜_3 𝒫^-1_3 ⊕⊕_i=1^8 𝒫_2 𝒜_2 𝒫^-1_2 ⊕𝒵


    = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕𝒵


    = L_Q^D.





*

    
    Our goal is to prove that L_Q^D 𝒟 = 𝒟(L_Q^D)^T, and we use <ref>. 

    Consider,
    
    L_Q^D 𝒟 = (𝒫𝒜𝒫^-1⊕𝒵)(𝒫𝒫^T ⊕𝒵)

    
    = (𝒫𝒜𝒫^-1𝒫𝒫^T) ⊕𝒵𝒵 = 𝒫𝒜𝒫^T ⊕𝒵

    
    = 𝒫(𝒫^T (𝒫^T)^-1)𝒜𝒫^T ⊕𝒵

    
    = (𝒫𝒫^T)((𝒫^-1)^T𝒜𝒫^T) ⊕𝒵

    
    = (𝒫𝒫^T)(𝒫𝒜𝒫^-1)^T ⊕𝒵

    
    = (𝒫𝒫^T ⊕𝒵)((𝒫𝒜𝒫^-1)^T ⊕𝒵)

    
    = (𝒫𝒫^T ⊕𝒵)((𝒫𝒜𝒫^-1) ⊕𝒵)^T

    
    = 𝒟(L_Q^D)^T.

    Note that 
    𝒟 = 𝒫𝒫^T ⊕𝒵 = 𝒫_9(𝒫_9)^T ⊕⊕_i=1^4 𝒫_6(𝒫_6)^T ⊕⊕_i=1^4 𝒫_4(𝒫_4)^T ⊕⊕_i=1^4 𝒫_3(𝒫_3)^T ⊕⊕_i=1^8 𝒫_2(𝒫_2)^T ⊕𝒵

    and that
    
    𝒫_2(𝒫_2)^T = [ q^8+1 1-q^4; 1-q^4     2 ].

    contains non-trivial values, completing the proof.

Note that the remaining entries of 𝒟 can be easily calculated from <ref>, and that 𝒫_4(𝒫_4)^T is displayed in <ref>.



§ ALGEBRAIC PROOFS
 



 §.§ W is an irreducible representation


We begin by proving Proposition <ref>. First, we must first establish a few definitions. 

    We define P to be the weight lattice, with λ∈ P called a weight. A vector v_λ is called a highest weight vector if E_1v_λ = E_2 v_λ = E_3v_λ = 0. Then, a _q(_6)-module M is a highest weight module if M=_q(_6)v_λ, and will be denoted M=V(λ).

Since _q(_6) can be triangularly decomposed into its three subalgebras generated by {E_i}, {q^H_i}, and {F_i} (sometimes known as the Cartan decomposition), M can be computed by multiplying compositions of F_is with a highest weight vector v_λ  as described in <cit.>. Note that, in the fundamental representation of _q(_6), each E_i has an empty first column and thus e_1 ∈^6 is a highest weight vector. Denote λ = λ_1L_1 + λ_2L_2 + λ_3L_3. Since e_1 has weight L_1, we can use the following equation from <cit.> to determine the dimension of V(L_1):

    dim(V(λ)) = 1/12∏_1 ≤ i < j ≤ 3 (λ_i - λ_j +j-i)(λ_i+λ_j+6-i-j).

Therefore V(L_1) is a 6-dimensional _q(_6)-module, and V(L_1) is thus the entire fundamental representation.

We will prove multiple statements partially using the theory of crystal bases as stated in <cit.>. We use the correspondence between a weight module V(λ) and its crystal base (ℒ(λ), ℬ(λ)), formed of the crystal lattice and crystal basis respectively. We also use the tensor product rule, Theorem 4.4.1 on pg 83 of <cit.>. 
   V(L_1) ⊗ V(L_1) ≅ V(2L_1) ⊕ V(L_1 + L_2) ⊕ V(0).


    First, define 1 = e_1, 2 = e_2, 3 = e_3, 1̅ = e_4, 2̅ = -e_5, 3̅ = e_6. Then, note that V(L_1) has crystal base as shown in Figure <ref>, with arrow index i corresponding to Kashiwara operator f_i. 

    

    Then, by the tensor product rule, we can draw the crystal graph of V ⊗ V as shown in Figure <ref>. 




    Thus, by reading off the disjoint connected components in the crystal graph of Figure <ref>, we see that
    
    V(L_1) ⊗ V(L_1) ≅ V(2L_1) ⊕ V(L_1 + L_2) ⊕ V(0).

    Note that, by applying the corresponding F_i, we can trace a path of arrows to each vertex in W and obtain the corresponding basis vector of W. This process is used to generate the compositions of F_is shown in Lemma <ref>.


Now, we move on to constructing V(2L_1) explicitly.
 The _q(_6)-module V(2L_1) is 20-dimensional with basis

    { e_1 ⊗ e_1   ,  e_1 ⊗ e_2 + q^-1e_2 ⊗ e_1, e_1⊗ e_3 + q^-1e_3 ⊗ e_1, 
       -e_1 ⊗ e_4 +q^-1e_2 ⊗ e_5 - q^-2e_4 ⊗ e_1+q^-1e_5 ⊗ e_2, e_1⊗ e_5 +q^-1e_5 ⊗ e_1, 
       - e_1 ⊗ e_6-q^-1e_6 ⊗ e_1,  (q+q^-1)e_2 ⊗ e_2,e_2 ⊗ e_3 + q^-1 e_3 ⊗ e_2, 
       -e_2 ⊗ e_4 -q^-1e_4 ⊗ e_2, e_2 ⊗ e_5 - q^-1e_3 ⊗ e_6+q^-2e_5 ⊗ e_2 -(q^2+1)e_6 ⊗ e_3,
       -e_2 ⊗ e_6 -q^_1e_6⊗ e_2,(q^-1+q)e_3 ⊗ e_3, 
       -(q^2 +2 +q^-2)e_3 ⊗ e_4 - (q+2q^-1+q^-3)e_4 ⊗ e_3, 
       (q^2 + 2+q^-2)e_3 ⊗ e_5 + (q+2q^-1+q^-3)e_5 ⊗ e_3,q^-2e_3 ⊗ e_6 -q^-1e_6 ⊗ e_3, 
       (q^4 + 4q^2 + 6 + 4q^-2 + q^-4)e_4 ⊗ e_4, 
       -(q^2+3+3q^-2+q^-4)e_4 ⊗ e_5 - (q^3 +3q+3q^-1+q^-3)e_5 ⊗ e_4  
       (q^-2+1)e_4 ⊗ e_6 +(q+q^-1)e_6 ⊗ e_4,(q^2+2+q^-2)e_5 ⊗ e_5, 
       -(q^-2+1)e_5 ⊗ e_6 - (q^-1+q)e_6 ⊗ e_5, (q^-1+q)e_6 ⊗ e_6}



    Define π_V(F_i) to be the projection of Δ(F_i) into V = ^6 ⊗^6. The tensor product assumes the role of the Kronecker product in this 36-dimensional representation V of _q(_6). Note that {π_V(E_i)} annihilates e_1 ⊗ e_1, making e_1 ⊗ e_1 a highest weight vector with highest weight 2L_1. To generate V(2L_1), we use Lemma <ref> to obtain the correct compositions of F_is applied to e_1 ⊗ e_1 to form each basis vector: we apply F_is corresponding to the arrows in the path from e_1 ⊗ e_1 to each e_i ⊗ e_j contained in the upper right-hand cycle as displayed Figure <ref>. A few examples are done below, and the rest are left to the reader. The notation F_ijk is used to shorten F_iF_jF_k.
    
        π_V(F_1)(e_1 ⊗e_1)    = (1 ⊗F_1)(e_1 ⊗e_1) + (F_1 ⊗q^-H_1)(e_1 ⊗e_1) 

           = e_1 ⊗e_2 + q^-1e_2 ⊗e_1 

        π_V(F_11) π_V(F_1F_1)(e_1 ⊗e_1)    = (q+q^-1)e_2 ⊗e_2  

        π_V(F_11332211) (e_1 ⊗e_1)    = (q^4 + 4q^2 + 6 + 4q^-2 + q^-4)e_4 ⊗e_4 
    
    After complete evaluation, we obtain the basis shown in Equation <ref> for V(2L_1). Note that the number of basis vectors generated agrees with the dimension formula in Equation <ref>.

Using Lemma <ref> and Lemma <ref>, we are now ready to prove Proposition <ref>.
*

    Note that the basis vectors of V(2L_1) span 
    W and are linearly independent. Since finite-dimensional highest weight modules of _q(_6) are irreducible representations, W is an irreducible representation.

As a sanity check, if C is a central element of _q(_6), we will see that π_V(C)|_W is a constant times Id_20 × 20.

    In 2020, Kuan, Landry, Lin, Park, and Zhou <cit.> found the following central element of _q(_6) (note that the fundamental representation of _2n presented in that paper is slightly incorrect, leading to some sign errors in its central elements <cit.>. The E_3 used in that paper is -E_3 under our notation). We write F_ij to shorten F_iF_j and r to represent q+q^-1. 
    
    C    = q^-4-2H_1-H_2-H_3
                    +q^-2-H_2-H_3
                    +q^H_2-H_3
                    +q^H_3-H_2
                    +q^2+H_2+H_3
       +q^4+2H_1+H_2+H_3
                    +r^2/q^3F_1q^-H_1-H_2-H_3E_1
                    +r^2/qF_2q^-H_3E_2 
       +r^2/qF_3q^-H_2E_3
                    +r^2qF_2q^H_3E_2
                    +r^2qF_3q^H_2E_3
                    +r^2q^3F_1q^H_1+H_2+H_3E_1 
       +r^2/q^3(qF_12-F_21)q^-H_1-H_3(qE_21-E_12) 
       +r^2/q^3(qF_13-F_31)q^-H_1-H_2(qE_31-E_13) 
       +r^2q(qF_21-F_12)q^H_1+H_3(qE_12-E_21) 
       +r^2q(qF_31-F_13)q^H_1+H_2(qE_13-E_31) 
       +r^2/q^3(q^2F_123-qF_213-qF_312+F_231)q^-H_1(q^2E_231-qE_312-qE_213+E_123) 
       +r^2/q(q^2F_231-qF_312-qF_213+F_123)q^H_1(q^2E_123-qE_213-qE_312+E_231) 
       +r^4/q^2((q^2+1)F_1231-qF_1312-qF_2131)((q^2+1)E_1231-qE_1312-qE_2131) 
       +r^4F_2F_3E_2E_3

    By some tedious 36× 36 matrix computations, one can find π_V(C). Then, construct the change of basis matrix Y with first 20 columns being the above-constructed basis of W and the last 16 columns extending to a basis of V. Then, the matrix Y^-1π_V(C)Y has an upper left 20 × 20 block of (q^8+q^2+2+q^-2+q^-8) Id_20 × 20 = π_V(C)|_W.




 §.§ Decompositions of W tensor W

In this section, we analyze the structure of W ⊗ W. This will allow us to deduce properties of π_W ⊗ W(C) in Section <ref>.  


  §.§.§ Decomposition of W tensor W into irreducible representations

In order to decompose W ⊗ W into a direct sum of irreducible representations, we would like to use some analog of the tensor product rule as done in Lemma <ref>. However, since our current crystal graph of W is three-dimensional, creating the crystal graph of W ⊗ W is very difficult. Therefore, we introduce Young tableaux to be able to visualize W ⊗ W, and use the corresponding tensor product rule to decompose W ⊗ W. An interested reader may reference <cit.> chapters 7 and 8 for all appropriate background.

The chain of correspondence is as follows: ℬ(2L_1) describes W, as seen in Lemma <ref>, and

    ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
            [thick, -] (00) – (20) – (21) – (01) – (00);
            [thick, -] (10) – (11);
        ) ℬ(𝒴)

describes ℬ(2L_1). Thus, the decomposition of ℬ(𝒴) ⊗ℬ(𝒴) dictates the structure of W ⊗ W. This brings us to Lemma <ref>. 
  W ⊗ W decomposes as follows into irreducible representations:
    
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0).

    As expected, the sum of the dimensions of these irreducible representations is 400 = dim(W⊗ W).


    As can be read off of Figure <ref>, 
    
    ℬ(2L_1) = { (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
            [thick, -] (00) – (20) – (21) – (01) – (00);
            [thick, -] (10) – (11);
            [circle] (ablah) at (.25,-.05) a;
            [circle] (ablah) at (.75,0) b;
        |  b ≽ a }

    We choose the English notation and convention of reading Young diagrams down columns from right to left. By the tensor product rule for Young diagrams (Theorem 8.6.6, pg 206 of <cit.>),
    
    ℬ(𝒴) ⊗ℬ(𝒴) ≅⊕_b_1 ⊗ b_2 ∈ Wℬ(𝒴[b_1, b_2]).

    To compute the right hand side, note that:
    
        ℬ(𝒴[v_1, v_1])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1.5, .25);
        (40) at (2, -.25);
        (41) at (2, .25);
        [thick, -] (00) – (40) – (41) – (01) – (00);
        [thick, -] (10) – (11);
        [thick, -] (20) – (21);
        [thick, -] (30) – (31);
    
    ) 

    ℬ(𝒴[v_2, v_1])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1.5, .25);
        (02) at (0, -.75);
        (12) at (.5, -.75);
        [thick, -] (00) – (30) – (31) – (01) – (00);
        [thick, -] (10) – (11);
        [thick, -] (20) – (21);
        [thick, -] (02) – (12);
        [thick, -] (01) – (02);
        [thick, -] (11) – (12);
    
    ) 

    ℬ(𝒴[v_2, v_2])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1, -.75);
        (02) at (0, -.75);
        (12) at (.5, -.75);
        [thick, -] (01) – (02) – (31) – (21) – (01) ;
        [thick, -] (00) – (20);
        [thick, -] (11) – (12);
    
    ) 

    ℬ(𝒴[v_1̅, v_1])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        [thick, -] (00) – (20) – (21) – (01) – (00);
        [thick, -] (10) – (11);
    
    ) 

    ℬ(𝒴[v_1̅,v_2])    = ℬ (  
    
        (11) at (.5,.25);
        (01) at (0,.25);
        (00) at (0,-.25);
        (10) at (.5,-.25);
        (20) at (1,-.25);
        (21) at (1, .25);
        (30) at (1.5, -.25);
        (31) at (1, -.75);
        (02) at (0, -.75);
        (12) at (.5, -.75);
        [thick, -] (01) – (02) – (12) – (11) – (01) ;
        [thick, -] (00) – (10);
    
    ) 

    ℬ(𝒴[v_1̅, v_1̅])    = ℬ(∅). 
    
    Thus, we have that
    
    ℬ(𝒴) ⊗ℬ(𝒴) ≅ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1.5, .25);
             (40) at (2, -.25);
             (41) at (2, .25);
            [thick, -] (00) – (40) – (41) – (01) – (00);
            [thick, -] (10) – (11);
            [thick, -] (20) – (21);
            [thick, -] (30) – (31);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1.5, .25);
             (02) at (0, -.75);
             (12) at (.5, -.75);
            [thick, -] (00) – (30) – (31) – (01) – (00);
            [thick, -] (10) – (11);
            [thick, -] (20) – (21);
            [thick, -] (02) – (12);
            [thick, -] (01) – (02);
            [thick, -] (11) – (12);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1, -.75);
             (02) at (0, -.75);
             (12) at (.5, -.75);
            [thick, -] (01) – (02) – (31) – (21) – (01) ;
            [thick, -] (00) – (20);
            [thick, -] (11) – (12);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
            [thick, -] (00) – (20) – (21) – (01) – (00);
            [thick, -] (10) – (11);
        )
        ⊕ℬ(  (11) at (.5,.25);
             (01) at (0,.25);
             (00) at (0,-.25);
             (10) at (.5,-.25);
             (20) at (1,-.25);
             (21) at (1, .25);
             (30) at (1.5, -.25);
             (31) at (1, -.75);
             (02) at (0, -.75);
             (12) at (.5, -.75);
            [thick, -] (01) – (02) – (12) – (11) – (01) ;
            [thick, -] (00) – (10);
        ) ⊕ℬ(∅)

    which implies that
     
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0).

    As for the equality of dimensions, note that
    
        dim(W ⊗W)    = 400 
 
           = 105+175+84+20+15+1 

           = ∑_λdimV(λ)
    
    using the dimension formula Equation <ref>. The order of dimensions in the above sum is the same order in which the irreducible representations are listed in the decomposition above.

We proceed to investigating our next trait of W ⊗ W.



  §.§.§ Decomposition of W tensor W into weight spaces

By definition, each highest-weight _q(_6)-module has a weight space decomposition. Therefore, we can express W ⊗ W as a direct sum of weight spaces using the Young tableaux found in Lemma <ref>. We are most interested in the dimension of each weight space, which will help deduce the shape of π_W ⊗ W(C). 


    W ⊗ W admits a decomposition into eighty-five weight spaces, one with dimension 22, twelve with dimension 12, six with dimension 8, twenty-four with dimension 4, twelve with dimension 3, twenty-four with dimension 2, and six with dimension 1.


    For ease of notation, define
    
    L_-i -L_i

    For 1 ≤ i ≤ 3. Recall that the weights associated with the standard basis vectors of the fundamental representation ^6 of _6 are as follows:
    
    
    It is a property of weights that if v and v' are vectors in a representation V with respective weights λ and λ', then v ⊗ v' is a vector in V ⊗ V with weight λ + λ'. Thus, we can observe from the crystal graph of W that the weights of basis vectors in W take the following forms, for i,j ∈{1,2,3,-1,-2,-3} with i,-i,j,-j all unique:
    
    {0,2L_i,L_i+L_j}

    Note that two basis vectors in W (e_1 ⊗ e_4 and e_2 ⊗ e_5) have weight 0, and each of the other eighteen basis vectors in W has a unique weight corresponding to one of the forms above. By considering possible sums of the weights above, the weights of vectors in W ⊗ W take one of the following forms, for i,j,k ∈{1,2,3,-1,-2,-3} with i,-i,j,-j,k,-k all unique:
    
    {0,L_i + L_j,2L_i,2L_i + L_j + L_k,2L_i + 2L_j,3L_i + L_j,4L_i}

    Now we count the number of weight spaces corresponding to each of the forms above.
    
        
  * There is evidently one weight space with weight 0.
        
  * There are 6 · 4 = 24 possible pairs (i,j). Since the pairs (i,j) and (j,i) yield the same weight, there are twelve weight spaces with a weight of the form L_i + L_j.
        
  * Since there are 6 possible values for i, there are six weight spaces with a weight of the form 2L_i.
        
  * There are 6 · 4 · 2 = 48 possible triplets (i,j,k). Since the triplets (i,j,k) and (i,k,j) yield the same weight, there are twenty-four weight spaces with a weight of the form 2L_i + L_j + L_k.
        
  * There are 6 · 4 = 24 possible pairs (i,j). Since the pairs (i,j) and (j,i) yield the same weight, there are twelve weight spaces with a weight of the form 2L_i + 2L_j.
        
  * There are 6 · 4 = 24 possible pairs (i,j). Since each pair yields a distinct weight, there are twenty-four weight spaces with a weight of the form 3L_i + L_j.
        
  * Since there are 6 possible values for i, there are six weight spaces with a weight of the form 4L_i.
    
    Now we count the dimension of each type of weight space. We can do so by counting the number of basis vectors w_1 ⊗ w_2 ∈ W ⊗ W such that the sum of the weights corresponding to w_1 and w_2 in W equals the desired weight. Denote the weights in W corresponding to w_1 and w_2 by λ_1 and λ_2, respectively.
    
        
  * If λ_1+λ_2 = 0, then either
        
            
  * λ_1 = 0. Then λ_2 = 0 as well. There are thus two options for each of w_1 and w_2, and thus four total possibilities in this case.
            
  * λ_1  0. Then there are eighteen possible values for w_1 with nonzero weights, and exactly one possible value for w_2 such that λ_2 = -λ_1.
        
        Thus, the dimension of the weight space of W ⊗ W with weight 0 is twenty-two.
        
  * If λ_1+λ_2 takes the form L_i+L_j, then either
        
            
  * λ_1 = L_i + L_j and λ_2 = 0; or λ_1 = 0 and λ_2 = L_i + L_j. Since the weight space of W with weight 0 has dimension 2, there are four total possibilities in this case.
            
  * λ_1 = L_i - L_j and λ_2 = 2L_j; or λ_1 = 2L_j and λ_2 = L_i - L_j; or λ_1 = -L_i + L_j and λ_2 = 2L_i; or λ_1 = 2L_i and λ_2 = -L_i + L_j. There are four total possibilities in this case.
            
  * λ_1 = L_i + L_k and λ_2 = -L_j - L_k; or λ_1 = -L_j - L_k and λ_2 = L_i + L_k. Since there are two possibilities for k once i and j are fixed, there are four total possibilities in this case.
        
        Thus, each weight space of W ⊗ W with a weight of the form L_i + L_j has dimension twelve.
        
  * If λ_1 + λ_2 takes the form 2L_i, then either
        
            
  * λ_1 = 2L_i and λ_2 = 0; or λ_1 = 0 and λ_2 = 2L_i. Since the weight space of W with weight 0 has dimension 2, there are four total possibilities in this case.
            
  * λ_1 = L_i + L_j and λ_2 = L_i - L_j. Since there are four possibilities for j  once i is fixed, there are four total possibilities in this case.
        
        Thus, each weight space of W ⊗ W with a weight of the form 2L_i has dimension eight.
        
  * If λ_1 + λ_2 takes the form 2L_i + L_j + L_k, then either
        
            
  * λ_1 = 2L_i and λ_2 = L_j + L_k
            
  * λ_1 = L_j + L_k and λ_2 = 2L_i
            
  * λ_1 = L_i + L_j and λ_2 = L_i + L_k
            
  * λ_1 = L_i + L_k and λ_2 = L_i + L_j
        
        Thus, each weight space of W ⊗ W with a weight of the form 2L_i + L_j + L_k has dimension four.
        
  * If λ_1 + λ_2 takes the form 2L_i + 2L_j, then either
        
            
  * λ_1 = 2L_i and λ_2 = 2L_j
            
  * λ_1 = 2L_j and λ_2 = 2L_i
            
  * λ_1 = L_i + L_j and λ_2 = L_i + L_j
        
        Thus, each weight space of W ⊗ W with a weight of the form 2L_i + 2L_j has dimension three.
        
  * If λ_1 + λ_2 takes the form 3L_i + L_j, then either
        
            
  * λ_1 = 2L_i and λ_2 = L_i + L_j
            
  * λ_1 = L_i + L_j and λ_2 = 2L_i
        
        Thus, each weight space of W ⊗ W with a weight of the form 3L_i + L_j has dimension two.
        
  * If λ_1 + λ_2 takes the form 4L_i, then λ_1 = 2L_i and λ_2 = 2L_i, so each weight space of W ⊗ W with a weight of the form 4L_i has dimension one.
    
    



    This is a direct consequence of Lemma <ref> and Lemma <ref>.




 §.§ Block form of pi(W tensor W)(C)
 




  §.§.§ Block sizes of pi(W tensor W)(C)

Since W ⊗ W can be decomposed into a direct sum of weight spaces, any matrix in W ⊗ W can be expressed as a direct sum of matrices in the weight spaces of W ⊗ W. This yields Proposition <ref>, restated as the following corollary of Lemma <ref>:

    π_W ⊗ W(C) admits a block diagonal decomposition into a direct sum of one 22 × 22 block, twelve 12 × 12 blocks, six 8 × 8 blocks, twenty-four 4 × 4 blocks, twelve 3 × 3 blocks, twenty-four 2 × 2 blocks, and six 1 × 1 blocks. The basis in respect to which results in π_W ⊗ W(C) blocked this way is included in Section <ref>.



Note that since _6 and 𝔰𝔩_4 are isomorphic as Lie algebras, computing the Kostka numbers for 𝔰𝔩_4 would also yield the block sizes found above. For example, the 22× 22 block can be calculated by 22=3+6+7+2+3+1 where each summand appears as K_λμ for the following choices of λ and μ.

    K_λμ = 7,      λ = (4,3,1,0), μ = (2,2,2,2)
    
    K_λμ = 6,      λ = (4,2,2,0), μ = (2,2,2,2) 
    
    K_λμ = 3,      λ = (4,4,0,0), μ = (2,2,2,2) 
    
    K_λμ = 2,      λ = (2,2,0,0), μ = (1,1,1,1) 
    
    K_λμ = 3,      λ = (2,1,1,0), μ = (1,1,1,1) 
    
    K_λμ = 1,      λ = (0,0,0,0), μ = (0,0,0,0)





  §.§.§ Entries of pi(W tensor W)(C)

Since the coproduct Δ is a homomorphism, π_^6 ⊗^6 ⊗^6 ⊗^6(C) can be computed from Definition <ref> by computing Δ^3(E_i), Δ^3(F_i), and Δ^3(q^H_i) for 1 ≤ i ≤ 3 and substituting them for E_i, F_i, and q^H_i, respectively, in the equation in Example <ref>. By Definition <ref>, for all 1 ≤ i ≤ 3,

    Δ^3 (E_i) = E_i ⊗ 1 ⊗ 1 ⊗ 1 + q^H_i⊗ E_i ⊗ 1 ⊗ 1 + q^H_1⊗ q^H_1⊗ E_i ⊗ 1 + q^H_1⊗ q^H_1⊗ q^H_1⊗ E_1 
       Δ^3 (F_i) = 1 ⊗ 1 ⊗ 1 ⊗ F_i + 1 ⊗ 1 ⊗ F_i ⊗ q^-H_i + 1 ⊗ F_i ⊗ q^-H_i⊗ q^-H_i + F_i ⊗ q^-H_i⊗ q^-H_i⊗ q^-H_i
       Δ^3 (q^H_i) = q^H_i⊗ q^H_i⊗ q^H_i⊗ q^H_i

Denote the matrix resulting from the above procedure by π_^1296 (C).


Denote the basis of W established in the proof of Lemma <ref> by B_W  (w_1,w_2,…,w_20). Recall the 36 × 36 change of basis matrix Y defined in Example <ref>, and note that a change of basis by Y maps w_i to the standard basis vector e_i of ^6 ⊗^6 for 1 ≤ i ≤ 20. Thus, a change of basis by Y ⊗ Y in ^6 ⊗^6 ⊗^6 ⊗^6 maps B_W ⊗ B_W to {e_i ⊗ e_j : 1 ≤ i,j ≤ 20}. Since for any 1 ≤ i,j ≤ 36, e_i ⊗ e_j is the (36(i-1)+j)th basis vector of ^6 ⊗^6 ⊗^6 ⊗^6, π_W ⊗ W(C) = π_^1296(C) |_W ⊗ W can be found by taking the (36(i-1)+j)th row and column of (Y^-1⊗ Y^-1)π_^1296(C)(Y ⊗ Y) for all 1 ≤ i,j ≤ 20. Since the e_i ⊗ e_jth row or column of π_W ⊗ W(C) corresponds directly to the (36(i-1)+j)th row or column of (Y^-1⊗ Y^-1)π_^1296(C)(Y ⊗ Y), we only need to determine a few of the values of (Y^-1⊗ Y^-1)π_^1296(C)(Y ⊗ Y) to compute some of the blocks of π_W ⊗ W(C). Explicit values of π_W ⊗ W(E_i),π_W ⊗ W(F_i),π_W ⊗ W(q^H_i) for 1 ≤ i ≤ 3 are given in Section <ref>



  §.§.§ Eigenvalues of pi(W tensor W)(C)

Given W ⊗ W's decomposition into highest weight _q(_6)-modules, we are also able to deduce the eigenvalues of π_W ⊗ W(C).

    π_W ⊗ W(C) has eigenvalues of
    
           q^12+q^2+2+q^-2+q^-12, 

           q^10+q^4 + 2 + q^-4+q^-10, 

           q^8 +q^6+2+q^-6+q^-8, 

           q^8+q^2+2+q^-2+q^-8 

           q^6 + q^4 +2 + q^-4+q^-6, 

           q^4+q^2+2+q^-2+q^-4
    
    with respective multiplicities of 105, 175, 84, 20, 15, 1.


     By Lemma <ref>,
     
    W ⊗ W ≅ V(4L_1) ⊕ V(3L_1+L_2) ⊕ V(2L_1+2L_2) ⊕ V(2L_1) ⊕ V(L_1+L_2) ⊕ V(0)

    with respective dimensions of 105, 175, 84, 20, 15, 1. Since each of these representations is irreducible, 
    π_W ⊗ W(C)|_V(λ) = cId
 for some c ∈[q,q^-1]. Also, by construction of each V(λ), v_λ is a basis vector of V(λ). Therefore, in changing the basis of W ⊗ W to compute π_W ⊗ W(C)|_V(λ), we will have 
    
    π_W ⊗ W(C)|_V(λ)(1,1) = ⟨ v_λ|π_W ⊗ W(C)|_V(λ)|v_λ⟩

    since we may express each highest weight vector v_λ as being a 400-dimensional vector with a 1 in the i^th position and 0s elsewhere. Thus, to find the c associated with each v_λ, we must only find the i^th diagonal entry of π_W ⊗ W(C). We will then have that c is an eigenvalue of multiplicity dim(V(λ)) of this matrix with changed basis. Since a change of basis does not impact the eigenvalues of the matrix and π_W ⊗ W(C) is a block matrix by Lemma <ref>, c is an eigenvalue of π_W ⊗ W(C) with multiplicity dim(V(λ)). 
    Next, as v_λ is a highest weight vector of weight λ, by definition,
    
    E_1v_λ = E_2v_λ = E_3v_λ = 0.

    Thus, 
    
    π_W ⊗ W(C)v_λ = π_W ⊗ W(q^-4-2H_1-H_2-H_3
                +q^-2-H_2-H_3
                +q^H_2-H_3
                +q^H_3-H_2
                +q^2+H_2+H_3 
                +q^4+2H_1+H_2+H_3)v_λ.

    Since each q^H_i is a diagonal matrix, we may use Section <ref> to find the corresponding diagonal entry to each highest weight vector in a feasible way computationally. For example, v_4L_1 = e_1^⊗ 2 where e_1 is 20-dimensional, and thus π_W ⊗ W(C) has an eigenvalue of 
    
    c    =  ⟨e_1^⊗2|π_W ⊗W(C)|e_1^⊗2 ⟩

       =⟨e_1^⊗2|π_W ⊗W(q^-4-2H_1-H_2-H_3
            +q^-2-H_2-H_3
            +q^H_2-H_3
            +q^H_3-H_2
            +q^2+H_2+H_3 
            +q^4+2H_1+H_2+H_3)| e_1^⊗2⟩

               = q^-4·q^-4 ·q^-4 ·1 ·1 + q^-2·1 ·1+1·1+1·1+q^2·1·1 +q^4·q^4·q^4·1·1 

           = q^12+q^2+2+q^-2+q^-12
    
    with multiplicity dim(V(4L_1)) = 105. The remaining eigenvalues can be found similarly.
    



    Recall that _6 is a rank 3 Lie algebra which is isomorphic to the rank 3 Lie algebra 𝔰𝔩_4. In this case, previous research in <cit.> allows us to express C using generators from both _6 and 𝔰𝔩_4 in terms of the above eigenvalues. We choose not to explore this, but someone who wishes to relate <cit.>'s central element C to the Harish Chandra isomorphism may reference <cit.>.    




 §.§ Type A Ground State Transformation

Following the path of previous research, we used Section <ref> as a starting point to form our ground state transformation. We perform a ground state transformation so that the resulting matrix has rows which sum to 0, allowing us to use this transformed matrix as a generator for a Markov process. We will then compare this generator to that of the probability section. 


    
    A ground state transformation is a map from a Hamiltonian matrix, H, to a matrix whose rows sum to 0. We define such a map to be of the following form, where G is a diagonal matrix and a is in the underlying field.
    
    G^-1HG - a Id

    We call G a ground state transformation matrix of H.


Section <ref> provides a ground state transformation that worked with Type A_n Lie algebras. However, as can be seen using the respective crystal graphs, Type D_n Lie algebras have more complicated behavior: the crystal graph for a fundamental representation of _q(𝔰𝔩_4) is shown in Figure <ref>.


As a result, the crystal graph for V ⊗ V of _q(𝔰𝔩_4) is the graph shown in Figure <ref>. Note that the cycle from the highest weight vector to the lowest weight vector is straightforward.
 

Instead, in _q(_6), the crystal graph for V ⊗ V shown in Figure <ref> was much less simple due to the more complex crystal graph for V shown in Figure <ref>. In this case, the shortest path from the highest weight vector to the lowest weight vector requires eight applications of F_is as shown in the third example within the proof of Lemma <ref>.

Thus, intuitively, when we apply the tensor product rule again to W, the path from the highest weight vector to the lowest weight vector should be even more intricate. Therefore, the ground state transformation in Section <ref> should include more complicated structures than solely permutations of F_1^k_1F_2^k_2F_3^k_3 in order to generate a complete matrix G. This result is formalized below.
*

    By the crystal graph for V shown in Figure <ref>, which is based on the fundamental representation defined in Definition <ref>, we see that we will not be able to compute e_4 without applying either F_1F_2F_3F_1 or F_1F_3F_2F_1.
    
Now, note that W is a subspace of Sym^2_q(^6); in particular, as shown in Lemma <ref>, W has basis elements of the form:

    { p_i(q)   e_i ⊗e_i, p_i(q)e_i ⊗e_j+p_j(q)e_j ⊗e_i, |i ≠j, (i,j) ≠(1,4),(2,5),(3,6), p_j  Laurent polynomial} ∪

       { -e_1 ⊗e_4 +q^-1e_2 ⊗e_5 - q^-2e_4 ⊗e_1+q^-1e_5 ⊗e_2, e_2 ⊗e_5 - q^-1e_3 ⊗e_6+q^-2e_5 ⊗e_2 -(q^2+1)e_6 ⊗e_3}

Thus, applying a sequence of F_is to obtain, for example, 

    ⟨ e_1 ⊗ e_2| π_W(Δ(F_1))| e_1 ⊗ e_1 ⟩ = 1

will also yield

    ⟨ e_2 ⊗ e_1| π_W(Δ(F_1))| e_1 ⊗ e_1 ⟩ = q^-1

because e_1 ⊗ e_2+qe_2 ⊗ e_1 is a basis vector of W.

However, the composition of F_is required to obtain e_4 ⊗ e_4, Π_i (Δ F_i), will only produce a nonzero 

    ⟨ e_4 ⊗ e_4| π_W(Π_i (Δ(F_i)))| e_1 ⊗ e_1 ⟩

and result in 0 when taking, for any other (i,j) ≠ (4,4),

    ⟨ e_i ⊗ e_j| π_W(Π_i (Δ(F_i)))| e_1 ⊗ e_1 ⟩

as e_4 ⊗ e_4 does not appear in any other basis vector of W. Therefore, since e_4 ⊗ e_4 lies in the bottom right corner of Figure <ref>, it is clear that in order to obtain a nonzero value corresponding to e_4 ⊗ e_4, we must follow the arrows and apply a composition of the form

    ⟨ e_4 ⊗ e_4| π_W(Δ(F_1) ·Π_i (Δ(F_i)) ·Δ(F_1))| e_1 ⊗ e_1 ⟩

with, in particular, 

    ⟨ e_4 ⊗ e_4| π_W(Δ(F_1)Δ(F_1)Δ(F_3)Δ(F_3)Δ(F_2)Δ(F_2)Δ(F_1)  Δ(F_1))| e_1 ⊗ e_1 ⟩ = q^4 + 4q^2 + 6 + 4q^-2 + q^-4.

Finally, since W is a subset of a symmetric tensor, W ⊗ W will be as well. Therefore, we must again find a composition Π_i(Δ (F_i)) that produces exactly e_4^⊗ 4, and it will not be enough to produce any other vector. By the tensor product rule, this will yet again result in a nonzero value to an expression only in the following structure

    ⟨ e_4^⊗ 4| π_W(Δ(F_1) ·Π_i (Δ(F_i)) ·Δ(F_1))| e_1^⊗ 4⟩





 §.§ Finding a Markov Process Generator




  §.§.§ Partial Type A Ground State Transformation

 
    We define a partial Type A ground state transformation of π_W ⊗ W(C) to be a map
    
    π_W ⊗ W(C) ↦. ( G^-1π_W ⊗ W(C)G - aId) | _K

    where K is the vector subspace of W ⊗ W which is the span of all basis vectors u_i ⊗ u_j for which there exists a triple (k_1,k_2,k_3) of nonnegative integers satisfying
    
    ⟨ u_i ⊗ u_j | ∏_i=1^3 Δ(F_ℓ_i)^k_ℓ_i| e_1 ⊗ e_1 ⟩ 0

    where (ℓ_1,ℓ_2,ℓ_3) is some permutation of (1,2,3), and where G is a 400 × 400 diagonal matrix such that the diagonal entry of G corresponding to u_i ⊗ u_j is the unique nonzero value above.

Note that by Lemma <ref>, the partial Type A ground state transformation is not unique.



  §.§.§ Selecting a Ground State Transformation

We now extend the partial Type A ground state transformation to a ground state transformation of π_W ⊗ W(C). Define a  q^12 + q^2 + q^-2 + q^-12.




    Let G be a ground state transformation matrix for π_W ⊗ W(C), i.e., a diagonal matrix such that each row of a^-1G^-1π_W ⊗ W(C)G - Id sums to 0, and let g_i denote the ith diagonal entry of G for 1 ≤ i ≤ 400. Then the vector
    
    g⃗[   g_1;   g_2;     ⋮; g_400 ]

    satisfies π_W ⊗ W(C) g⃗ = ag⃗. This condition is both sufficient and necessary, i.e., if g⃗ is an eigenvector of π_W ⊗ W(G) with eigenvalue a, the diagonal matrix G with entries corresponding to g⃗ as above is such that the rows of a^-1G^-1π_W ⊗ WG - Id sum to 0.



    Define the matrix L by
    
    L  a^-1G^-1π_W ⊗ W(C)G - Id

    By definition, each row of L sums to 0. Then we have
    
    G^-1(π_W ⊗ W(C))G - aId   = aL 
    
            G^-1(π_W ⊗ W(C))G    = aL + aId
    π_W ⊗ W(C)G    = aGL + aG

    Since G is diagonal and each row of L sums to 0, each row of aGL sums to 0. Thus, the sum of the elements in each row of π_W ⊗ W(C) G equals the sum of the elements in the same row of aG. Defining g_i as above, we can write the sum of the ith row of π_W ⊗ W(C)G as ∑_j=1^400π_W ⊗ W(C)_i,j g_j, and the sum of the ith row of aG as ag_i. Thus, for all 1 ≤ i ≤ 400,
    
    ∑_j=1^400π_W ⊗ W(C)_i,j g_j    = ag_i

    This is equivalent to saying that π_W ⊗ W(C) g⃗ = ag⃗.
    
    
    Reversing the steps above shows that the condition π_W ⊗ W(C)g⃗ = ag⃗ is sufficient as well as necessary.


Thus, any partial Type A ground state transformation matrix G of π_W ⊗ W(C) which satisfies the condition in Lemma <ref> is a ground state transformation matrix of π_W ⊗ W(C). Finding diagonal entries of G as described in Definition <ref>  and solving the condition in Lemma <ref> as a system of linear equations yields a solution space 𝒮 with 37 unknowns.


    Given any G ∈𝒮,
    
    a^-1G^-1HG - Id

    is a generator for a Markov process.


    Immediate by the definition of 𝒮.

Note that the proof of Theorem <ref> is an immediate consequence of 𝒮 being nonempty. We give one such generator in Section <ref> by setting all unknowns to 0 for ease of computation. One detail to note is that this generator has dimension 196 = 14^2, which suggests that, when interpreted as an interacting particle system, the process allows for 14 different states — all but four particles of the same class — at each site.



 §.§ Difference with Probability Generator

To relate our algebraic findings to those of probability, we end by comparing the two generators. First, note that the 2 × 2 and 3 × 3 blocks of the algebraically-produced Markov generator in Section <ref> are a constant multiple of the 2 × 2 and 3 × 3 blocks of the probabilistically-produced Markov generator in Section <ref>. However, we notice that any generator matrix given by a ground state transformation as described in Definition <ref> will not produce a generator matrix for a Markov process matching the probabilistically-produced generator matrix. We now prove Proposition <ref>.


    We will show that no block of any ground state transformation of π_W ⊗ W(C) can be a constant multiple of the 4× 4 block of L_Q in Section <ref> corresponding to the communicating class

    ( ⟨ 3 , 0⟩, ⟨1 , 2⟩, ⟨2 , 1⟩, ⟨0 , 3⟩).

    Thus, L_Q must be different from any Markov generator produced from π_W ⊗ W(C).

    Note that the above communicating class contains one class 1 particle and one class 2 particle per Section <ref>. As proven in Proposition 2.5 of <cit.>, this communicating class must correspond to the weight 
    
    2 λ_i + λ_j +λ_k

    found in Section <ref>. As discussed in Lemma <ref>, there are 24 weight spaces of dimension 4 associated with 2 λ_i + λ_j +λ_k, and thus 2 λ_i + λ_j +λ_k corresponds to the twenty-four 4 × 4 blocks in π_W ⊗ W. It is therefore sufficient to show that none of the 4 × 4 blocks of π_W ⊗ W(C), which are listed in Section <ref>, can be mapped to the 4 × 4 block in L_Q corresponding to the above communicating class.

    By referencing Section <ref>, we notice that each 4 × 4 block does not contain any zero entries. However, every 4 × 4 block listed in Section <ref> has an off-diagonal entry which takes the value of 0.

    Since conjugation by a diagonal matrix will not change a 0-entry in a matrix and the 0-entries are off-diagonal, Definition <ref> confirms that no ground state transformation of π_W ⊗ W(C) will result in L_Q.


The computationally-heavy verification which involves disregarding the relationship between weights and particles and instead checking 4 × 4 blocks after trying all possible ground state transformations on π_W ⊗ W(C) will yield the same result, as this proof is equivalent to the one detailed above.



§ APPENDIX
 


 §.§ Code for Probabilistic Sections

The SymPy code used to verify the generator matrix L_Q, limits of component generator matrices, stationary distributions, spectral gaps, and matrices of Markov duality can be found at the authors' https://github.com/evaengel/stochastic-fusionGitHub repository. Because the largest communicating classes only have size 9, the matrices are small enough that the calculations can be done by hand. Those details are omitted from this paper, and the SymPy code only serves as an independent verification.



 §.§ List of Communicating Classes by Size
 
The following communicating classes are for the two-γ lattice site Type D ASEP.

The ordered 9-state communicating class follows:

    ( ⟨ 3, 3 ⟩,⟨ 0, 33 ⟩, ⟨ 33,0 ⟩,⟨ 1,32 ⟩,⟨ 32,1 ⟩,⟨ 2,31 ⟩, ⟨ 31,2 ⟩,⟨ 11, 22 ⟩,⟨ 22, 11 ⟩)


The ordered 6-state communicating classes follow:

    ( ⟨ 1 , 3⟩, ⟨3 , 1⟩, ⟨0 , 31⟩, ⟨31 , 0⟩, ⟨2 , 11⟩, ⟨11 , 2 ⟩)


    ( ⟨ 2 , 3⟩, ⟨3 , 2⟩, ⟨0 , 32⟩, ⟨32 , 0⟩, ⟨1 , 22⟩, ⟨22 , 1⟩)


    ( ⟨ 3 , 31⟩, ⟨31 , 3⟩, ⟨1 , 33⟩, ⟨33 , 1⟩, ⟨11 , 32⟩, ⟨32 , 11⟩)


    ( ⟨ 3 , 32⟩, ⟨32 , 3⟩, ⟨2 , 33⟩, ⟨33 , 2⟩, ⟨22 , 31⟩, ⟨31 , 22⟩)


The ordered 4-state communicating classes follow:

    ( ⟨ 3 , 0⟩, ⟨1 , 2⟩, ⟨2 , 1⟩, ⟨0 , 3⟩)


    (⟨ 3 , 11⟩, ⟨1 , 31⟩, ⟨11 , 3⟩, ⟨31 , 1⟩)


    (⟨ 3 , 22⟩, ⟨2 , 32⟩, ⟨22 , 3⟩, ⟨32 , 2⟩)


    (⟨ 33 , 3⟩, ⟨31 , 32⟩, ⟨32 , 31⟩, ⟨3 , 33⟩)


The ordered 3-state communicating classes follow:

    (⟨ 1 , 1⟩, ⟨0 , 11⟩, ⟨11 , 0⟩)


    (⟨ 2 , 2⟩, ⟨0 , 22⟩, ⟨22 , 0⟩)


    (⟨ 31 , 31⟩, ⟨11 , 33⟩, ⟨33 , 11⟩)


    (⟨ 32 , 32⟩, ⟨22 , 33⟩, ⟨33 , 22⟩)


The ordered 2-state communicating classes follow:

    (⟨ 0 , 1⟩, ⟨1 , 0⟩)


    (⟨ 0 , 2⟩, ⟨2 , 0⟩)


    (⟨ 1 , 11⟩, ⟨11 , 1⟩)


    (⟨ 2 , 22⟩, ⟨22 , 2⟩)


    (⟨ 11 , 31⟩, ⟨31 , 11⟩)


    (⟨ 22 , 32⟩, ⟨32 , 22⟩)


    (⟨ 31 , 33⟩, ⟨33 , 31⟩)


    (⟨ 32 , 33⟩, ⟨33 , 32⟩)


The ordered 1-state communicating classes follow:

    (⟨ 0 , 0⟩)


    (⟨ 11 , 11⟩)


    (⟨ 22 , 22⟩)


    (⟨ 33 , 33⟩)




 §.§ Presenting LQ


L_Q is the direct sum of one 9 × 9 block

    ℒ_9 =


    [
    [    *  D_1  D_2 D_10 D_11 D_10 D_11 D_21 D_21;  D_3    *    0 D_12    0 D_13    0    0    0;  D_4    0    *    0 D_14    0 D_14    0    0;  D_5  D_6    0    *    0 D_15 D_16 D_22    0;  D_7    0  D_8    0    * D_15 D_18    0 D_23;  D_5  D_6    0 D_15 D_16    *    0    0 D_22;  D_7    0  D_8 D_17 D_18    0    * D_23    0;  D_9    0    0 D_19    0    0 D_20    *    0;  D_9    0    0    0 D_20 D_19    0    0    *;      ]]


where 

    D_1 = q^4n+q^4-2q^2n+2/(q^12+4q^10+6q^8+4q^6+q^4)q^2n   D_2 = q^12+q^4n+8-2q^2n+10/(q^8+4q^6+6q^4+4q^2+1)q^2n)



    D_3 = q^4n-2q^2n+2+q^4/q^2n   D_4 =q^4n-2q^2n+2+q^4/q^2n+4



    D_5 = q^4n(2q^2-1)+2q^2n(q^6-q^4+q^2)+2q^6-q^8/(q^4+q^2)q^2n   D_6 = 2q^2n-q^2+1/(q^4+2q^2+1)q^2n



    D_7 = q^4n(2q^2-1)+2q^2n(q^6-q^4+q^2)+2q^6-q^8/(q^6+q^4)q^2n   D_8 = q^2n(q^4-q^2)+2q^4/q^4+2q^2+1



    D_9 = q^4n-2q^2n+2+q^4/q^2n+2   D_10 = 2q^2n(q^6-q^4+q^2)+q^4n(2q^2-1)+2q^6-q^8/(q^10+3q^8+3q^6+q^4)q^2n



    D_11 = q^4n(2q^4-q^2)+2q^2n(q^8-q^6+q^4)+2q^8-q^10/(q^6+3q^4+3q^2+1)q^2n   D_12 = 2q^2n+2-q^2-q^4/q^2n



    D_13 = 2q^2n+2-q^4+q^2/q^2n   D_14 = q^2n(q^2-1)+2q^2/q^4



    D_15 = q^4n-2q^2n+2+q^4/(q^6+2q^4+q^2)q^2n   D_16 = q^4n+4-2q^2n+6+q^8/(q^4+2q^2+1)q^2n



    D_17 = q^4n-2q^2n+2+q^4/(q^8+2q^6+q^4)q^2n   D_18 = q^4n+2-2q^2n+4+q^6/(q^4+2q^2+1)q^2n



    D_19 = 2q^2n+1-q^2/q^2n   D_20 = q^2n(q^2-1)+2q^2/q^2



    D_21 = q^6+q^4n+2-2q^2n+4/(q^8+4q^6+6q^4+4q^2+1)q^2n   D_22 = 2q^2n+6+q^6-q^8/(q^4+2q^2+1)q^2n



    D_23 = (q^2-1)q^2n+2q^2/q^8+2q^6+q^4



and ℒ_9 corresponds to the 9-state communicating class listed in Section <ref>,

four 6 × 6 blocks

    ℒ_6 =


    [
    [    *  D_1  D_2 D_14 D_15 D_16;  D_3    *  D_4 D_17 D_18 D_19;  D_5  D_6    *    0 D_20    0;  D_7  D_8    0    *    0 D_21;  D_9 D_10 D_11    0    *    0; D_12 D_13    0 D_22    0    *;      ]]


where 


    D_1 = 2q^6+(2q^2-1)q^4n+2q^2n(q^6-q^4+q^2)-q^8/(q^4+2q^2+1)q^2n   D_2 = q^4n+2q^2n+4+q^4+q^2-q^6/(q^8+3q^6+3q^4+q^2)q^2n



    D_3 = 2q^6+(2q^2-1)q^4n+2q^2n(q^6-q^4+q^2)-q^8/(q^8+2q^6+q^4)q^2n   D_4 = q^4+q^4n-2q^2n+2/(q^10+3q^8+3q^6+q^4)q^2n



    D_5 = q^4n+2q^2n+4+q^4+q^2-q^6/(q^2+1)q^2n   D_6 = q^6+q^4n+2-2q^2n+4/(q^2+1)



    D_7 = q^4+q^4n-2q^2n+2/(q^6+q^4)q^2n   D_8 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/(q^6+q^4)q^2n



    D_9 =  q^4+q^4n-2q^2n+2/(q^4+q^2)q^2n   D_10 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/(q^4+q^2)q^2n



    D_11 = 2q^2n-q^2+1/(q^2+1)q^2n   D_12 = q^4n+2q^2n+4/(q^4+q^2)q^2n



    D_13 = q^4n-2q^2n+2+q^4/(q^2+1)q^2n   D_14 = q^10+q^4n+6-2q^2n+8/(q^6+3q^4+3q^2+1)q^2n



    D_15 = q^4+q^4n-2q^2n+2/(q^6+3q^4+3q^2+1)q^2n   D_16 = q^8+q^6+q^4n+4+2q^2n+8-q^10/(q^6+3q^4+3q^2+1)q^2n



    D_17 = q^8+(q^6+q^4-q^2)q^4n+2q^2n+4/(q^6+3q^4+3q^2+1)q^2n   D_18 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/(q^10+3q^8+3q^6+q^4)q^2n)



    D_19 = q^6+q^4n+2-2q^2n+4/(q^6+3q^4+3q^2+1)q^2n   D_20 = 2q^2n+4+q^4-q^6/(q^2+1)q^2n



    D_21 = (q^2-1)q^2n+2q^2/q^6+q^4   D_22 = (q^2-1)q^2n+2q^2/q^2+1


and ℒ_6 corresponds to the 6-state communicating classes listed in Section <ref>,

four 4 × 4 blocks

    ℒ_4 = 1/(q^4+2q^2+1)q^2nM


where 


    M =


    [
    [   * D_1 D_1 D_6; D_2   * D_3 D_7; D_2 D_3   * D_7; D_4 D_5 D_5   *;     ]]


where 


    D_1 = q^6+(q^4+q^2-1)q^4n+2q^2n+2/q^4   D_2 = q^6+(q^4+q^2-1)q^4n+2q^2n+2



    D_3 = q^4+q^4n-2q^2n+2   D_4 = q^8+q^4n+4-2q^2n+6



    D_5 = q^4n+2+2q^2n+6+q^6+q^4-q^8    D_6 = q^4+q^4n-2q^2n+2/q^4



    D_7 = 2q^2n+4+q^4n+q^4+q^2-q^6/q^2


and ℒ_4 corresponds to the 4-state communicating classes listed in Section <ref>,

four 3 × 3 blocks

    ℒ_3 =  [                           * q^2+q^4n/(q^6+2q^4+q^2)q^2n q^6+q^4n+4/(q^4+2q^2+1)q^2n;               q^2+q^4n/q^2n                           *                           0;             q^2+q^4n/q^2n+2                           0                           * ]

corresponding to the 3-state communicating classes listed in Section <ref>,
eight 2 × 2 blocks

    ℒ_2 = [                      * q^4+q^4n+2/(q^2+1)q^2n; q^2+q^4n/(q^4+q^2)q^2n                      * ]

corresponding to the 2-state communicating classes listed in Section <ref>,

and four 1 × 1 blocks with entry 0 corresponding to the 1-state communicating classes listed in Section <ref>. Here, the diagonal entries are choosen so that the rows sum to 0. To summarize, the generator matrix is

    L^D_Q = ℒ_9 ⊕⊕_i=1^4 ℒ_6 ⊕⊕_i=1^4 ℒ_4 ⊕⊕_i=1^4 ℒ_3 ⊕⊕_i=1^8 ℒ_2 ⊕⊕_i=1^4 [0]

with respect to the ordered basis that is the concatenation of all of the above communicating classes in the order they are presented.



 §.§ Ordering of States and Permutations of 
Lm and Lq




  §.§.§ Gamma State Ordering

The assumed ordering of states for four Γ lattice sites is Ω⊗Ω. To be more concise we represent the state (x_1,x_2,x_3,x_4) as x_1x_2x_3x_4.

Assumed Ordering:

(3030,   3021,   3003,   3012,   3010,   3001,   3020,   3002,   3031,   3013,   3032,   3023,   3000,   3011,   3022,   3033,   2130,   2121,   2103,   2112,   2110,   2101,   2120,   2102,   2131,   2113,   2132,   2123,   2100,   2111,   2122,   2133,   0330,   0321,   0303,   0312,   0310,   0301,   0320,   0302,   0331,   0313,   0332,   0323,   0300,   0311,   0322,   0333,   1230,   1221,   1203,   1212,   1210,   1201,   1220,   1202,   1231,   1213,   1232,   1223,   1200,   1211,   1222,   1233,   1030,   1021,   1003,   1012,   1010,   1001,   1020,   1002,   1031,   1013,   1032,   1023,   1000,   1011,   1022,   1033,   0130,   0121,   0103,   0112,   0110,   0101,   0120,   0102,   0131,   0113,   0132,   0123,   0100,   0111,   0122,   0133,   2030,   2021,   2003,   2012,   2010,   2001,   2020,   2002,   2031,   2013,   2032,   2023,   2000,   2011,   2022,   2033,   0230,   0221,   0203,   0212,   0210,   0201,   0220,   0202,   0231,   0213,   0232,   0223,   0200,   0211,   0222,   0233,   3130,   3121,   3103,   3112,   3110,   3101,   3120,   3102,   3131,   3113,   3132,   3123,   3100,   3111,   3122,   3133,   1330,   1321,   1303,   1312,   1310,   1301,   1320,   1302,   1331,   1313,   1332,   1323,   1300,   1311,   1322,   1333,   3230,   3221,   3203,   3212,   3210,   3201,   3220,   3202,   3231,   3213,   3232,   3223,   3200,   3211,   3222,   3233,   2330,   2321,   2303,   2312,   2310,   2301,   2320,   2302,   2331,   2313,   2332,   2323,   2300,   2311,   2322,   2333,   0030,   0021,   0003,   0012,   0010,   0001,   0020,   0002,   0031,   0013,   0032,   0023,   0000,   0011,   0022,   0033,   1130,   1121,   1103,   1112,   1110,   1101,   1120,   1102,   1131,   1113,   1132,   1123,   1100,   1111,   1122,   1133,   2230,   2221,   2203,   2212,   2210,   2201,   2220,   2202,   2231,   2213,   2232,   2223,   2200,   2211,   2222,   2233,   3330,   3321,   3303,   3312,   3310,   3301,   3320,   3302,   3331,   3313,   3332,   3323,   3300,   3311,   3322,   3333 )

Diagonal Ordering:

(3300,   3210,   3030,   3120,   3100,   3010,   3200,   3020,   3310,   3130,   3320,   3230,   3000,   3110,   3220,   3330,   2301,   2211,   2031,   2121,   2101,   2011,   2201,   2021,   2311,   2131,   2321,   2231,   2001,   2111,   2221,   2331,   0303,   0213,   0033,   0123,   0103,   0013,   0203,   0023,   0313,   0133,   0323,   0233,   0003,   0113,   0223,   0333,   1302,   1212,   1032,   1122,   1102,   1012,   1202,   1022,   1312,   1132,   1322,   1232,   1002,   1112,   1222,   1332,   1300,   1210,   1030,   1120,   1100,   1010,   1200,   1020,   1310,   1130,   1320,   1230,   1000,   1110,   1220,   1330,   0301,   0211,   0031,   0121,   0101,   0011,   0201,   0021,   0311,   0131,   0321,   0231,   0001,   0111,   0221,   0331,   2300,   2210,   2030,   2120,   2100,   2010,   2200,   2020,   2310,   2130,   2320,   2230,   2000,   2110,   2220,   2330,   0302,   0212,   0032,   0122,   0102,   0012,   0202,   0022,   0312,   0132,   0322,   0232,   0002,   0112,   0222,   0332,   3301,   3211,   3031,   3121,   3101,   3011,   3201,   3021,   3311,   3131,   3321,   3231,   3001,   3111,   3221,   3331,   1303,   1213,   1033,   1123,   1103,   1013,   1203,   1023,   1313,   1133,   1323,   1233,   1003,   1113,   1223,   1333,   3302,   3212,   3032,   3122,   3102,   3012,   3202,   3022,   3312,   3132,   3322,   3232,   3002,   3112,   3222,   3332,   2303,   2213,   2033,   2123,   2103,   2013,   2203,   2023,   2313,   2133,   2323,   2233,   2003,   2113,   2223,   2333,   0300,   0210,   0030,   0120,   0100,   0010,   0200,   0020,   0310,   0130,   0320,   0230,   0000,   0110,   0220,   0330,   1301,   1211,   1031,   1121,   1101,   1011,   1201,   1021,   1311,   1131,   1321,   1231,   1001,   1111,   1221,   1331,   2302,   2212,   2032,   2122,   2102,   2012,   2202,   2022,   2312,   2132,   2322,   2232,   2002,   2112,   2222,   2332,   3303,   3213,   3033,   3123,   3103,   3013,   3203,   3023,   3313,   3133,   3323,   3233,   3003,   3113,   3223,   3333 )




  §.§.§ gamma State Ordering

The assumed ordering of states for two γ lattice sites is χ⊗χ. To be more concise we represent the state (x_1,x_2) as x_1_x_2.

Assumed Ordering:

(0_0,   0_1,   0_2,   0_11,   0_3,   0_22,   0_31,   0_32,   0_33,   1_0,   1_1,   1_2,   1_11,   1_3,   1_22,   1_31,   1_32,   1_33,   2_0,   2_1,   2_2,   2_11,   2_3,   2_22,   2_31,   2_32,   2_33,   11_0,   11_1,   11_2,   11_11,   11_3,   11_22,   11_31,   11_32,   11_33,   3_0,   3_1,   3_2,   3_11,   3_3,   3_22,   3_31,   3_32,   3_33,   22_0,   22_1,   22_2,   22_11,   22_3,   22_22,   22_31,   22_32,   22_33,   31_0,   31_1,   31_2,   31_11,   31_3,   31_22,   31_31,   31_32,   31_33,   32_0,   32_1,   32_2,   32_11,   32_3,   32_22,   32_31,   32_32,   32_33,   33_0,   33_1,   33_2,   33_11,   33_3,   33_22,   33_31,   33_32,   33_33 )

Diagonal Ordering:

 (3_3,   0_33,   33_0,   1_32,   32_1,   2_31,   31_2,   11_22,   22_11,   1_3,   3_1,   0_31,   31_0,   2_11,   11_2,   2_3,   3_2,   0_32,   32_0,   1_22,   22_1,   3_31,   31_3,   1_33,   33_1,   11_32,   32_11,   3_32,   32_3,   2_33,   33_2,   22_31,   31_22,   3_0,   1_2,   2_1,   0_3,   31_1,   11_3,   3_11,   1_31,   32_2,   22_3,   3_22,   2_32,   33_3,   31_32,   32_31,   3_33,   1_1,   0_11,   11_0,   2_2,   0_22,   22_0,   31_31,   11_33,   33_11,   32_32,   22_33,   33_22,   0_1,   1_0,   0_2,   2_0,   1_11,   11_1,   2_22,   22_2,   11_31,   31_11,   22_32,   32_22,   31_33,   33_31,   32_33,   33_32,   0_0,   11_11,   22_22,   33_33 )




 §.§ Limits of Components of 
Lp2
 

Recall that L^(2)_p is given in section 2.2 “Probabilistic definitions” of <cit.> as the direct sum of three distinct block matrices L_1, L_2, and four 1 × 1 zero matrices.

We list the limits for q^-2n L_1 and q^-2n L_2 for q > 1. The limits follow similarly when multiplying  L_1 and L_2 by q^2n for 0 < q < 1.

Take q > 1.


    lim_n →∞ (q^-2n L_1) = 
    [ -2q^2-1/q^4   q^2-1/q^4       1/q^4   q^2-1/q^4; q^2 - 1/q^2          -1           0       1/q^2;           1           0          -1           0; q^2 - 1/q^2       1/q^2           0          -1;             ]


    lim_n →∞ (q^-2n L_2) = 
    [ -1/q^2  1/q^2;      1     -1;        ]




 §.§ Diagonalization for Communicating Classes


We show that for each communicating class ℒ_i where i ∈{3,4,6,9} the matrix of right eigenvectors is invertible for q > 0 and q ≠ 1 , and therefore that all the eigenvectors are linearly independent which implies diagonalizability.

Consider  𝒫_9,

    𝒫_9 =


    [                                     1             - q^4/q^8 + q^6 + q^2 + 1             - q^4/q^8 + q^6 + q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1                   q^2/q^4 + 2 q^2 + 1     - q^4 + 2 q^2 - 1/q^4 + 2 q^2 + 1;                                     1                           q^6/q^6 + 1                           q^6/q^6 + 1                           q^4/q^4 - 1                           q^4/q^4 - 1                                   q^6                         - q^6/q^2 - 1                         - q^6/q^2 - 1                                 - q^4;                                     1                             1/q^6 + 1                             1/q^6 + 1                           - 1/q^4 - 1                           - 1/q^4 - 1                                 1/q^6                           1/q^6 - q^4                           1/q^6 - q^4                               - 1/q^4;                                     1 q^10 - q^6 + q^4/q^10 - q^6 + q^4 - 1            - q^6/q^10 - q^6 + q^4 - 1   q^6 + q^4 - q^2/q^6 + q^4 - q^2 - 1               q^4/q^6 + q^4 - q^2 - 1                         - q^4/q^2 + 1 - q^8 + q^6 + q^4/q^6 + q^4 - q^2 - 1               q^4/q^6 + q^4 - q^2 - 1                   - q^4 + q^2/q^2 + 1;                                     1              q^4/q^10 - q^6 + q^4 - 1  - q^6 + q^4 - 1/q^10 - q^6 + q^4 - 1             - q^2/q^6 + q^4 - q^2 - 1     q^4 - q^2 - 1/q^6 + q^4 - q^2 - 1                         - 1/q^4 + q^2             - q^2/q^6 + q^4 - q^2 - 1 - q^4 - q^2 + 1/q^8 + q^6 - q^4 - q^2                     q^2 - 1/q^4 + q^2;                                     1            - q^6/q^10 - q^6 + q^4 - 1 q^10 - q^6 + q^4/q^10 - q^6 + q^4 - 1               q^4/q^6 + q^4 - q^2 - 1   q^6 + q^4 - q^2/q^6 + q^4 - q^2 - 1                         - q^4/q^2 + 1               q^4/q^6 + q^4 - q^2 - 1 - q^8 + q^6 + q^4/q^6 + q^4 - q^2 - 1                   - q^4 + q^2/q^2 + 1;                                     1  - q^6 + q^4 - 1/q^10 - q^6 + q^4 - 1              q^4/q^10 - q^6 + q^4 - 1     q^4 - q^2 - 1/q^6 + q^4 - q^2 - 1             - q^2/q^6 + q^4 - q^2 - 1                         - 1/q^4 + q^2 - q^4 - q^2 + 1/q^8 + q^6 - q^4 - q^2             - q^2/q^6 + q^4 - q^2 - 1                     q^2 - 1/q^4 + q^2;                                     1                                     1                                     0                                     1                                     0                                     1                                     1                                     0                                     1;                                     1                                     0                                     1                                     0                                     1                                     1                                     0                                     1                                     1 ]


It follows that 

    (𝒫_9) = (q^4 + 1)^5(q^2 - q + 1)^5(q^2 + q + 1)^5/q^6(q - 1)^3(q + 1)^3(q^2 + 1)^9(q^4 - q^2 + 1)
 
For q > 0 and q ≠ 1, the (𝒫_9) is not 0 and is defined. Therefore, 𝒫_9 is invertible for q > 0 and q ≠ 1.

Consider  𝒫_6,

    𝒫_6 = [                                   1                       - q^4/q^2 + 1             q^4/q^6 + q^4 - q^2 - 1 q^6 + q^4 - q^2/q^6 + q^4 - q^2 - 1                       - q^4/q^2 + 1                 - q^4 + q^2/q^2 + 1;                                   1                       - q^4/q^2 + 1   q^4 - q^2 - 1/q^6 + q^4 - q^2 - 1           - q^2/q^6 + q^4 - q^2 - 1                           1/q^2 + 1                   q^2 - 1/q^4 + q^2;                                   1                                 q^6                         q^4/q^4 - 1                         q^4/q^4 - 1                                 q^6                               - q^4;                                   1                                   1                         - 1/q^4 - 1                         - 1/q^4 - 1                             - 1/q^4                             - 1/q^4;                                   1                                 q^6                                   1                                   0                               - q^2                                   1;                                   1                                   1                                   0                                   1                                   1                                   1 ]


It follows that 

    (𝒫_6) = -(q^4+1)^4(q^2-q+1)^2(q^2+q+1)^2/q^4(q-1)(q+1)(q^2+1)^3
 
For q > 0 and q ≠ 1, the (𝒫_6) is not 0 and is defined. Therefore, 𝒫_6 is invertible for q > 0 and q ≠ 1.

Consider 𝒫_4,

    𝒫_4 = [         1         0    -1/q^4     1/q^8;         1        -1 q^4-1/q^4    -1/q^4;         1         1         0    -1/q^4;         1         0         1         1 ]

It follows that 

    (𝒫_4) = - q^12+3q^8+3q^4+1/q^12
 
For q > 0, the (𝒫_4) is not 0. Therefore, 𝒫_4 is invertible for q > 0.

Consider 𝒫_3,

    𝒫_3 = [             1    -q^4/q^2+1 q^2-q^4/q^2+1;             1           q^6          -q^4;             1             1             1 ]

It follows that 

    (𝒫_3) = q^10+q^8+2q^6+q^4+q^2/q^2+1
 
For q > 0, the (𝒫_3) is not 0. Therefore, 𝒫_3 is invertible for q > 0.



 §.§ Ordering of Basis Vectors to Block pi(W tensor W)(C)
 
Denote v_i to be the i^th basis vector of W. Then, the blocked matrix π_W ⊗ W(C) is with respect to ordered basis: 

       v_1 ⊗v_1,
v_1 ⊗v_20,
v_2 ⊗v_19,
v_4 ⊗v_17,
v_5 ⊗v_16,
v_8 ⊗v_13,
v_9 ⊗v_9,
v_3 ⊗v_18,
v_6 ⊗v_15, 
 
   v_7 ⊗v_14,
v_9 ⊗v_11,
v_11 ⊗v_9,
v_11 ⊗v_11,
v_13 ⊗v_8,
v_10 ⊗v_12,
v_14 ⊗v_7,
v_16 ⊗v_5,
v_12 ⊗v_10, 
 
   v_15 ⊗v_6,
v_17 ⊗v_4,
v_18 ⊗v_3,
v_19 ⊗v_2,
v_20 ⊗v_1,
v_1 ⊗v_13,
v_2 ⊗v_9,
v_2 ⊗v_11,
v_4 ⊗v_7, 
 
   v_5 ⊗v_6,
v_8 ⊗v_3,
v_9 ⊗v_2,
v_3 ⊗v_8,
v_6 ⊗v_5,
v_7 ⊗v_4,
v_11 ⊗v_2,
v_13 ⊗v_1,
v_1 ⊗v_16, 
 
   v_2 ⊗v_14,
v_4 ⊗v_9,
v_4 ⊗v_11,
v_5 ⊗v_10,
v_8 ⊗v_6,
v_9 ⊗v_4,
v_6 ⊗v_8,
v_11 ⊗v_4,
v_10 ⊗v_5, 
 
   v_14 ⊗v_2,
v_16 ⊗v_1,
v_1 ⊗v_17,
v_2 ⊗v_15,
v_4 ⊗v_12,
v_5 ⊗v_9,
v_5 ⊗v_11,
v_8 ⊗v_7,
v_9 ⊗v_5, 
 
   v_7 ⊗v_8,
v_11 ⊗v_5,
v_12 ⊗v_4,
v_15 ⊗v_2,
v_17 ⊗v_1,
v_1 ⊗v_19,
v_2 ⊗v_18,
v_4 ⊗v_15,
v_5 ⊗v_14, 
 
   v_8 ⊗v_9,
v_8 ⊗v_11,
v_9 ⊗v_8,
v_11 ⊗v_8,
v_14 ⊗v_5,
v_15 ⊗v_4,
v_18 ⊗v_2,
v_19 ⊗v_1,
v_2 ⊗v_16, 
 
   v_3 ⊗v_14,
v_4 ⊗v_13,
v_6 ⊗v_9,
v_6 ⊗v_11,
v_7 ⊗v_10,
v_9 ⊗v_6,
v_11 ⊗v_6,
v_13 ⊗v_4,
v_14 ⊗v_3, 
 
   v_16 ⊗v_2,
v_10 ⊗v_7,
v_2 ⊗v_17,
v_3 ⊗v_15,
v_5 ⊗v_13,
v_6 ⊗v_12,
v_7 ⊗v_9,
v_7 ⊗v_11,
v_9 ⊗v_7, 
 
   v_11 ⊗v_7,
v_13 ⊗v_5,
v_12 ⊗v_6,
v_15 ⊗v_3,
v_17 ⊗v_2,
v_2 ⊗v_20,
v_3 ⊗v_19,
v_6 ⊗v_17,
v_7 ⊗v_16, 
 
   v_9 ⊗v_13,
v_11 ⊗v_13,
v_13 ⊗v_9,
v_13 ⊗v_11,
v_16 ⊗v_7,
v_17 ⊗v_6,
v_19 ⊗v_3,
v_20 ⊗v_2,
v_4 ⊗v_19, 
 
   v_6 ⊗v_18,
v_8 ⊗v_16,
v_9 ⊗v_14,
v_10 ⊗v_15,
v_11 ⊗v_14,
v_14 ⊗v_9,
v_14 ⊗v_11,
v_16 ⊗v_8,
v_18 ⊗v_6, 
 
   v_19 ⊗v_4,
v_15 ⊗v_10,
v_4 ⊗v_20,
v_6 ⊗v_19,
v_9 ⊗v_16,
v_10 ⊗v_17,
v_11 ⊗v_16,
v_14 ⊗v_13,
v_16 ⊗v_9, 
 
   v_13 ⊗v_14,
v_16 ⊗v_11,
v_17 ⊗v_10,
v_19 ⊗v_6,
v_20 ⊗v_4,
v_5 ⊗v_19,
v_7 ⊗v_18,
v_8 ⊗v_17,
v_9 ⊗v_15, 
 
   v_11 ⊗v_15,
v_12 ⊗v_14,
v_15 ⊗v_9,
v_15 ⊗v_11,
v_17 ⊗v_8,
v_14 ⊗v_12,
v_18 ⊗v_7,
v_19 ⊗v_5,
v_5 ⊗v_20, 
 
   v_7 ⊗v_19,
v_9 ⊗v_17,
v_11 ⊗v_17,
v_12 ⊗v_16,
v_15 ⊗v_13,
v_17 ⊗v_9,
v_13 ⊗v_15,
v_17 ⊗v_11,
v_16 ⊗v_12, 
 
   v_19 ⊗v_7,
v_20 ⊗v_5,
v_8 ⊗v_20,
v_9 ⊗v_19,
v_11 ⊗v_19,
v_14 ⊗v_17,
v_15 ⊗v_16,
v_18 ⊗v_13,
v_19 ⊗v_9, 
 
   v_13 ⊗v_18,
v_16 ⊗v_15,
v_17 ⊗v_14,
v_19 ⊗v_11,
v_20 ⊗v_8,
v_1 ⊗v_9,
v_2 ⊗v_8,
v_4 ⊗v_5,
v_5 ⊗v_4, 
 
   v_8 ⊗v_2,
v_9 ⊗v_1,
v_1 ⊗v_11,
v_11 ⊗v_1,
v_2 ⊗v_13,
v_3 ⊗v_9,
v_3 ⊗v_11,
v_6 ⊗v_7,
v_7 ⊗v_6, 
 
   v_9 ⊗v_3,
v_11 ⊗v_3,
v_13 ⊗v_2,
v_4 ⊗v_16,
v_6 ⊗v_14,
v_9 ⊗v_10,
v_10 ⊗v_9,
v_10 ⊗v_11,
v_11 ⊗v_10, 
 
   v_14 ⊗v_6,
v_16 ⊗v_4,
v_5 ⊗v_17,
v_7 ⊗v_15,
v_9 ⊗v_12,
v_11 ⊗v_12,
v_12 ⊗v_9,
v_12 ⊗v_11,
v_15 ⊗v_7, 
 
   v_17 ⊗v_5,
v_8 ⊗v_19,
v_9 ⊗v_18,
v_11 ⊗v_18,
v_14 ⊗v_15,
v_15 ⊗v_14,
v_18 ⊗v_9,
v_18 ⊗v_11,
v_19 ⊗v_8, 
 
   v_9 ⊗v_20,
v_13 ⊗v_19,
v_16 ⊗v_17,
v_17 ⊗v_16,
v_19 ⊗v_13,
v_20 ⊗v_9,
v_11 ⊗v_20,
v_20 ⊗v_11,
v_1 ⊗v_6, 
 
   v_2 ⊗v_4,
v_4 ⊗v_2,
v_6 ⊗v_1,
v_1 ⊗v_7,
v_2 ⊗v_5,
v_5 ⊗v_2,
v_7 ⊗v_1,
v_1 ⊗v_14,
v_4 ⊗v_8, 
 
   v_8 ⊗v_4,
v_14 ⊗v_1,
v_1 ⊗v_15,
v_5 ⊗v_8,
v_8 ⊗v_5,
v_15 ⊗v_1,
v_2 ⊗v_6,
v_3 ⊗v_4,
v_4 ⊗v_3, 
 
   v_6 ⊗v_2,
v_2 ⊗v_7,
v_3 ⊗v_5,
v_5 ⊗v_3,
v_7 ⊗v_2,
v_2 ⊗v_10,
v_4 ⊗v_6,
v_6 ⊗v_4,
v_10 ⊗v_2, 
 
   v_2 ⊗v_12,
v_5 ⊗v_7,
v_7 ⊗v_5,
v_12 ⊗v_2,
v_3 ⊗v_16,
v_6 ⊗v_13,
v_13 ⊗v_6,
v_16 ⊗v_3,
v_3 ⊗v_17, 
 
   v_7 ⊗v_13,
v_13 ⊗v_7,
v_17 ⊗v_3,
v_4 ⊗v_14,
v_8 ⊗v_10,
v_10 ⊗v_8,
v_14 ⊗v_4,
v_4 ⊗v_18,
v_8 ⊗v_14, 
 
   v_14 ⊗v_8,
v_18 ⊗v_4,
v_5 ⊗v_15,
v_8 ⊗v_12,
v_12 ⊗v_8,
v_15 ⊗v_5,
v_5 ⊗v_18,
v_8 ⊗v_15,
v_15 ⊗v_8, 
 
   v_18 ⊗v_5,
v_6 ⊗v_16,
v_10 ⊗v_13,
v_13 ⊗v_10,
v_16 ⊗v_6,
v_6 ⊗v_20,
v_13 ⊗v_16,
v_16 ⊗v_13,
v_20 ⊗v_6, 
 
   v_7 ⊗v_17,
v_12 ⊗v_13,
v_13 ⊗v_12,
v_17 ⊗v_7,
v_7 ⊗v_20,
v_13 ⊗v_17,
v_17 ⊗v_13,
v_20 ⊗v_7,
v_10 ⊗v_19, 
 
   v_14 ⊗v_16,
v_16 ⊗v_14,
v_19 ⊗v_10,
v_12 ⊗v_19,
v_15 ⊗v_17,
v_17 ⊗v_15,
v_19 ⊗v_12,
v_14 ⊗v_19,
v_16 ⊗v_18, 
 
   v_18 ⊗v_16,
v_19 ⊗v_14,
v_14 ⊗v_20,
v_16 ⊗v_19,
v_19 ⊗v_16,
v_20 ⊗v_14,
v_15 ⊗v_19,
v_17 ⊗v_18,
v_18 ⊗v_17, 
 
   v_19 ⊗v_15,
v_15 ⊗v_20,
v_17 ⊗v_19,
v_19 ⊗v_17,
v_20 ⊗v_15,
v_1 ⊗v_3,
v_2 ⊗v_2,
v_3 ⊗v_1,
v_1 ⊗v_10, 
 
   v_4 ⊗v_4,
v_10 ⊗v_1,
v_1 ⊗v_12,
v_5 ⊗v_5,
v_12 ⊗v_1,
v_1 ⊗v_18,
v_8 ⊗v_8,
v_18 ⊗v_1,
v_3 ⊗v_10, 
 
   v_6 ⊗v_6,
v_10 ⊗v_3,
v_3 ⊗v_12,
v_7 ⊗v_7,
v_12 ⊗v_3,
v_3 ⊗v_20,
v_13 ⊗v_13,
v_20 ⊗v_3,
v_10 ⊗v_18, 
 
   v_14 ⊗v_14,
v_18 ⊗v_10,
v_10 ⊗v_20,
v_16 ⊗v_16,
v_20 ⊗v_10,
v_12 ⊗v_18,
v_15 ⊗v_15,
v_18 ⊗v_12,
v_12 ⊗v_20, 
 
   v_17 ⊗v_17,
v_20 ⊗v_12,
v_18 ⊗v_20,
v_19 ⊗v_19,
v_20 ⊗v_18,
v_1 ⊗v_2,
v_2 ⊗v_1,
v_1 ⊗v_4,
v_4 ⊗v_1, 
 
   v_1 ⊗v_5,
v_5 ⊗v_1,
v_1 ⊗v_8,
v_8 ⊗v_1,
v_2 ⊗v_3,
v_3 ⊗v_2,
v_3 ⊗v_6,
v_6 ⊗v_3,
v_3 ⊗v_7, 
 
   v_7 ⊗v_3,
v_3 ⊗v_13,
v_13 ⊗v_3,
v_4 ⊗v_10,
v_10 ⊗v_4,
v_5 ⊗v_12,
v_12 ⊗v_5,
v_6 ⊗v_10,
v_10 ⊗v_6, 
 
   v_7 ⊗v_12,
v_12 ⊗v_7,
v_8 ⊗v_18,
v_18 ⊗v_8,
v_10 ⊗v_14,
v_14 ⊗v_10,
v_10 ⊗v_16,
v_16 ⊗v_10,
v_12 ⊗v_15, 
 
   v_15 ⊗v_12,
v_12 ⊗v_17,
v_17 ⊗v_12,
v_13 ⊗v_20,
v_20 ⊗v_13,
v_14 ⊗v_18,
v_18 ⊗v_14,
v_15 ⊗v_18,
v_18 ⊗v_15, 
 
   v_16 ⊗v_20,
v_20 ⊗v_16,
v_17 ⊗v_20,
v_20 ⊗v_17,
v_18 ⊗v_19,
v_19 ⊗v_18,
v_19 ⊗v_20,
v_20 ⊗v_19,
v_3 ⊗v_3, 
 
   v_10 ⊗v_10,
v_12 ⊗v_12,
v_18 ⊗v_18,
v_20 ⊗v_20





 §.§ Algebraically-produced Markov Generator
 
This generator is produced as described in Section <ref>, with the additional step of deleting the ith row and column of the matrix a^-1G^-1π_W ⊗ W(C)G - Id if g_i = 0, or if row i or column i of the matrix contains a negative off-diagonal entry, as those lack a probabilistic interpretation.

The authors note that the polynomials below may be expressible as q-hypergeometric series, but we do not investigate this possibility in this paper. Recall that a = q^12 + q^2 + q^-2 + q^-12, and let the matrix entry * denote, as in Section <ref>, -1 times the sum of the other elements in its row. We may express the Markov generator produced algebraically as the direct sum of one 9 × 9 block:

    a^-2[    *  B_1  B_1  B_2    0    0    0    0    0;  B_3    *  B_4  B_5  B_6  B_7    0    0    0;  B_3  B_4    *  B_5    0    0  B_6  B_7    0;  B_8  B_9  B_9    * B_10 B_11 B_10 B_11 B_12;    0 B_13    0 B_14    * B_15    0    0    0;    0 B_16    0 B_17 B_18    *    0 B_19 B_20;    0    0 B_13 B_14    0    0    * B_15    0;    0    0 B_16 B_17    0 B_19 B_18    * B_20;    0    0    0 B_21    0 B_22    0 B_22    * ]

where

    B_1    =
    q^22a^2-q^20a^2-2q^18a^2+2q^16a^2+3q^14a^2-q^12a^2-4q^10a^2+2q^6a^2
    
        B_2    =
    q^20a^2-2q^16a^2-2q^14a^2+q^12a^2+4q^10a^2+q^8a^2-2q^6a^2-2q^4a^2+1a^2
    
        B_3    =
    q^24a^2-3q^22a^2+3q^20a^2-q^18a^2+2q^16a^2-4q^14a^2+2q^12a^2
    
        B_4    =
    q^22a^2-2q^20a^2+q^18a^2-2q^16a^2+4q^14a^2-2q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_5    =
    a^2 ·2q^16-3q^14-q^12+2q^10+2q^8-q^6-3q^4+2q^2/q^20-q^16+q^12+q^10+q^8-q^4+1
    
        B_6    =
    q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+2q^10a^2-4q^8a^2+2q^6a^2
    
        B_7    =
    q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+4q^8a^2-2q^6a^2+q^4a^2-2q^2a^2+1a^2
    
        B_8    =
    q^24a^2-4q^22a^2+6q^20a^2-4q^18a^2+q^16a^2
    
        B_9    =
    2q^22a^2-5q^20a^2+3q^18a^2+q^16a^2+q^14a^2-4q^12a^2+2q^10a^2
    
        B_10   =
    q^18a^2-2q^16a^2-q^14a^2+4q^12a^2-q^10a^2-2q^8a^2+q^6a^2
    
        B_11   =
    2q^14a^2-4q^12a^2+q^10a^2+q^8a^2+3q^6a^2-5q^4a^2+2q^2a^2
    
        B_12   =
    q^8a^2-4q^6a^2+6q^4a^2-4q^2a^2+1a^2
    
        B_13   =
    q^24a^2-q^22a^2-2q^20a^2+2q^18a^2+3q^16a^2-q^14a^2-4q^12a^2+2q^8a^2
    
        B_14   =
    q^22a^2-2q^18a^2-2q^16a^2+q^14a^2+4q^12a^2+q^10a^2-2q^8a^2-2q^6a^2+q^2a^2
    
        B_15   =
    2q^16a^2-4q^12a^2-q^10a^2+3q^8a^2+2q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_16   =
    q^24a^2-2q^22a^2+q^20a^2-2q^18a^2+4q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+q^8a^2
    
        B_17   =
    a^2 ·2q^18-3q^16-q^14+2q^12+2q^10-q^8-3q^6+2q^4/q^20-q^16+q^12+q^10+q^8-q^4+1
    
        B_18   =
    2q^18a^2-4q^16a^2+2q^14a^2-q^12a^2+3q^10a^2-3q^8a^2+q^6a^2
    
        B_19   =
    q^18a^2-2q^16a^2+q^14a^2-2q^12a^2+4q^10a^2-2q^8a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_20   =
    2q^12a^2-4q^10a^2+2q^8a^2-q^6a^2+3q^4a^2-3q^2a^2+1a^2
    
        B_21   =
    q^24a^2-2q^20a^2-2q^18a^2+q^16a^2+4q^14a^2+q^12a^2-2q^10a^2-2q^8a^2+q^4a^2
    
        B_22   =
    2q^18a^2-4q^14a^2-q^12a^2+3q^10a^2+2q^8a^2-2q^6a^2-q^4a^2+q^2a^2

a 6 × 6 block:

    a^-2[    *  B_1  B_2  B_3    0  B_4;  B_1    *  B_2    0  B_3  B_4;  B_5  B_5    *    0    0  B_6;  B_7    0    0    *  B_1  B_8;    0  B_7    0  B_1    *  B_8;  B_9  B_9 B_10 B_11 B_11    * ]

where

    B_1    =
    q^20a^2-2q^18a^2+q^16a^2-2q^14a^2+4q^12a^2-2q^10a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_2    =
    2q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-3q^4a^2+q^2a^2
    
        B_3    =
    q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_4    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_5    =
    2q^18a^2-2q^16a^2-2q^14a^2+q^12a^2+2q^10a^2-2q^6a^2+q^4a^2
    
        B_6    =
    q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_7    =
    q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_8    =
    q^18a^2-2q^16a^2+3q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_9    =
    q^22a^2-2q^20a^2+2q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_10   =
    -q^18a^2+3q^16a^2-3q^14a^2+q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2
    
        B_11   =
    q^20a^2-2q^18a^2+2q^16a^2-3q^14a^2+3q^12a^2-q^10a^2+q^8a^2-2q^6a^2+q^4a^2

two 6 × 6 blocks:

    a^-2[    *  B_1  B_2  B_3    0    0;  B_4    *  B_5  B_6    0    0;  B_7  B_8    *  B_9    0  B_3; B_10 B_11 B_12    * B_13 B_14;    0    0    0 B_15    * B_16;    0    0 B_17 B_18 B_19    * ]

where

    B_1    =
    q^20a^2-3q^18a^2+3q^16a^2-q^14a^2+2q^12a^2-4q^10a^2+2q^8a^2
    
        B_2    =
    q^18a^2-2q^16a^2+q^14a^2-2q^12a^2+4q^10a^2-2q^8a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_3    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_4    =
    q^24a^2-2q^22a^2+2q^18a^2+q^16a^2-2q^14a^2-2q^12a^2+2q^10a^2
    
        B_5    =
    2q^16a^2-2q^14a^2-2q^12a^2+q^10a^2+2q^8a^2-2q^4a^2+q^2a^2
    
        B_6    =
    q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_7    =
    q^24a^2-2q^22a^2+q^20a^2-2q^18a^2+4q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+q^8a^2
    
        B_8    =
    2q^18a^2-4q^16a^2+2q^14a^2-q^12a^2+3q^10a^2-3q^8a^2+q^6a^2
    
        B_9    =
    q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_10   =
    q^24a^2-3q^22a^2+3q^20a^2-2q^18a^2+3q^16a^2-3q^14a^2+q^12a^2
    
        B_11   =
    q^22a^2-2q^20a^2+2q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_12   =
    -q^18a^2+3q^16a^2-3q^14a^2+q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2
    
        B_13   =
    q^18a^2-2q^16a^2+q^12a^2+q^10a^2-2q^6a^2+q^4a^2
    
        B_14   =
    q^14a^2-q^12a^2-q^10a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_15   =
    q^20a^2-q^18a^2-q^16a^2-q^14a^2+2q^12a^2+2q^10a^2-q^8a^2-q^6a^2-q^4a^2+q^2a^2
    
        B_16   =
    2q^14a^2-2q^12a^2-2q^10a^2+q^8a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_17   =
    q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_18   =
    q^20a^2-2q^18a^2+3q^16a^2-4q^14a^2+2q^12a^2-q^10a^2+3q^8a^2-2q^6a^2-q^4a^2+q^2a^2
    
        B_19   =
    2q^16a^2-4q^14a^2+2q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2

a 6 × 6 block:

    a^-2[   * B_1 B_1   0   0   0; B_2   * B_3 B_4   0   0; B_2 B_5   *   0 B_4   0;   0 B_6   0   * B_7 B_8;   0   0 B_6 B_7   * B_8;   0   0   0 B_9 B_9   * ]

where

    B_1    =
    q^22a^2-2q^20a^2+2q^16a^2+q^14a^2-2q^12a^2-2q^10a^2+2q^8a^2
    
        B_2    =
    q^24a^2-3q^22a^2+3q^20a^2-q^18a^2+2q^16a^2-4q^14a^2+2q^12a^2
    
        B_3    =
    q^22a^2-2q^20a^2+q^18a^2-2q^16a^2+4q^14a^2-2q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_4    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_5    =
    q^22a^2-2q^20a^2+q^18a^2-2q^16a^2+4q^14a^2-2q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_6    =
    q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_7    =
    q^18a^2-2q^16a^2+q^14a^2-2q^12a^2+4q^10a^2-2q^8a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_8    =
    2q^12a^2-4q^10a^2+2q^8a^2-q^6a^2+3q^4a^2-3q^2a^2+1a^2
    
        B_9    =
    2q^16a^2-2q^14a^2-2q^12a^2+q^10a^2+2q^8a^2-2q^4a^2+q^2a^2

four 6 × 6 blocks:

    a^-2[    *  B_1  B_2  B_3    0    0;  B_4    *  B_5  B_6    0    0;  B_7  B_8    *  B_9 B_10 B_11; B_12 B_13 B_14    * B_15 B_16;    0    0 B_17 B_18    * B_19;    0    0 B_20 B_21 B_22    * ]

where

    B_1    =
    q^20a^2-2q^18a^2+2q^14a^2+q^12a^2-2q^10a^2-2q^8a^2+2q^6a^2
    
        B_2    =
    q^22a^2-3q^18a^2+q^16a^2+2q^14a^2+q^12a^2-2q^10a^2-q^8a^2+q^6a^2-q^4a^2+q^2a^2
    
        B_3    =
    q^18a^2-q^16a^2-q^14a^2-q^12a^2+2q^10a^2+2q^8a^2-q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_4    =
    q^24a^2-2q^22a^2+2q^18a^2+q^16a^2-2q^14a^2-2q^12a^2+2q^10a^2
    
        B_5    =
    q^22a^2-q^20a^2-q^18a^2-q^16a^2+2q^14a^2+2q^12a^2-q^10a^2-q^8a^2-q^6a^2+q^4a^2
    
        B_6    =
    q^20a^2-q^18a^2+q^16a^2-q^14a^2-2q^12a^2+q^10a^2+2q^8a^2+q^6a^2-3q^4a^2+1a^2
    
        B_7    =
    q^24a^2-2q^22a^2+2q^18a^2-q^14a^2-q^12a^2+q^10a^2
    
        B_8    =
    q^20a^2-2q^18a^2+q^14a^2+q^12a^2-2q^8a^2+q^6a^2
    
        B_9    =
    q^18a^2-q^16a^2-2q^12a^2+3q^10a^2-q^8a^2+2q^6a^2-4q^4a^2+2q^2a^2
    
        B_10   =
    q^18a^2-2q^16a^2+q^14a^2-q^12a^2+3q^10a^2-3q^8a^2+2q^6a^2-2q^4a^2+q^2a^2
    
        B_11   =
    q^12a^2-3q^10a^2+3q^8a^2-2q^6a^2+3q^4a^2-3q^2a^2+1a^2
    
        B_12   =
    q^24a^2-3q^22a^2+3q^20a^2-2q^18a^2+3q^16a^2-3q^14a^2+q^12a^2
    
        B_13   =
    q^22a^2-2q^20a^2+2q^18a^2-3q^16a^2+3q^14a^2-q^12a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_14   =
    2q^22a^2-4q^20a^2+2q^18a^2-q^16a^2+3q^14a^2-2q^12a^2-q^8a^2+q^6a^2
    
        B_15   =
    q^18a^2-2q^16a^2+q^12a^2+q^10a^2-2q^6a^2+q^4a^2
    
        B_16   =
    q^14a^2-q^12a^2-q^10a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_17   =
    q^24a^2-3q^20a^2+q^18a^2+2q^16a^2+q^14a^2-2q^12a^2-q^10a^2+q^8a^2-q^6a^2+q^4a^2
    
        B_18   =
    q^20a^2-q^18a^2-q^16a^2-q^14a^2+2q^12a^2+2q^10a^2-q^8a^2-q^6a^2-q^4a^2+q^2a^2
    
        B_19   =
    2q^14a^2-2q^12a^2-2q^10a^2+q^8a^2+2q^6a^2-2q^2a^2+1a^2
    
        B_20   =
    q^24a^2-q^22a^2-q^20a^2-q^18a^2+2q^16a^2+2q^14a^2-q^12a^2-q^10a^2-q^8a^2+q^6a^2
    
        B_21   =
    q^22a^2-q^20a^2+q^18a^2-q^16a^2-2q^14a^2+q^12a^2+2q^10a^2+q^8a^2-3q^6a^2+q^2a^2
    
        B_22   =
    2q^18a^2-2q^16a^2-2q^14a^2+q^12a^2+2q^10a^2-2q^6a^2+q^4a^2

a 4 × 4 block:

    a^-2[   * B_1 B_2 B_3; B_1   * B_2 B_3; B_4 B_4   * B_3; B_5 B_5 B_6   * ]

where

    B_1    =
    q^20a^2-2q^18a^2+q^16a^2-2q^14a^2+4q^12a^2-2q^10a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_2    =
    2q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-3q^4a^2+q^2a^2
    
        B_3    =
    q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_4    =
    2q^16a^2-4q^14a^2+2q^12a^2-q^10a^2+3q^8a^2-3q^6a^2+q^4a^2
    
        B_5    =
    q^22a^2-q^20a^2-q^16a^2+2q^12a^2-q^8a^2-q^6a^2+q^4a^2
    
        B_6    =
    -q^18a^2+2q^16a^2-2q^12a^2+2q^8a^2-2q^4a^2+q^2a^2

two 4 × 4 blocks:

    a^-2[   * B_1 B_1 B_2; B_3   * B_4 B_5; B_3 B_4   * B_5; B_6 B_7 B_7   * ]

where

    B_1    =
    q^20a^2-q^18a^2-q^16a^2+2q^12a^2-q^8a^2-q^4a^2+q^2a^2
    
        B_2    =
    q^14a^2-2q^12a^2+q^8a^2+q^6a^2-2q^2a^2+1a^2
    
        B_3    =
    q^24a^2-q^22a^2-2q^20a^2+3q^18a^2-q^16a^2+2q^14a^2-4q^12a^2+3q^10a^2-2q^8a^2+q^6a^2
    
        B_4    =
    q^20a^2-2q^18a^2+q^16a^2-2q^14a^2+4q^12a^2-2q^10a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_5    =
    q^18a^2-2q^16a^2+3q^14a^2-4q^12a^2+2q^10a^2-q^8a^2+3q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_6    =
    q^24a^2-2q^22a^2+q^18a^2+q^16a^2-2q^12a^2+q^10a^2
    
        B_7    =
    q^22a^2-q^20a^2-q^16a^2+2q^12a^2-q^8a^2-q^6a^2+q^4a^2

two 4 × 4 blocks:

    a^-2[   * B_1 B_1 B_2; B_3   * B_4 B_5; B_3 B_4   * B_5; B_6 B_7 B_7   * ]

where

    B_1    = q^20a^2-q^18a^2-2q^16a^2+3q^14a^2-q^12a^2+2q^10a^2-4q^8a^2+3q^6a^2-2q^4a^2+q^2a^2
    
        B_2    = q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+4q^8a^2-2q^6a^2+q^4a^2-2q^2a^2+1a^2
    
        B_3    = q^24a^2-q^22a^2-2q^20a^2+2q^18a^2+2q^16a^2-q^14a^2-2q^12a^2+q^8a^2
    
        B_4    = q^20a^2-q^18a^2-2q^16a^2+q^14a^2+2q^12a^2+q^10a^2-2q^8a^2-q^6a^2+q^4a^2
    
        B_5    = q^16a^2-2q^12a^2-q^10a^2+2q^8a^2+2q^6a^2-2q^4a^2-q^2a^2+1a^2
    
        B_6    = q^24a^2-2q^22a^2+q^20a^2-2q^18a^2+4q^16a^2-2q^14a^2+q^12a^2-2q^10a^2+q^8a^2
    
        B_7    = q^22a^2-2q^20a^2+3q^18a^2-4q^16a^2+2q^14a^2-q^12a^2+3q^10a^2-2q^8a^2-q^6a^2+q^4a^2

eight 4 × 4 blocks:

    a^-2[   * B_1 B_2   0; B_3   * B_4 B_5; B_3 B_6   * B_5;   0 B_7 B_1   * ]

where

    B_1    = q^22a^2-q^20a^2-q^18a^2+q^16a^2+q^8a^2-q^6a^2-q^4a^2+q^2a^2
    
        B_2    = q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_3    = q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_4    = q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_5    = q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_6    = q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2
    
        B_7    = q^24a^2-q^22a^2-q^20a^2+q^18a^2+q^10a^2-q^8a^2-q^6a^2+q^4a^2

four 4 × 4 blocks:

    a^-2[   * B_1 B_1 B_2; B_3   *   0 B_4; B_3   0   * B_4; B_5 B_6 B_6   * ]

where

    B_1    = q^20a^2-2q^18a^2+q^16a^2+q^6a^2-2q^4a^2+q^2a^2
    
        B_2    = q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_3    = q^24a^2-q^22a^2-q^20a^2+q^18a^2+q^10a^2-q^8a^2-q^6a^2+q^4a^2
    
        B_4    = q^20a^2-q^18a^2-q^16a^2+q^14a^2+q^6a^2-q^4a^2-q^2a^2+1a^2
    
        B_5    = q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_6    = q^22a^2-2q^20a^2+q^18a^2+q^8a^2-2q^6a^2+q^4a^2

eight 3 × 3 blocks:

    a^-2[  B_1 -B_1    0;  B_2  B_3  B_4;    0  B_5 -B_5 ]

where

    B_1    = -q^22a^2+2q^18a^2-q^14a^2-q^8a^2+2q^4a^2-1a^2
    
        B_2    = q^24a^2-2q^22a^2+q^20a^2+q^10a^2-2q^8a^2+q^6a^2
    
        B_3    = -q^24a^2+2q^22a^2-q^20a^2-q^18a^2+2q^16a^2-q^14a^2-q^10a^2+2q^8a^2-q^6a^2-q^4a^2+2q^2a^2-1a^2
    
        B_4    = q^18a^2-2q^16a^2+q^14a^2+q^4a^2-2q^2a^2+1a^2
    
        B_5    = q^24a^2-2q^20a^2+q^16a^2+q^10a^2-2q^6a^2+q^2a^2

sixteen 2 × 2 blocks:

    a^-2[  B_1 -B_1;  B_2 -B_2 ]

where

    B_1    = -q^20a^2+q^18a^2+q^16a^2-q^14a^2-q^6a^2+q^4a^2+q^2a^2-1a^2
    
        B_2    = q^24a^2-q^22a^2-q^20a^2+q^18a^2+q^10a^2-q^8a^2-q^6a^2+q^4a^2

and five 1 × 1 blocks with entry 0.


Note that multiplying the 3 × 3 and 2 × 2 blocks of the algebraic generator by the scalar

    a^2/q^12·q^2-1+q^-2/(q-q^-1)^2 (q+q^-1) (q^7+q^-7)

yields the 3 × 3 and 2 × 2 blocks of L_Q (as described in Section <ref>) when n=2. Note additionally that no larger block of L_Q (with n=2) is a constant multiple of a block of the algebraic generator.





 §.§ Type A Ground State Transformation
 
The following notes were provided to us by Dr. Jeffrey Kuan to aid in our calculation of the ground state transformation.

Suppose C is a central element and H= (π_V ⊗π_V) (Δ (C)). Let v ∈ V be a highest-weight vector of a highest-weight representation V. Then,

    H(v ⊗ v) = av⊗ v

for some constant a. By the definition of a central element,

    Δ(F_j^k) C = C Δ(F_j^k).

Given unit basis vectors u_i ⊗ u_j ∈ V ⊗ V, let (k_1, k_2, k_3) ∈ℤ_≥ 0^3 such that

    ⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩≠ 0.

F_3, F_2, F_1 may need to be in a different order, but the values of k= (k_1, k_2, k_3) should be the same. 

Set G: V ⊗ V → V ⊗ V by 

    G(u_i ⊗ u_j) =⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩ u_i ⊗ u_j

where k is the above term. 

Now, set S = G^-1HG. 

    a^-1S-Id has rows that sum to 0.


    For fixed u_i ⊗ u_j, we have that the sum of the row indexed by u_i ⊗ u_j equals
    
    ∑_w_1,w_2⟨ u_i ⊗ u_j | H | w_1 ⊗ w_2 ⟩⟨ w_1 ⊗ w_2 | G | w_1 ⊗ w_2 ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩

    which then equals
    
    ∑_w_1,w_2[ ⟨ u_i ⊗ u_j | H | w_1 ⊗ w_2 ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩∑_k⟨ w_1 ⊗ w_2 |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩]

    By commutation between the coproduct and H, this must then equal
    
    ∑_k⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1)H | v ⊗ v ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩

    Finally, this equals
    
    ∑_ka⟨ u_i ⊗ u_j |Δ(F_3^k_3) ·Δ(F_2^k_2) ·Δ(F_1^k_1) | v ⊗ v ⟩/⟨ u_i ⊗ u_j | G | u_i ⊗ u_j ⟩ = a

    Multiplying by a^-1 and subtracting the identity, we obtain a matrix whose rows sum to 0.



 §.§ Explicit values of Ei,Fi,qHi in W tensor W


Now we give explicitly the values of π_W ⊗ W(E_i), π_W ⊗ W(F_i), π_W ⊗ W(q^H_i) for 1 ≤ i ≤ 3. When performing the calculations, the decomposition of W⊗ W into irreducible representations and weight spaces significantly reduces the computational effort, allowing all computations to be done by hand. The matrices below are given for references only, and are strictly speaking not necessary for the proof. In the expressions below, E_i,j denotes the 400 × 400 matrix with all entries 0 except for the (i,j)th entry, which is a 1.


    π_W ⊗ W(E_1) =    (q^3+q)(E_1,2+E_2,3+E_8,9+E_19,20+E_141,142+E_142,143+E_148,149+E_159,160+E_341,342
       +E_342,343+E_348,349+E_359,360) 
    
        +   (q^2+1)(E_61,62+E_62,63+E_68,69+E_79,80+E_81,82+E_82,83+E_88,89+E_99,100+E_261,262
       +E_262,263+E_268,269+E_279,280+E_281,282+E_282,283+E_288,289+E_299,300) 
    
        +   (q^2)(E_4,6+E_5,7+E_8,11+E_9,13+E_14,16+E_15,17+E_18,19+E_144,146+E_145,147+E_148,151
       +E_149,153+E_154,156+E_155,157+E_158,159+E_344,346+E_345,347+E_348,351+E_349,353
       +E_354,356+E_355,357+E_358,359)
    
        +   (q)(E_64,66+E_65,67+E_68,71+E_69,73+E_74,76+E_75,77+E_78,79+E_84,86+E_85,87+E_88,91
       +E_89,93+E_94,96+E_95,97+E_98,99+E_264,266+E_265,267+E_268,271+E_269,273+E_274,276
       +E_275,277+E_278,279+E_284,286+E_285,287+E_288,291+E_289,293+E_294,296+E_295,297+E_298,299) 
    
        +   (q+q^-1)( ∑_i=1^40 E_i,i+20 + ∑_i=141^160 E_i,i+20 +∑_361^400E_i,i+20 + E_22,23+E_28,29+E_39,40+E_362,363+E_379,380. 
       +E_162,163+E_168,169+E_179,180+E_181,182+E_182,183+E_188,189+E_199,200+E_201,202
       . +E_202,203+E_208,209+E_219,220+E_221,222+E_222,223+E_228,229+E_239,240+E_361,362) 
    
        +   (1) ( ∑_i=61^100 E_i,i+40 + ∑_i=141^160 E_i,i+60+∑_i=161^180 E_i,i+80 +∑_i=261^300 E_i,i+40 +∑_i=341^360E_i,i+20. 
       +E_24,26+E_25,27+E_28,31+E_29,33+E_34,36+E_35,37+E_38,39
       +E_164,166+E_165,167+E_169,173+E_174,176+E_175,177+E_178,179
       +E_184,186+E_185,187+E_188,191+E_189,193+E_194,196+E_195,197+E_198,199+E_204,206
       +E_205,207+E_208,211+E_209,213+E_214,216+E_215,217+E_218,219+E_224,226+E_225,227
       +E_228,231+E_229,233+E_234,236+E_235,237+E_238,239
       .+E_364,366+E_365,367+E_368,371+E_369,373+E_374,376+E_375,377+E_378,379)) 
    
        +   (q^-1)(E_104,106+E_105,107+E_108,111+E_109,113+E_114,116+E_115,117+E_118,119+E_124,126
       +E_125,127+E_128,131+E_129,133+E_134,136+E_135,137+E_138,139+E_304,306+E_305,307
       +E_308,311+E_309,313+E_314,316+E_315,317+E_318,319+E_324,326+E_325,327+E_328,331
       +E_329,333+E_334,336+E_335,337+E_338,339) 
    
        +   (q^-2)(E_44,46+E_45,47+E_48,51+E_49,53+E_54,56+E_55,57+E_58,59+E_244,246+E_245,247
       +E_248,251+E_249,253+E_254,256+E_255,257+E_258,259+E_384,386+E_385,387+E_388,391
       +E_389,393+E_394,396+E_395,397+E_398,399) 
    
        +   (1+q^-2)(E_101,102+E_102,103+E_108,109+E_119,120+E_121,122+E_122,123+E_128,129+E_139,140
       +E_301,302+E_302,303+E_308,309+E_319,320+E_321,322+E_322,323+E_328,329+E_339,340) 
    
        +   (q^-1+q^-3)(E_41,42+E_42,43+E_48,49+E_59,60+E_241,242+E_242,243+E_248,249+E_259,260
       +E_381,382+E_382,383+E_388,389+E_399,400)



    π_W ⊗ W(E_2)    =
    (q^3 + q)(E_46,50+E_47,51+E_51,54+E_52,55+E_53,56+E_55,58+E_126,130+E_127,131+E_131,134
       +E_132,135+E_133,136+E_135,138+E_226,230+E_227,231+E_231,234+E_232,235+E_233,236+E_235,238)
    
    +   (q^2 + 1)(E_26,30+E_27,31+E_31,34+E_32,35+E_33,36+E_35,38+E_86,90+E_87,91+E_91,94+E_92,95
       +E_93,96+E_95,98+E_246,250+E_247,251+E_251,254+E_252,255+E_253,256+E_255,258+E_326,330
       +E_327,331+E_331,334+E_332,335+E_333,336+E_335,338)
    
    +   (q^2)(E_42,44+E_43,46+E_45,48+E_47,49+E_57,59+E_122,124+E_123,126+E_125,128+E_127,129
       +E_137,139+E_222,224+E_223,226+E_225,228+E_227,229+E_237,239)
    
    +   (q)(E_22,24+E_23,26+E_25,28+E_27,29+E_37,39+E_82,84+E_83,86+E_85,88+E_87,89+E_97,99
       +E_242,244+E_243,246+E_245,248+E_247,249+E_257,259+E_322,324+E_323,326+E_325,328
       +E_327,329+E_337,339)
    
    +   (q + q^-1)(∑_i=101^140 E_i, i+80 +∑_i=201^260  E_i,i+60 + ∑_i=281 E_i,i+60 + E_6,10+E_7,11+E_11,14+E_12,15+E_13,16.
       +E_15,18+E_106,110+E_107,111+E_111,114+E_112,115+E_113,116+E_115,118+E_166,170+E_167,171
       +E_171,174+E_172,175+E_173,176 +E_175,178+E_206,210
    +E_207,211+E_211,214+E_212,215
       +E_213,216+E_215,218+E_286,290+E_287,291+E_291,294+E_292,295+E_293,296+E_295,298
       .+E_386,390+E_387,391+E_391,394+E_392,395+E_393,396+E_395,398)
    
    +   (1)(∑_i=21^40E_i,i+40 +∑_i=41^60E_i,i+60∑_i=81^100E_i,i+60 + ∑_i=121^140E_i,i+40 +∑_i=321^340 E_i,i+40 +E_2,4+E_3,6+E_5,8. 
       +E_7,9+E_17,19+E_102,104+E_103,106+E_105,108+E_107,109+E_117,119+E_162,164+E_163,166
       +E_165,168+E_167,169+E_177,179+E_202,204+E_203,206+E_205,208 +E_207,209+E_217,219+E_282,284
       +E_283,286+E_285,288+E_287,289+E_297,299+E_382,384+E_383,386 +E_385,388+E_387,389+E_397,399)
    
    +   (q^-1)(E_62,64+E_63,66+E_65,68+E_67,69+E_77,79+E_142,144+E_143,146+E_145,148+E_147,149
       +E_157,159+E_302,304+E_303,306+E_305,308+E_307,309+E_317,319+E_362,364+E_363,366
       +E_365,368+E_367,369+E_377,379)
    
    +   (q^-2)(E_182,184+E_183,186+E_185,188+E_187,189+E_197,199+E_262,264+E_263,266+E_265,268
       +E_267,269+E_277,279+E_342,344+E_343,346+E_345,348+E_347,349+E_357,359)
    
    +   (1 + q^-2)(E_66,70+E_67,71+E_71,74+E_72,75+E_73,76+E_75,78+E_146,150+E_147,151+E_151,154
       +E_152,155+E_153,156+E_155,158+E_306,310+E_307,311+E_311,314+E_312,315+E_313,316
       +E_315,318+E_366,370+E_367,371+E_371,374+E_372,375+E_373,376+E_375,378)
    
    +   (q^-1 + q^-3)(E_186,190+E_187,191+E_191,194+E_192,195+E_193,196+E_195,198+E_266,270+E_267,271
       +E_271,274+E_272,275+E_273,276+E_275,278+E_346,350+E_347,351+E_351,354+E_352,355
       +E_353,356+E_355,358)



    π_W ⊗ W(E_3)    =
    (-q^3 - q)(E_46,51+E_47,52+E_50,54+E_51,55+E_53,57+E_54,58+E_106,111+E_107,112+E_110,114
       +E_111,115+E_113,117+E_114,118+E_186,191+E_187,192+E_190,194+E_191,195+E_193,197+E_194,198)
    
    +   (-q^2 - 1)(E_26,31+E_27,32+E_30,34+E_31,35+E_33,37+E_34,38+E_66,71+E_67,72+E_70,74+E_71,75
       +E_73,77+E_74,78+E_246,251+E_247,252+E_250,254+E_251,255+E_253,257+E_254,258+E_306,311
       +E_307,312+E_310,314+E_311,315+E_313,317+E_314,318)
    
    +   (-q^2)(E_42,45+E_43,47+E_44,48+E_46,49+E_56,59+E_102,105+E_103,107+E_104,108+E_106,109
       +E_116,119+E_182,185+E_183,187+E_184,188+E_186,189+E_196,199)
    
    +   (-q)(E_22,25+E_23,27+E_24,28+E_26,29+E_36,39+E_62,65+E_63,67+E_64,68+E_66,69+E_76,79
       +E_242,245+E_243,247+E_244,248+E_246,249+E_256,259+E_302,305+E_303,307+E_304,308
       +E_306,309+E_316,319)
    
    +   (-q - q^-1)(∑_i=101^140E_i,i+100+∑_i=181^220E_i,i+80+ ∑_i=241^280E_i,i+80+E_6,11+E_7,12+E_10,14+E_11,15.
       +E_13,17+E_14,18+E_126,131+E_127,132+E_130,134+E_131,135+E_133,137+E_134,138+E_166,171
       +E_167,172+E_170,174+E_171,175+E_173,177+E_174,178+E_206,211+E_207,212+E_210,214+E_211,215
       +E_213,217+E_214,218+E_267,272+E_270,274+E_271,275+E_273,277+E_274,278+E_386,391+E_387,392
       +E_390,394+E_391,395+E_393,397+E_394,398)
    
    +   (-1)(∑_i=21^40 + ∑_i=41 E_i,i+80+E_i,i+60^80+∑_i=101^120E_i,i+60+∑_i=301^320E_i,i+60+E_2,5+E_3,7+E_4,8+E_6,9. 
       ++E_16,19+E_122,125+E_123,127+E_124,128+E_126,129+E_136,139+E_162,165+E_163,167
       +E_164,168+E_166,169+E_176,179+E_202,205+E_203,207+E_204,208+E_206,209+E_216,219+E_262,265
       +E_263,267+E_264,268+E_266,269+E_276,279+E_382,385+E_383,387+E_384,388+E_386,389+E_396,399)
    
    +   (-q^-1)(E_82,85+E_83,87+E_84,88+E_86,89+E_96,99+E_142,145+E_143,147+E_144,148+E_146,149
       +E_156,159+E_322,325+E_323,327+E_324,328+E_326,329+E_336,339+E_362,365+E_363,367
       +E_364,368+E_366,369+E_376,379)
    
    +   (-q^-2)(E_222,225+E_223,227+E_224,228+E_226,229+E_236,239+E_282,285+E_283,287+E_284,288
       +E_286,289+E_296,299+E_342,345+E_343,347+E_344,348+E_346,349+E_356,359)
    
    +   (-1 - q^-2)(E_86,91+E_87,92+E_90,94+E_91,95+E_93,97+E_94,98+E_146,151+E_147,152+E_150,154
       +E_151,155+E_153,157+E_154,158+E_326,331+E_327,332+E_330,334+E_331,335+E_333,337
       +E_334,338+E_366,371+E_367,372+E_370,374+E_371,375+E_373,377+E_374,378)
    
    +   (-q^-1 - q^-3)(E_226,231+E_227,232+E_230,234+E_231,235+E_233,237+E_234,238+E_286,291+E_287,292
       +E_290,294+E_291,295+E_293,297+E_294,298+E_346,351+E_347,352+E_350,354+E_351,355
       +E_353,357+E_354,358)




    π_W ⊗ W(F_1)    =
    (q^3 + q)(E_243,163+E_253,173+E_260,180+E_363,343+E_373,353+E_380,360)
    
    +   (q^2 + 1)(E_246,166+E_247,167+E_256,176+E_257,177+E_366,346+E_367,347+E_376,356+E_377,357)
    
    +   (q^2)(E_23,3+E_33,13+E_40,20+E_43,23+E_53,33+E_60,40+E_103,63+E_113,73+E_120,80+E_123,83
       +E_133,93+E_140,100+E_163,143+E_173,153+E_180,160+E_243,203+E_253,213+E_260,220+E_303,263
       +E_313,273+E_320,280+E_323,283+E_333,293+E_340,300+E_383,363+E_393,373+E_400,380)
    
    +   (q)(E_26,6+E_27,7+E_36,16+E_37,17+E_46,26+E_47,27+E_56,36+E_57,37+E_106,66+E_107,67+E_116,76
       +E_117,77+E_126,86+E_127,87+E_136,96+E_137,97+E_166,146+E_167,147+E_176,156+E_177,157
       +E_246,206+E_247,207+E_256,216+E_257,217+E_306,266+E_307,267+E_316,276+E_317,277+E_326,286
       +E_327,287+E_336,296+E_337,297+E_386,366+E_387,367+E_396,376+E_397,377)
    
    +   (q + q^-1)(E_13,9+E_19,18+E_33,29+E_39,38+E_53,49+E_59,58+E_73,69+E_79,78+E_93,89+E_99,98
       +E_113,109+E_119,118+E_133,129+E_139,138+E_153,149+E_159,158+E_173,169+E_179,178+E_193,189
       +E_199,198+E_213,209+E_219,218+E_233,229+E_239,238+E_242,162+E_249,169+E_250,170+E_251,171
       +E_252,172+E_253,249+E_259,179+E_259,258+E_273,269+E_279,278+E_293,289+E_299,298+E_313,309
       +E_319,318+E_333,329+E_339,338+E_353,349+E_359,358+E_362,342+E_369,349+E_370,350+E_371,351
       +E_372,352+E_373,369+E_379,359+E_379,378+E_393,389+E_399,398)
    
    +   (1)(E_2,1+E_3,2+E_6,4+E_7,5+E_9,8+E_13,11+E_16,14+E_17,15+E_20,19+E_22,2+E_22,21+E_23,22
       +E_26,24+E_27,25+E_29,9+E_29,28+E_30,10+E_31,11+E_32,12+E_33,31+E_36,34+E_37,35+E_39,19
       +E_40,39+E_42,22+E_42,41+E_43,42+E_46,44+E_47,45+E_49,29+E_49,48+E_50,30+E_51,31+E_52,32
       +E_53,51+E_56,54+E_57,55+E_59,39+E_60,59+E_62,61+E_63,62+E_66,64+E_67,65+E_69,68+E_73,71
       +E_76,74+E_77,75+E_80,79+E_82,81+E_83,82+E_86,84+E_87,85+E_89,88+E_93,91+E_96,94+E_97,95
       +E_100,99+E_102,62+E_102,101+E_103,102+E_106,104+E_107,105+E_109,69+E_109,108+E_110,70
       +E_111,71+E_112,72+E_113,111+E_116,114+E_117,115+E_119,79+E_120,119+E_122,82+E_122,121
       +E_123,122+E_126,124+E_127,125+E_129,89+E_129,128+E_130,90+E_131,91+E_132,92+E_133,131
       +E_136,134+E_137,135+E_139,99+E_140,139+E_142,141+E_143,142+E_146,144+E_147,145+E_149,148
       +E_153,151+E_156,154+E_157,155+E_160,159+E_162,142+E_162,161+E_163,162+E_166,164+E_167,165
       +E_169,149+E_169,168+E_170,150+E_171,151+E_172,152+E_173,171+E_176,174+E_177,175+E_179,159
       +E_180,179+E_182,181+E_183,182+E_186,184+E_187,185+E_189,188+E_193,191+E_196,194+E_197,195
       +E_200,199+E_202,201+E_203,202+E_206,204+E_207,205+E_209,208+E_213,211+E_216,214+E_217,215
       +E_220,219+E_222,221+E_223,222+E_226,224+E_227,225+E_229,228+E_233,231+E_236,234+E_237,235
       +E_240,239+E_242,202+E_242,241+E_243,242+E_246,244+E_247,245+E_249,209+E_249,248+E_250,210
       +E_251,211+E_252,212+E_253,251+E_256,254+E_257,255+E_259,219+E_260,259+E_262,261+E_263,262
       +E_266,264+E_267,265+E_269,268+E_273,271+E_276,274+E_277,275+E_280,279+E_282,281+E_283,282
       +E_286,284+E_287,285+E_289,288+E_293,291+E_296,294+E_297,295+E_300,299+E_302,262+E_302,301
       +E_303,302+E_306,304+E_307,305+E_309,269+E_309,308+E_310,270+E_311,271+E_312,272+E_313,311
       +E_316,314+E_317,315+E_319,279+E_320,319+E_322,282+E_322,321+E_323,322+E_326,324+E_327,325
       +E_329,289+E_329,328+E_330,290+E_331,291+E_332,292+E_333,331+E_336,334+E_337,335+E_339,299
       +E_340,339+E_342,341+E_343,342+E_346,344+E_347,345+E_349,348+E_353,351+E_356,354+E_357,355
       +E_360,359+E_362,361+E_363,362+E_366,364+E_367,365+E_369,368+E_373,371+E_376,374+E_377,375
       +E_380,379+E_382,362+E_382,381+E_383,382+E_386,384+E_387,385+E_389,369+E_389,388+E_390,370
       +E_391,371+E_392,372+E_393,391+E_396,394+E_397,395+E_399,379+E_400,399)
    
    +   (q^-1)(E_24,4+E_25,5+E_34,14+E_35,15+E_44,24+E_45,25+E_54,34+E_55,35+E_104,64+E_105,65
       +E_114,74+E_115,75+E_124,84+E_125,85+E_134,94+E_135,95+E_164,144+E_165,145+E_174,154
       +E_175,155+E_244,204+E_245,205+E_254,214+E_255,215+E_304,264+E_305,265+E_314,274
       +E_315,275+E_324,284+E_325,285+E_334,294+E_335,295+E_384,364+E_385,365+E_394,374+E_395,375)
    
    +   (q^-2)(E_21,1+E_28,8+E_38,18+E_41,21+E_48,28+E_58,38+E_101,61+E_108,68+E_118,78+E_121,81
       +E_128,88+E_138,98+E_161,141+E_168,148+E_178,158+E_241,201+E_248,208+E_258,218+E_301,261
       +E_308,268+E_318,278+E_321,281+E_328,288+E_338,298+E_381,361+E_388,368+E_398,378)
    
    +   (1 + q^-2)(E_244,164+E_245,165+E_254,174+E_255,175+E_364,344+E_365,345+E_374,354+E_375,355)
    
    +   (q^-1 + q^-3)(E_241,161+E_248,168+E_258,178+E_361,341+E_368,348+E_378,358)




    π_W ⊗ W(F_1) =
       (q^3 + q)(E_243,163+E_253,173+E_260,180+E_363,343+E_373,353+E_380,360)
    
    +   (q^2 + 1)(E_246,166+E_247,167+E_256,176+E_257,177+E_366,346+E_367,347+E_376,356+E_377,357)
    
    +   (q^2)(E_23,3+E_33,13+E_40,20+E_43,23+E_53,33+E_60,40+E_103,63+E_113,73+E_120,80+E_123,83
       +E_133,93+E_140,100+E_163,143+E_173,153+E_180,160+E_243,203+E_253,213+E_260,220+E_303,263
       +E_313,273+E_320,280+E_323,283+E_333,293+E_340,300+E_383,363+E_393,373+E_400,380)
    
    +   (q)(∑_i=1^4 E_10i+16,10i-4 + E_10i+17,10i-3. 
       . + E_106,66 + E_107,67 + E_116,76) 
    
    +   (q + q^-1)(∑_i=1^20 E_20i-7,20i-11 + E_20i-1,20i-2. 
       + ∑_i=1^2 E_120i+122,180i-18 + E_120i+129,180i-11 + E_120i+130,180i-10 + E_120i+131,180i-9
       . + E_120i+132,180i-8 + E_120i+139,180i-1) 
    
    +   (1)(∑_i=1^20 E_20i-18,20i-19 + E_20i-17,20i-18 + E_20i-11,20i-12 +E_20i,20i-1 + E_20i-14,20i-16. 
       + E_20i-13,20i-15 + E_20i-7,20i-9 + E_20i-4,20i-6 + E_20i-3,20i-5
       + ∑_i ∈{1,2,8,19} E_20i+2,20i-18 + E_20i+9,20i-11 + E_20i+10,20i-10 + E_20i+11,20i-9 + E_20i+12,20i-8
       + E_20i+19,20i-1
       + ∑_i ∈{1,2,8,11,12} E_20i+82,20i+42 + E_20i+89,20i+49 + E_20i+90,20i+50 + E_20i+91,20i+51 + E_20i+92,20i+52
       . + E_20i+99,20i+59) 
    
    +   (q^-1)(∑_i ∈{1,2,3,4,15,16,37,38} E_10i+14,10i-6 + E_10i+15,10i-5. 
       . + ∑_i ∈{1,2,3,4,15,16,21,22,23,24} E_10i+94,10i+54 + E_10i+95,10i+55) 
    
    +   (q^-2)(E_21,1+E_28,8+E_38,18+E_41,21+E_48,28+E_58,38+E_101,61+E_108,68+E_118,78+E_121,81
       +E_128,88+E_138,98+E_161,141+E_168,148+E_178,158+E_241,201+E_248,208+E_258,218+E_301,261
       +E_308,268+E_318,278+E_321,281+E_328,288+E_338,298+E_381,361+E_388,368+E_398,378)
    
    +   (1 + q^-2)(E_244,164+E_245,165+E_254,174+E_255,175+E_364,344+E_365,345+E_374,354+E_375,355)
    
    +   (q^-1 + q^-3)(E_241,161+E_248,168+E_258,178+E_361,341+E_368,348+E_378,358)




    π_W ⊗ W(F_2)    =
    (q^3 + q)(E_110,50+E_114,54+E_118,58)
    
    +   (q^2 + 1)(E_104,44+E_108,48+E_116,56+E_119,59)
    
    +   (q^2)(E_70,30+E_74,34+E_78,38+E_150,90+E_154,94+E_158,98+E_190,110+E_194,114+E_198,118+E_210,130
       +E_214,134+E_218,138+E_270,210+E_274,214+E_278,218+E_290,230+E_294,234+E_298,238+E_350,290
       +E_354,294+E_358,298+E_370,330+E_374,334+E_378,338)
    
    +   (q)(E_64,24+E_68,28+E_76,36+E_79,39+E_144,84+E_148,88+E_156,96+E_159,99+E_184,104+E_188,108
       +E_196,116+E_199,119+E_204,124+E_208,128+E_216,136+E_219,139+E_264,204+E_268,208+E_276,216
       +E_279,219+E_284,224+E_288,228+E_296,236+E_299,239+E_344,284+E_348,288+E_356,296+E_359,299
       +E_364,324+E_368,328+E_376,336+E_379,339)
    
    +   (q + q^-1)(E_6,3+E_26,23+E_46,43+E_66,63+E_86,83+E_101,41+E_106,46+E_106,103+E_109,49+E_111,51
       +E_115,55+E_120,60+E_126,123+E_146,143+E_166,163+E_186,183+E_206,203+E_226,223+E_246,243
       +E_266,263+E_286,283+E_306,303+E_326,323+E_346,343+E_366,363+E_386,383)
    
    +   (1)(E_4,2+E_8,5+E_10,6+E_11,7+E_14,11+E_15,12+E_18,15+E_19,17+E_24,22+E_28,25+E_30,26
       +E_31,27+E_34,31+E_35,32+E_38,35+E_39,37+E_44,42+E_48,45+E_50,46+E_51,47+E_54,51
       +E_55,52+E_58,55+E_59,57+E_61,21+E_64,62+E_66,26+E_68,65+E_69,29+E_70,66+E_71,31
       +E_71,67+E_74,71+E_75,35+E_75,72+E_78,75+E_79,77+E_80,40+E_84,82+E_88,85+E_90,86
       +E_91,87+E_94,91+E_95,92+E_98,95+E_99,97+E_104,102+E_108,105+E_110,106+E_111,107
       +E_114,111+E_115,112+E_118,115+E_119,117+E_124,122+E_128,125+E_130,126+E_131,127+E_134,131
       +E_135,132+E_138,135+E_139,137+E_141,81+E_144,142+E_146,86+E_148,145+E_149,89+E_150,146
       +E_151,91+E_151,147+E_154,151+E_155,95+E_155,152+E_158,155+E_159,157+E_160,100+E_164,162
       +E_168,165+E_170,166+E_171,167+E_174,171+E_175,172+E_178,175+E_179,177+E_181,101+E_184,182
       +E_186,106+E_188,185+E_189,109+E_190,186+E_191,111+E_191,187+E_194,191+E_195,115+E_195,192
       +E_198,195+E_199,197+E_200,120+E_201,121+E_204,202+E_206,126+E_208,205+E_209,129+E_210,206
       +E_211,131+E_211,207+E_214,211+E_215,135+E_215,212+E_218,215+E_219,217+E_220,140+E_224,222
       +E_228,225+E_230,226+E_231,227+E_234,231+E_235,232+E_238,235+E_239,237+E_244,242+E_248,245
       +E_250,246+E_251,247+E_254,251+E_255,252+E_258,255+E_259,257+E_261,201+E_264,262+E_266,206
       +E_268,265+E_269,209+E_270,266+E_271,211+E_271,267+E_274,271+E_275,215+E_275,272+E_278,275
       +E_279,277+E_280,220+E_281,221+E_284,282+E_286,226+E_288,285+E_289,229+E_290,286+E_291,231
       +E_291,287+E_294,291+E_295,235+E_295,292+E_298,295+E_299,297+E_300,240+E_304,302+E_308,305
       +E_310,306+E_311,307+E_314,311+E_315,312+E_318,315+E_319,317+E_324,322+E_328,325+E_330,326
       +E_331,327+E_334,331+E_335,332+E_338,335+E_339,337+E_341,281+E_344,342+E_346,286+E_348,345
       +E_349,289+E_350,346+E_351,291+E_351,347+E_354,351+E_355,295+E_355,352+E_358,355+E_359,357
       +E_360,300+E_361,321+E_364,362+E_366,326+E_368,365+E_369,329+E_370,366+E_371,331+E_371,367
       +E_374,371+E_375,335+E_375,372+E_378,375+E_379,377+E_380,340+E_384,382+E_388,385+E_390,386
       +E_391,387+E_394,391+E_395,392+E_398,395+E_399,397)
    
    +   (q^-1)(E_62,22+E_65,25+E_73,33+E_77,37+E_142,82+E_145,85+E_153,93+E_157,97+E_182,102+E_185,105
       +E_193,113+E_197,117+E_202,122+E_205,125+E_213,133+E_217,137+E_262,202+E_265,205+E_273,213
       +E_277,217+E_282,222+E_285,225+E_293,233+E_297,237+E_342,282+E_345,285+E_353,293+E_357,297
       +E_362,322+E_365,325+E_373,333+E_377,337)
    
    +   (q^-2)(E_63,23+E_67,27+E_72,32+E_143,83+E_147,87+E_152,92+E_183,103+E_187,107+E_192,112
       +E_203,123+E_207,127+E_212,132+E_263,203+E_267,207+E_272,212+E_283,223+E_287,227
       +E_292,232+E_343,283+E_347,287+E_352,292+E_363,323+E_367,327+E_372,332)
    
    +   (1 + q^-2)(E_102,42+E_105,45+E_113,53+E_117,57)
    
    +   (q^-1 + q^-3)(E_103,43+E_107,47+E_112,52)
    
    +   (q^2/q + q^-1)(E_270,170+E_274,174+E_278,178+E_310,250+E_314,254+E_318,258)
    
    +   (q/q + q^-1)(E_264,164+E_268,168+E_276,176+E_279,179+E_304,244+E_308,248+E_316,256+E_319,259)
    
    +   (1/q + q^-1)(E_14,9+E_16,13+E_34,29+E_36,33+E_54,49+E_56,53+E_74,69+E_76,73+E_94,89+E_96,93
       +E_114,109+E_116,113+E_134,129+E_136,133+E_154,149+E_156,153+E_174,169+E_176,173+E_194,189
       +E_196,193+E_214,209+E_216,213+E_234,229+E_236,233+E_254,249+E_256,253+E_261,161+E_266,166
       +E_269,169+E_271,171+E_274,269+E_275,175+E_276,273+E_280,180+E_294,289+E_296,293+E_301,241
       +E_306,246+E_309,249+E_311,251+E_314,309+E_315,255+E_316,313+E_320,260+E_334,329+E_336,333
       +E_354,349+E_356,353+E_374,369+E_376,373+E_394,389+E_396,393)
    
    +   (q^-1/q + q^-1)(E_262,162+E_265,165+E_273,173+E_277,177+E_302,242+E_305,245+E_313,253+E_317,257)
    
    +   (q^-2/q + q^-1)(E_263,163+E_267,167+E_272,172+E_303,243+E_307,247+E_312,252)




    π_W ⊗ W(F_2) =
       (q^3 + q)(E_110,50+E_114,54+E_118,58)
    
    +   (q^2 + 1)(E_104,44+E_108,48+E_116,56+E_119,59)
    
    +   (q^2)(∑_i ∈{1,16} E_20i+50,20i+10 + E_20i+54,20i+14 + E_20i+58,20i+18.
       +∑_i ∈{4,10,11,14} E_20i+70,20i+10 + E_20i+74,20i+14 + E_20i+78,20i+18
       .+∑_i ∈{5,6} E_20i+90,20i+10 + E_20i+94,20i+14 + E_20i+98,20i+18) 
    
    +   (q)(∑_i ∈{1,16} E_20i+44,20i+4 + E_20i+48,20i+8 + E_20i+56,20i+16 + E_E_20i+59,20i+19.
       +∑_i ∈{4,10,11,14} E_20i+64,20i+4 + E_20i+68,20i+8 + E_20i+76,20i+16 + E_E_20i+79,20i+19
       . + ∑_i ∈{5,6} E_20i+84,20i+4 + E_20i+88,20i+8 + E_20i+96,20i+16 + E_E_20i+99,20i+19)
    
    +   (q+q^-1) ( ∑_i=1^20 E_20i-14,20i-17.
       . + ∑_j ∈{1,6,9,11,15,20} E_j+100,j+40)
    
    +   (1)( ∑_i=1^20 E_20i-16,20i-18 + E_20i-12,20i-15 + E_20i-10,20i-14 + E_20i-9,20i-13 + E_20i-6,20i-9. 
       + E_20i-5,20i-8 + E_20i-2,20i-5 +  E_20i-1,20i-3
       +∑_i ∈{1,16} E_20i+41,20i+1 + E_20i+46,20i+6 + E_20i+49,20i+9 + E_20i+51,20i+11 + E_20i+55,20i+15
       +E_20i+60,20i+20
       +∑_i ∈{4,10,11,14} E_20i+61,20i+1 + E_20i+66,20i+6 + E_20i+69,20i+9 + E_20i+71,20i+11 + E_20i+75,20i+15
       +E_20i+80,20i+20
       +∑_i ∈{5,6} E_20i+81,20i+1 + E_20i+86,20i+6 + E_20i+89,20i+9 + E_20i+91,20i+11 + E_20i+95,20i+15
       .+E_20i+100,20i+20) 
    
    +   (q^-1)( ∑_i ∈{1,16} E_20i+42,20i+2 + E_20i+45,20i+5 + E_20i+53,20i+13 + E_20i+57,20i+17.
       +∑_i ∈{4,10,11,14} E_20i+62,20i+2 + E_20i+65,20i+5 + E_20i+73,20i+13 + E_20i+77,20i+17
       . +∑_i ∈{5,6} E_20i+82,20i+2 + E_20i+85,20i+5 + E_20i+93,20i+13 + E_20i+97,20i+17) 
    
    +   (q^-2)( ∑_i ∈{1,16} E_20i+43,20i+3 + E_20i+47,20i+7 + E_20i+52,20i+12.
       +∑_i ∈{4,10,11,14} E_20i+63,20i+3 + E_20i+67,20i+7 + E_20i+72,20i+12
       . +∑_i ∈{5,6} E_20i+83,20i+3 + E_20i+87,20i+7 + E_20i+92,20i+12) 
    
    +   (1 + q^-2)(E_102,42+E_105,45+E_113,53+E_117,57)
    
    +   (q^-1 + q^-3)(E_103,43+E_107,47+E_112,52)
    
    +   (q^2/q + q^-1)(E_270,170+E_274,174+E_278,178+E_310,250+E_314,254+E_318,258)
    
    +   (q/q + q^-1)(E_264,164+E_268,168+E_276,176+E_279,179+E_304,244+E_308,248+E_316,256+E_319,259)
    
    +   (1/q + q^-1)( ∑_i=1^20 E_20i-6,20i-11 + E_20i-4,20i-7.
       . + ∑_j ∈{1,6,9,11,15,20} E_j+300,j+240 + E_j+260,j+160) 
    
    +   (q^-1/q + q^-1)(E_262,162+E_265,165+E_273,173+E_277,177+E_302,242+E_305,245+E_313,253+E_317,257)
    
    +   (q^-2/q + q^-1)(E_263,163+E_267,167+E_272,172+E_303,243+E_307,247+E_312,252)




    π_W ⊗ W(F_3)    =
    -(q^3 + q)(E_132,52+E_135,55+E_138,58)
    
    -   (q^2 + 1)(E_125,45+E_128,48+E_137,57+E_139,59)
    
    -   (q^2)(E_92,32+E_95,35+E_98,38+E_152,72+E_155,75+E_158,78+E_212,112+E_215,115+E_218,118+E_232,132
       +E_235,135+E_238,138+E_272,192+E_275,195+E_278,198+E_292,212+E_295,215+E_298,218+E_352,272
       +E_355,275+E_358,278+E_372,312+E_375,315+E_378,318)
    
    -   (q)(E_85,25+E_88,28+E_97,37+E_99,39+E_145,65+E_148,68+E_157,77+E_159,79+E_205,105+E_208,108
       +E_217,117+E_219,119+E_225,125+E_228,128+E_237,137+E_239,139+E_265,185+E_268,188+E_277,197
       +E_279,199+E_285,205+E_288,208+E_297,217+E_299,219+E_345,265+E_348,268+E_357,277+E_359,279
       +E_365,305+E_368,308+E_377,317+E_379,319)
    
    -   (q + q^-1)(E_7,3+E_27,23+E_47,43+E_67,63+E_87,83+E_107,103+E_121,41+E_127,47+E_127,123
       +E_129,49+E_131,51+E_134,54+E_140,60+E_147,143+E_167,163+E_187,183+E_207,203
       +E_227,223+E_247,243+E_267,263+E_287,283+E_307,303+E_327,323+E_347,343+E_367,363+E_387,383)
    
    -   (1)(E_5,2+E_8,4+E_11,6+E_12,7+E_14,10+E_15,11+E_18,14+E_19,16+E_25,22+E_28,24+E_31,26
       +E_32,27+E_34,30+E_35,31+E_38,34+E_39,36+E_45,42+E_48,44+E_51,46+E_52,47+E_54,50
       +E_55,51+E_58,54+E_59,56+E_65,62+E_68,64+E_71,66+E_72,67+E_74,70+E_75,71+E_78,74
       +E_79,76+E_81,21+E_85,82+E_87,27+E_88,84+E_89,29+E_91,31+E_91,86+E_92,87+E_94,34
       +E_94,90+E_95,91+E_98,94+E_99,96+E_100,40+E_105,102+E_108,104+E_111,106+E_112,107
       +E_114,110+E_115,111+E_118,114+E_119,116+E_125,122+E_128,124+E_131,126+E_132,127+E_134,130
       +E_135,131+E_138,134+E_139,136+E_141,61+E_145,142+E_147,67+E_148,144+E_149,69+E_151,71
       +E_151,146+E_152,147+E_154,74+E_154,150+E_155,151+E_158,154+E_159,156+E_160,80+E_165,162
       +E_168,164+E_171,166+E_172,167+E_174,170+E_175,171+E_178,174+E_179,176+E_185,182+E_188,184
       +E_191,186+E_192,187+E_194,190+E_195,191+E_198,194+E_199,196+E_201,101+E_205,202+E_207,107
       +E_208,204+E_209,109+E_211,111+E_211,206+E_212,207+E_214,114+E_214,210+E_215,211+E_218,214
       +E_219,216+E_220,120+E_221,121+E_225,222+E_227,127+E_228,224+E_229,129+E_231,131+E_231,226
       +E_232,227+E_234,134+E_234,230+E_235,231+E_238,234+E_239,236+E_240,140+E_245,242+E_248,244
       +E_251,246+E_252,247+E_254,250+E_255,251+E_258,254+E_259,256+E_261,181+E_265,262+E_267,187
       +E_268,264+E_269,189+E_271,191+E_271,266+E_272,267+E_274,194+E_274,270+E_275,271+E_278,274
       +E_279,276+E_280,200+E_281,201+E_285,282+E_287,207+E_288,284+E_289,209+E_291,211+E_291,286
       +E_292,287+E_294,214+E_294,290+E_295,291+E_298,294+E_299,296+E_300,220+E_305,302+E_308,304
       +E_311,306+E_312,307+E_314,310+E_315,311+E_318,314+E_319,316+E_325,322+E_328,324+E_331,326
       +E_332,327+E_334,330+E_335,331+E_338,334+E_339,336+E_341,261+E_345,342+E_347,267+E_348,344
       +E_349,269+E_351,271+E_351,346+E_352,347+E_354,274+E_354,350+E_355,351+E_358,354+E_359,356
       +E_360,280+E_361,301+E_365,362+E_367,307+E_368,364+E_369,309+E_371,311+E_371,366+E_372,367
       +E_374,314+E_374,370+E_375,371+E_378,374+E_379,376+E_380,320+E_385,382+E_388,384+E_391,386
       +E_392,387+E_394,390+E_395,391+E_398,394+E_399,396)
    
    -   (q^-1)(E_82,22+E_84,24+E_93,33+E_96,36+E_142,62+E_144,64+E_153,73+E_156,76+E_202,102+E_204,104
       +E_213,113+E_216,116+E_222,122+E_224,124+E_233,133+E_236,136+E_262,182+E_264,184+E_273,193
       +E_276,196+E_282,202+E_284,204+E_293,213+E_296,216+E_342,262+E_344,264+E_353,273+E_356,276
       +E_362,302+E_364,304+E_373,313+E_376,316)
    
    -   (q^-2)(E_83,23+E_86,26+E_90,30+E_143,63+E_146,66+E_150,70+E_203,103+E_206,106+E_210,110+E_223,123
       +E_226,126+E_230,130+E_263,183+E_266,186+E_270,190+E_283,203+E_286,206+E_290,210+E_343,263
       +E_346,266+E_350,270+E_363,303+E_366,306+E_370,310)
    
    -   (1 + q^-2)(E_122,42+E_124,44+E_133,53+E_136,56)
    
    -   (q^-1 + q^-3)(E_123,43+E_126,46+E_130,50)
    
    -   (q^2/q + q^-1)(E_292,172+E_295,175+E_298,178+E_332,252+E_335,255+E_338,258)
    
    -   (q/q + q^-1)(E_285,165+E_288,168+E_297,177+E_299,179+E_325,245+E_328,248+E_337,257+E_339,259)
    
    -   (1/q + q^-1)(E_15,9+E_17,13+E_35,29+E_37,33+E_55,49+E_57,53+E_75,69+E_77,73+E_95,89+E_97,93
       +E_115,109+E_117,113+E_135,129+E_137,133+E_155,149+E_157,153+E_175,169+E_177,173+E_195,189
       +E_197,193+E_215,209+E_217,213+E_235,229+E_237,233+E_255,249+E_257,253+E_275,269+E_277,273
       +E_281,161+E_287,167+E_289,169+E_291,171+E_294,174+E_295,289+E_297,293+E_300,180+E_315,309
       +E_317,313+E_321,241+E_327,247+E_329,249+E_331,251+E_334,254+E_335,329+E_337,333+E_340,260
       +E_355,349+E_357,353+E_375,369+E_377,373+E_395,389+E_397,393)
    
    -   (q^-1/q + q^-1)(E_282,162+E_284,164+E_293,173+E_296,176+E_322,242+E_324,244+E_333,253+E_336,256)
    
    -   (q^-2/q + q^-1)(E_283,163+E_286,166+E_290,170+E_323,243+E_326,246+E_330,250)




    π_W ⊗ W(F_3) =
    -   (q^3 + q)(E_132,52+E_135,55+E_138,58)
    
    -   (q^2 + 1)(E_125,45+E_128,48+E_137,57+E_139,59)
    
    -   (q^2)( ∑_i ∈{1,15} E_20i+72,20i+12 + E_20i+75,20i+15 + E_20i+78,20i+18.
       + ∑_i ∈{3,9,10,13} E_20i+92,20i+12 + E_20i+95,20i+15 + E_20i+98,20i+18
       .+ ∑_i ∈{5,6} E_20i+112,20i+12 + E_20i+115,20i+15 + E_20i+118,20i+18)
    
    -   (q)( ∑_i ∈{1,15} E_20i+65,20i+5 + E_20i+68,20i+8 + E_20i+77,20i+17 + E_20i+79,20i+19.
       + ∑_i ∈{3,9,10,13} E_20i+85,20i+5 + E_20i+88,20i+8 + E_20i+97,20i+17 + E_20i+99,20i+19
       . + ∑_i ∈{5,6} E_20i+105,20i+5 + E_20i+108,20i+8 + E_20i+117,20i+17 + E_20i+119,20i+19)
    
    -   (q+q^-1)( ∑_i=1^20 E_20i-13,20i-17.
       . + ∑_j ∈{1,7,9,11,14,20} E_j+120,j+40)
       -(1)( ∑_i=1^20 E_20i-15,20i-18 + E_20i-12,20i-16 + E_20i-9,20i-14 + E_20i-8,20i-13 + E_20i-6,20i-10.
        + E_20i-5,20i-9 + E_20i-2,20i-6 + E_20i-1,20i-4
       + ∑_i ∈{1,15} E_20i+61,20i+1 + E_20i+67,20i+7 + E_20i+69,20i+9 + E_20i+71,20i+11 + E_20i+74,20i+14
       + E_20i+80,20i+20
       + ∑_i ∈{3,9,10,13} E_20i+81,20i+1 + E_20i+87,20i+7 + E_20i+89,20i+9 + E_20i+91,20i+11 + E_20i+94,20i+14
       + E_20i+80,20i+20
       + ∑_i ∈{5,6} E_20i+101,20i+1 + E_20i+107,20i+7 + E_20i+109,20i+9 + E_20i+111,20i+11 + E_20i+114,20i+14
       .+ E_20i+120,20i+20)
    
    -   (q^-1)( ∑_i ∈{1,15} E_20i+62,20i+2 + E_20i+64,20i+4 + E_20i+73,20i+13 + E_20i+76,20i+16.
       + ∑_i ∈{3,9,10,13} E_20i+82,20i+2 + E_20i+84,20i+4 + E_20i+93,20i+13 + E_20i+96,20i+16
       . + ∑_i ∈{5,6} E_20i+102,20i+2 + E_20i+104,20i+4 + E_20i+113,20i+13 + E_20i+116,20i+16)
    
    -   (q^-2)( ∑_i ∈{1,15} E_20i+63,20i+3 + E_20i+66,20i+6 + E_20i+70,20i+10.
       + ∑_i ∈{3,9,10,13} E_20i+83,20i+3 + E_20i+86,20i+6 + E_20i+90,20i+10
       . + ∑_i ∈{1,15} E_20i+103,20i+3 + E_20i+106,20i+6 + E_20i+110,20i+10)
    
    -   (1 + q^-2)(E_122,42+E_124,44+E_133,53+E_136,56)
    
    -   (q^-1 + q^-3)(E_123,43+E_126,46+E_130,50)
    
    -   (q^2/q + q^-1)(E_292,172+E_295,175+E_298,178+E_332,252+E_335,255+E_338,258)
    
    -   (q/q + q^-1)(E_285,165+E_288,168+E_297,177+E_299,179+E_325,245+E_328,248+E_337,257+E_339,259)
    
    -   (1/q + q^-1)( ∑_i=1^20 E_20i-5,20i-11 + E_20i-3,20i-7.
       . + ∑_j ∈{1,7,9,11,14,20} E_j+280,j+160 + E_j+320,j+240) 
    
    -   (q^-1/q + q^-1)(E_282,162+E_284,164+E_293,173+E_296,176+E_322,242+E_324,244+E_333,253+E_336,256)
    
    -   (q^-2/q + q^-1)(E_283,163+E_286,166+E_290,170+E_323,243+E_326,246+E_330,250)



    π_W ⊗ W(q^H_1)    =
    (q^4)(E_1,1+E_8,8+E_18,18+E_141,141+E_148,148+E_158,158+E_341,341+E_348,348+E_358,358)
    
    +   (q^3)(E_4,4+E_5,5+E_14,14+E_15,15+E_61,61+E_68,68+E_78,78+E_81,81+E_88,88+E_98,98+E_144,144
       +E_145,145+E_154,154+E_155,155+E_261,261+E_268,268+E_278,278+E_281,281+E_288,288+E_298,298
       +E_344,344+E_345,345+E_354,354+E_355,355)
    
    +   (q^2)(E_2,2+E_9,9+E_10,10+E_11,11+E_12,12+E_19,19+E_21,21+E_28,28+E_38,38+E_64,64+E_65,65
       +E_74,74+E_75,75+E_84,84+E_85,85+E_94,94+E_95,95+E_142,142+E_149,149+E_150,150+E_151,151
       +E_152,152+E_159,159+E_161,161+E_168,168+E_178,178+E_181,181+E_188,188+E_198,198+E_201,201
       +E_208,208+E_218,218+E_221,221+E_228,228+E_238,238+E_264,264+E_265,265+E_274,274+E_275,275
       +E_284,284+E_285,285+E_294,294+E_295,295+E_342,342+E_349,349+E_350,350+E_351,351+E_352,352
       +E_359,359+E_361,361+E_368,368+E_378,378)
    
    +   (q)(E_6,6+E_7,7+E_16,16+E_17,17+E_24,24+E_25,25+E_34,34+E_35,35+E_62,62+E_69,69+E_70,70
       +E_71,71+E_72,72+E_79,79+E_82,82+E_89,89+E_90,90+E_91,91+E_92,92+E_99,99+E_101,101
       +E_108,108+E_118,118+E_121,121+E_128,128+E_138,138+E_146,146+E_147,147+E_156,156+E_157,157
       +E_164,164+E_165,165+E_174,174+E_175,175+E_184,184+E_185,185+E_194,194+E_195,195+E_204,204
       +E_205,205+E_214,214+E_215,215+E_224,224+E_225,225+E_234,234+E_235,235+E_262,262+E_269,269
       +E_270,270+E_271,271+E_272,272+E_279,279+E_282,282+E_289,289+E_290,290+E_291,291+E_292,292
       +E_299,299+E_301,301+E_308,308+E_318,318+E_321,321+E_328,328+E_338,338+E_346,346+E_347,347
       +E_356,356+E_357,357+E_364,364+E_365,365+E_374,374+E_375,375)
    
    +   (1)(E_3,3+E_13,13+E_20,20+E_22,22+E_29,29+E_30,30+E_31,31+E_32,32+E_39,39+E_41,41+E_48,48
       +E_58,58+E_66,66+E_67,67+E_76,76+E_77,77+E_86,86+E_87,87+E_96,96+E_97,97+E_104,104
       +E_105,105+E_114,114+E_115,115+E_124,124+E_125,125+E_134,134+E_135,135+E_143,143+E_153,153
       +E_160,160+E_162,162+E_169,169+E_170,170+E_171,171+E_172,172+E_179,179+E_182,182+E_189,189
       +E_190,190+E_191,191+E_192,192+E_199,199+E_202,202+E_209,209+E_210,210+E_211,211+E_212,212
       +E_219,219+E_222,222+E_229,229+E_230,230+E_231,231+E_232,232+E_239,239+E_241,241+E_248,248
       +E_258,258+E_266,266+E_267,267+E_276,276+E_277,277+E_286,286+E_287,287+E_296,296+E_297,297
       +E_304,304+E_305,305+E_314,314+E_315,315+E_324,324+E_325,325+E_334,334+E_335,335+E_343,343
       +E_353,353+E_360,360+E_362,362+E_369,369+E_370,370+E_371,371+E_372,372+E_379,379+E_381,381
       +E_388,388+E_398,398)
    
    +   (q^-1)(E_26,26+E_27,27+E_36,36+E_37,37+E_44,44+E_45,45+E_54,54+E_55,55+E_63,63+E_73,73
       +E_80,80+E_83,83+E_93,93+E_100,100+E_102,102+E_109,109+E_110,110+E_111,111+E_112,112
       +E_119,119+E_122,122+E_129,129+E_130,130+E_131,131+E_132,132+E_139,139+E_166,166+E_167,167
       +E_176,176+E_177,177+E_186,186+E_187,187+E_196,196+E_197,197+E_206,206+E_207,207+E_216,216
       +E_217,217+E_226,226+E_227,227+E_236,236+E_237,237+E_244,244+E_245,245+E_254,254+E_255,255
       +E_263,263+E_273,273+E_280,280+E_283,283+E_293,293+E_300,300+E_302,302+E_309,309+E_310,310
       +E_311,311+E_312,312+E_319,319+E_322,322+E_329,329+E_330,330+E_331,331+E_332,332+E_339,339
       +E_366,366+E_367,367+E_376,376+E_377,377+E_384,384+E_385,385+E_394,394+E_395,395)
    
    +   (q^-2)(E_23,23+E_33,33+E_40,40+E_42,42+E_49,49+E_50,50+E_51,51+E_52,52+E_59,59+E_106,106
       +E_107,107+E_116,116+E_117,117+E_126,126+E_127,127+E_136,136+E_137,137+E_163,163+E_173,173
       +E_180,180+E_183,183+E_193,193+E_200,200+E_203,203+E_213,213+E_220,220+E_223,223+E_233,233
       +E_240,240+E_242,242+E_249,249+E_250,250+E_251,251+E_252,252+E_259,259+E_306,306+E_307,307
       +E_316,316+E_317,317+E_326,326+E_327,327+E_336,336+E_337,337+E_363,363+E_373,373+E_380,380
       +E_382,382+E_389,389+E_390,390+E_391,391+E_392,392+E_399,399)
    
    +   (q^-3)(E_46,46+E_47,47+E_56,56+E_57,57+E_103,103+E_113,113+E_120,120+E_123,123+E_133,133
       +E_140,140+E_246,246+E_247,247+E_256,256+E_257,257+E_303,303+E_313,313+E_320,320
       +E_323,323+E_333,333+E_340,340+E_386,386+E_387,387+E_396,396+E_397,397)
    
    +   (q^-4)(E_43,43+E_53,53+E_60,60+E_243,243+E_253,253+E_260,260+E_383,383+E_393,393+E_400,400)



    π_W ⊗ W(q^H_2)    =
    (q^4)(E_43,43+E_47,47+E_52,52+E_123,123+E_127,127+E_132,132+E_223,223+E_227,227+E_232,232)
    
    +   (q^3)(E_23,23+E_27,27+E_32,32+E_42,42+E_45,45+E_53,53+E_57,57+E_83,83+E_87,87+E_92,92+E_122,122
       +E_125,125+E_133,133+E_137,137+E_222,222+E_225,225+E_233,233+E_237,237+E_243,243+E_247,247
       +E_252,252+E_323,323+E_327,327+E_332,332)
    
    +   (q^2)(E_3,3+E_7,7+E_12,12+E_22,22+E_25,25+E_33,33+E_37,37+E_41,41+E_46,46+E_49,49+E_51,51
       +E_55,55+E_60,60+E_82,82+E_85,85+E_93,93+E_97,97+E_103,103+E_107,107+E_112,112+E_121,121
       +E_126,126+E_129,129+E_131,131+E_135,135+E_140,140+E_163,163+E_167,167+E_172,172+E_203,203
       +E_207,207+E_212,212+E_221,221+E_226,226+E_229,229+E_231,231+E_235,235+E_240,240+E_242,242
       +E_245,245+E_253,253+E_257,257+E_283,283+E_287,287+E_292,292+E_322,322+E_325,325+E_333,333
       +E_337,337+E_383,383+E_387,387+E_392,392)
    
    +   (q)(E_2,2+E_5,5+E_13,13+E_17,17+E_21,21+E_26,26+E_29,29+E_31,31+E_35,35+E_40,40+E_44,44
       +E_48,48+E_56,56+E_59,59+E_63,63+E_67,67+E_72,72+E_81,81+E_86,86+E_89,89+E_91,91
       +E_95,95+E_100,100+E_102,102+E_105,105+E_113,113+E_117,117+E_124,124+E_128,128+E_136,136
       +E_139,139+E_143,143+E_147,147+E_152,152+E_162,162+E_165,165+E_173,173+E_177,177+E_202,202
       +E_205,205+E_213,213+E_217,217+E_224,224+E_228,228+E_236,236+E_239,239+E_241,241+E_246,246
       +E_249,249+E_251,251+E_255,255+E_260,260+E_282,282+E_285,285+E_293,293+E_297,297+E_303,303
       +E_307,307+E_312,312+E_321,321+E_326,326+E_329,329+E_331,331+E_335,335+E_340,340+E_363,363
       +E_367,367+E_372,372+E_382,382+E_385,385+E_393,393+E_397,397)
    
    +   (1)(E_1,1+E_6,6+E_9,9+E_11,11+E_15,15+E_20,20+E_24,24+E_28,28+E_36,36+E_39,39+E_50,50
       +E_54,54+E_58,58+E_62,62+E_65,65+E_73,73+E_77,77+E_84,84+E_88,88+E_96,96+E_99,99
       +E_101,101+E_106,106+E_109,109+E_111,111+E_115,115+E_120,120+E_130,130+E_134,134+E_138,138
       +E_142,142+E_145,145+E_153,153+E_157,157+E_161,161+E_166,166+E_169,169+E_171,171+E_175,175
       +E_180,180+E_183,183+E_187,187+E_192,192+E_201,201+E_206,206+E_209,209+E_211,211+E_215,215
       +E_220,220+E_230,230+E_234,234+E_238,238+E_244,244+E_248,248+E_256,256+E_259,259+E_263,263
       +E_267,267+E_272,272+E_281,281+E_286,286+E_289,289+E_291,291+E_295,295+E_300,300+E_302,302
       +E_305,305+E_313,313+E_317,317+E_324,324+E_328,328+E_336,336+E_339,339+E_343,343+E_347,347
       +E_352,352+E_362,362+E_365,365+E_373,373+E_377,377+E_381,381+E_386,386+E_389,389+E_391,391
       +E_395,395+E_400,400)
    
    +   (q^-1)(E_4,4+E_8,8+E_16,16+E_19,19+E_30,30+E_34,34+E_38,38+E_61,61+E_66,66+E_69,69+E_71,71
       +E_75,75+E_80,80+E_90,90+E_94,94+E_98,98+E_104,104+E_108,108+E_116,116+E_119,119+E_141,141
       +E_146,146+E_149,149+E_151,151+E_155,155+E_160,160+E_164,164+E_168,168+E_176,176+E_179,179
       +E_182,182+E_185,185+E_193,193+E_197,197+E_204,204+E_208,208+E_216,216+E_219,219+E_250,250
       +E_254,254+E_258,258+E_262,262+E_265,265+E_273,273+E_277,277+E_284,284+E_288,288+E_296,296
       +E_299,299+E_301,301+E_306,306+E_309,309+E_311,311+E_315,315+E_320,320+E_330,330+E_334,334
       +E_338,338+E_342,342+E_345,345+E_353,353+E_357,357+E_361,361+E_366,366+E_369,369+E_371,371
       +E_375,375+E_380,380+E_384,384+E_388,388+E_396,396+E_399,399)
    
    +   (q^-2)(E_10,10+E_14,14+E_18,18+E_64,64+E_68,68+E_76,76+E_79,79+E_110,110+E_114,114+E_118,118
       +E_144,144+E_148,148+E_156,156+E_159,159+E_170,170+E_174,174+E_178,178+E_181,181+E_186,186
       +E_189,189+E_191,191+E_195,195+E_200,200+E_210,210+E_214,214+E_218,218+E_261,261+E_266,266
       +E_269,269+E_271,271+E_275,275+E_280,280+E_290,290+E_294,294+E_298,298+E_304,304+E_308,308
       +E_316,316+E_319,319+E_341,341+E_346,346+E_349,349+E_351,351+E_355,355+E_360,360+E_364,364
       +E_368,368+E_376,376+E_379,379+E_390,390+E_394,394+E_398,398)
    
    +   (q^-3)(E_70,70+E_74,74+E_78,78+E_150,150+E_154,154+E_158,158+E_184,184+E_188,188+E_196,196
       +E_199,199+E_264,264+E_268,268+E_276,276+E_279,279+E_310,310+E_314,314+E_318,318+E_344,344
       +E_348,348+E_356,356+E_359,359+E_370,370+E_374,374+E_378,378)
    
    +   (q^-4)(E_190,190+E_194,194+E_198,198+E_270,270+E_274,274+E_278,278+E_350,350+E_354,354+E_358,358)



    π_W ⊗ W(q^H_3)    =
    (q^4)(E_43,43+E_46,46+E_50,50+E_103,103+E_106,106+E_110,110+E_183,183+E_186,186+E_190,190)
    
    +   (q^3)(E_23,23+E_26,26+E_30,30+E_42,42+E_44,44+E_53,53+E_56,56+E_63,63+E_66,66+E_70,70+E_102,102
       +E_104,104+E_113,113+E_116,116+E_182,182+E_184,184+E_193,193+E_196,196+E_243,243+E_246,246
       +E_250,250+E_303,303+E_306,306+E_310,310)
    
    +   (q^2)(E_3,3+E_6,6+E_10,10+E_22,22+E_24,24+E_33,33+E_36,36+E_41,41+E_47,47+E_49,49+E_51,51
       +E_54,54+E_60,60+E_62,62+E_64,64+E_73,73+E_76,76+E_101,101+E_107,107+E_109,109+E_111,111
       +E_114,114+E_120,120+E_123,123+E_126,126+E_130,130+E_163,163+E_166,166+E_170,170+E_181,181
       +E_187,187+E_189,189+E_191,191+E_194,194+E_200,200+E_203,203+E_206,206+E_210,210+E_242,242
       +E_244,244+E_253,253+E_256,256+E_263,263+E_266,266+E_270,270+E_302,302+E_304,304+E_313,313
       +E_316,316+E_383,383+E_386,386+E_390,390)
    
    +   (q)(E_2,2+E_4,4+E_13,13+E_16,16+E_21,21+E_27,27+E_29,29+E_31,31+E_34,34+E_40,40+E_45,45
       +E_48,48+E_57,57+E_59,59+E_61,61+E_67,67+E_69,69+E_71,71+E_74,74+E_80,80+E_83,83
       +E_86,86+E_90,90+E_105,105+E_108,108+E_117,117+E_119,119+E_122,122+E_124,124+E_133,133
       +E_136,136+E_143,143+E_146,146+E_150,150+E_162,162+E_164,164+E_173,173+E_176,176+E_185,185
       +E_188,188+E_197,197+E_199,199+E_202,202+E_204,204+E_213,213+E_216,216+E_241,241+E_247,247
       +E_249,249+E_251,251+E_254,254+E_260,260+E_262,262+E_264,264+E_273,273+E_276,276+E_301,301
       +E_307,307+E_309,309+E_311,311+E_314,314+E_320,320+E_323,323+E_326,326+E_330,330+E_363,363
       +E_366,366+E_370,370+E_382,382+E_384,384+E_393,393+E_396,396)
    
    +   (1)(E_1,1+E_7,7+E_9,9+E_11,11+E_14,14+E_20,20+E_25,25+E_28,28+E_37,37+E_39,39+E_52,52
       +E_55,55+E_58,58+E_65,65+E_68,68+E_77,77+E_79,79+E_82,82+E_84,84+E_93,93+E_96,96
       +E_112,112+E_115,115+E_118,118+E_121,121+E_127,127+E_129,129+E_131,131+E_134,134+E_140,140
       +E_142,142+E_144,144+E_153,153+E_156,156+E_161,161+E_167,167+E_169,169+E_171,171+E_174,174
       +E_180,180+E_192,192+E_195,195+E_198,198+E_201,201+E_207,207+E_209,209+E_211,211+E_214,214
       +E_220,220+E_223,223+E_226,226+E_230,230+E_245,245+E_248,248+E_257,257+E_259,259+E_261,261
       +E_267,267+E_269,269+E_271,271+E_274,274+E_280,280+E_283,283+E_286,286+E_290,290+E_305,305
       +E_308,308+E_317,317+E_319,319+E_322,322+E_324,324+E_333,333+E_336,336+E_343,343+E_346,346
       +E_350,350+E_362,362+E_364,364+E_373,373+E_376,376+E_381,381+E_387,387+E_389,389+E_391,391
       +E_394,394+E_400,400)
    
    +   (q^-1)(E_5,5+E_8,8+E_17,17+E_19,19+E_32,32+E_35,35+E_38,38+E_72,72+E_75,75+E_78,78+E_81,81
       +E_87,87+E_89,89+E_91,91+E_94,94+E_100,100+E_125,125+E_128,128+E_137,137+E_139,139+E_141,141
       +E_147,147+E_149,149+E_151,151+E_154,154+E_160,160+E_165,165+E_168,168+E_177,177+E_179,179
       +E_205,205+E_208,208+E_217,217+E_219,219+E_222,222+E_224,224+E_233,233+E_236,236+E_252,252
       +E_255,255+E_258,258+E_265,265+E_268,268+E_277,277+E_279,279+E_282,282+E_284,284+E_293,293
       +E_296,296+E_312,312+E_315,315+E_318,318+E_321,321+E_327,327+E_329,329+E_331,331+E_334,334
       +E_340,340+E_342,342+E_344,344+E_353,353+E_356,356+E_361,361+E_367,367+E_369,369+E_371,371
       +E_374,374+E_380,380+E_385,385+E_388,388+E_397,397+E_399,399)
    
    +   (q^-2)(E_12,12+E_15,15+E_18,18+E_85,85+E_88,88+E_97,97+E_99,99+E_132,132+E_135,135+E_138,138
       +E_145,145+E_148,148+E_157,157+E_159,159+E_172,172+E_175,175+E_178,178+E_212,212+E_215,215
       +E_218,218+E_221,221+E_227,227+E_229,229+E_231,231+E_234,234+E_240,240+E_272,272+E_275,275
       +E_278,278+E_281,281+E_287,287+E_289,289+E_291,291+E_294,294+E_300,300+E_325,325+E_328,328
       +E_337,337+E_339,339+E_341,341+E_347,347+E_349,349+E_351,351+E_354,354+E_360,360+E_365,365
       +E_368,368+E_377,377+E_379,379+E_392,392+E_395,395+E_398,398)
    
    +   (q^-3)(E_92,92+E_95,95+E_98,98+E_152,152+E_155,155+E_158,158+E_225,225+E_228,228+E_237,237
       +E_239,239+E_285,285+E_288,288+E_297,297+E_299,299+E_332,332+E_335,335+E_338,338+E_345,345
       +E_348,348+E_357,357+E_359,359+E_372,372+E_375,375+E_378,378)
    
    +   (q^-4)(E_232,232+E_235,235+E_238,238+E_292,292+E_295,295+E_298,298+E_352,352+E_355,355+E_358,358)




 §.§ Listing 4x4 Blocks of pi(W tensor W)(C)


The twenty-four 4 × 4 blocks of π_W ⊗ W(C) take the following forms:


Ten blocks take the form

    [  B_1  B_2  B_3    0;  B_4  B_5  B_3  B_6;  B_7  B_3  B_8  B_9;    0 B_10 B_11 B_12 ]

Four blocks take the form

    [  B_1  B_7  B_6    0; B_13  B_5  B_3  B_3;  B_2  B_3  B_8 B_10;    0  B_9 B_14 B_12 ]

Four blocks take the form

    [ B_15  B_7  B_7  B_3;  B_3 B_16    0 B_10;  B_3    0 B_16 B_10;  B_3  B_6  B_6 B_17 ]

Four blocks take the form

    [ B_15  B_2  B_2  B_3;  B_6 B_16    0  B_9;  B_6    0 B_16  B_9;  B_3  B_3  B_3 B_17 ]

And two blocks take the form

    [  B_1 B_18 B_19    0; B_20  B_5  B_3 B_21; B_22  B_3  B_8 B_23;    0 B_24 B_25 B_12 ]

Where

    B_1    = q^12-q^10+2q^6+2+q^-2-q^-4+2q^-8
    
        B_2    = q^10-2q^8+q^6+q^-4-2q^-6+q^-8
    
        B_3    = q^9-2q^7+q^5+q^-5-2q^-7+q^-9
    
        B_4    = q^12-q^10-q^8+q^6+q^-2-q^-4-q^-6+q^-8
    
        B_5    = 2q^10-2q^8+q^6+q^4+2+2q^-4-2q^-6+q^-8+q^-10
    
        B_6    = q^10-q^8-q^6+q^4+q^-4-q^-6-q^-8+q^-10
    
        B_7    = q^11-q^9-q^7+q^5+q^-3-q^-5-q^-7+q^-9
    
        B_8    = q^10+q^8-2q^6+2q^4+2+q^-4+q^-6-2q^-8+2q^-10
    
        B_9    = q^9-q^7-q^5+q^3+q^-5-q^-7-q^-9+q^-11
    
        B_10   = q^8-2q^6+q^4+q^-6-2q^-8+q^-10
    
        B_11   = q^7-2q^5+q^3+q^-7-2q^-9+q^-11
    
        B_12   = 2q^8-q^4+q^2+2+2q^-6-q^-10+q^-12
    
        B_13   = q^11-2q^9+q^7+q^-3-2q^-5+q^-7
    
        B_14   = q^8-q^6-q^4+q^2+q^-6-q^-8-q^-10+q^-12
    
        B_15   = q^12-2q^8+3q^6+2+q^-2-2q^-6+3q^-8
    
        B_16   = q^10+q^4+2+q^-4+q^-10
    
        B_17   = 3q^8-2q^6+q^2+2+3q^-6-2q^-8+q^-12
    
        B_18   = q^9-3q^7+4q^5-4q^3+4q-4q^-4/q^--3q^-5+q^-7
    
        B_19   = q^8-3q^6+4q^4-4q^2+4-4q^-2+4q^-4-3q^-6+q^-8
    
        B_20   = q^13-2q^9+q^5+q^-2/q^-+q^-9
    
        B_21   = q^11-2q^7+q^3+q^-3-2q^-7+q^-11
    
        B_22   = q^12-2q^8+q^4+q^-2-2q^-6+q^-10
    
        B_23   = q^10-2q^6+q^2+q^-4-2q^-8+q^-12
    
        B_24   = q^7-3q^5+4q^3-4q+4q^-4/q^-+4q^-5-3q^-7+q^-9
    
        B_25   = q^6-3q^4+4q^2-4+4q^-2-4q^-4+4q^-6-3q^-8+q^-10










