Quivers for SL(2) tilting modules
Daniel Tubbenhauer Paul Wedrich
math.RT math.QA

Quivers for SL2 tilting modules]
Quivers for SL2 tilting modules
Daniel Tubbenhauer and Paul Wedrich]Daniel Tubbenhauer and Paul Wedrich

D.T.: Institut für Mathematik, Universität Zürich, 
Winterthurerstrasse 190, Campus Irchel, Office Y27J32, CH-8057 Zürich, 
Switzerland, www.dtubbenhauer.comwww.dtubbenhauer.com
daniel.tubbenhauer@math.uzh.ch

P.W.: Mathematical Sciences Institute, The Australian National University, 
Hanna Neumann Building,
Canberra ACT 2601, Australia, http://paul.wedrich.atpaul.wedrich.at
p.wedrich@gmail.com

Mathematics Subject Classification 2010. Primary: 20G05, 20C20; Secondary: 16D90, 17B10, 20G40.
Keywords. Modular representation theory, tilting modules, 
diagrammatic algebra, generators and relations, Temperley–Lieb, positive characteristic.


Using diagrammatic methods, we define a quiver with relations depending on a prime 𝗉 and 
show that the associated path algebra describes 
the category of tilting modules for 
SL_2 in characteristic 𝗉.
Along the way we obtain a presentation for 
morphisms between 𝗉-Jones–Wenzl projectors.


[
    [
    December 30, 2023
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toc






§ INTRODUCTION


Let  denote an algebraically closed field and =() the additive, 
-linear category of (left-)tilting modules for the algebraic group . 
This category can be described as the full subcategory of -modules 
which is monoidally generated by the vector representation (1)≅^2, and which is closed under
taking finite direct sums and direct summands.

The purpose of this paper is to give a generators and relations
presentation of  by identifying it with the category of projective
modules for the path algebra of an explicitly described quiver with relations.
This quiver can be interpreted as the semi-infinite Ringel dual of
 in the sense of <cit.>. For  of characteristic zero this is
trivial as  is semisimple, and the indecomposable tilting modules are
indeed the simple modules. The quantum analog at a complex root of unity is
related to the zigzag algebra with vertex set [0] and a starting condition, see
<cit.>. 

The focus of this paper is on the case of positive characteristic , for which we represent
 as a quotient =_ of the path algebra of an infinite, fractal-like quiver, 
a truncation of which is illustrated for =3 in figure:main-3.





 §.§ The main result


From now on let  be an algebraically closed field of characteristic >0, and 
let  be the corresponding special linear group.
Recall that the indecomposable tilting modules for  are classified (up to isomorphism) by their highest weight 
v-1∈[0], and we pick a collection of representatives denoted by (v-1).


There is an algebra isomorphism

    ⊕_v,w∈_((v-1),(w-1)),

which sends the constant path on the vertex v-1 to the idempotent for the summand (v-1).


Let  denote the
category of finitely-generated, projective (right-)modules for . By semi-infinite Ringel duality <cit.>, we have the following consequence of theorem:main.


There is an equivalence of additive, -linear categories

    ^',

sending indecomposable tilting modules to indecomposable projectives.


Several classical facts about -modules are reflected in the
presentation of the algebra . For example, a path from v-1 to w-1
can only be non-zero in  if (v-1) and (w-1) share a
common Weyl factor.
More specifically, if the -adic expansion
v=a_j,…,a_0=∑_i=0^ja_i^i has exactly
r+1 non-zero digits, then (v-1) has exactly 2^r Weyl factors
(w-1) where w is obtained by negating some of the non-zero digits a_i
for i<j. In this case,  contains r arrows from v-1 to those
w-1<v-1 that are obtained by negating a single digit.

Our assignment of morphisms to arrows uses the Temperley–Lieb
category. In contrast to other descriptions of morphisms between indecomposable
tilting modules for , this presentation of  is well-adapted to
study  as a monoidal category.

The Weyl factors in
indecomposable tilting modules are illustrated in the lines v, ,
 in figure:main-3, where the colors distinguish arrows in
different blocks, the connected components of the quiver, and each reddish
number corresponds to the unique simple tilting module in its block.



 §.§ The algebra Z in a nutshell


We define the algebra  as a quotient of 
the path algebra of an infinite, fractal-like quiver over the prime field ⊂. (In particular, we can always extend the algebra  to an algebra over .)
We will use this introduction to sketch the main features of  
and relegate the precise statement to theorem:main-tl-section.






  * The underlying quiver. 
We identify the vertex set with [0] and the constant path at the vertex v-1 
will be denoted [v-1] (it corresponds to (v-1)). 
If v=a_j,…,a_0, then for every 
digit a_i≠ 0 
with i≠ j there is a pair of arrows

    [i][v-1]
    (v-1)→(v[i]-1),
        [i][v[i]-1]
    (v[i]-1)→ (v-1),

where v[i]=a_j,…,a_i+1,-a_i,a_i-1,…,a_0
=v-2a_i^i.


  * Some relations.
Up to some additional rules in special cases
(which we ignore for the sake of this introduction),
there are five types of relations among paths, 
which hold whenever both sides are defined and satisfy certain admissibility conditions.





  * Idempotents.
[v-1][w-1]=δ_v,w[v-1],
[w-1]F[v-1]=F[v-1]
and [w-1]F[v-1]=[w-1]F, 
where F is a word in the generators starting at v-1 and ending at w-1.
(Throughout, we use such relations to absorb all but one idempotent in each string of generators.)


  * Nilpotency. [i]^2[v-1]=[i]^2[v-1]=0.


  * Far-commutativity. 
[i][j][v-1]=[j][i][v-1], 
[i][j][v-1]=[j][i][v-1], as well as 
[i][j][v-1]=[j][i][v-1] whenever |i-j|>1.


  * Adjacency relations.
[i+1][i][v-1]=[i][i+1][v-1] and 
[i][i+1][v-1]=[i+1][i][v-1], 
and scaled versions [i+1][i][v-1]=^'[i][i+1][v-1] 
and [i][i+1][v-1]=^''[i+1][i][v-1].


  * Zigzag. 
[i][i][v-1]=[i][i][v-1] 
+[i+1][i][i][i+1][v-1].



Here , ^', ^'' and  are scalars 
that depend on  and the digit a_i+1.


  * Hom spaces. 
For v,w∈ the -vector space  
[w-1][v-1] is spanned by paths of the 
form [w-1][j_k]⋯[j_1][i_1]⋯[i_l][v-1] with j_k>⋯>j_1,i_1<⋯<i_l,
paths that descend before ascending again. In particular, we have [v-1][v-1]≅ 
whenever v=a^j for 1≤ a<, 
which reflects the fact that the corresponding tilting module (v-1) is simple.


  * Endomorphism algebras. 
Let v>0 have r+1 non-zero digits with indices i_r+1>⋯>i_1. Then we have
the following identifications of -algebras

    [v-1][v-1]
    ≅[][[i_r][i_r],…,[i_1][i_1]]
    /
    ⟨([i_r][i_r])^2,…,
    ([i_1][i_1])^2⟩.
 
This leads to a description of the endomorphism algebra of (v-1) 
which could have been expected from Donkin's tensor product 
theorem <cit.>.

 

We would like to highlight that we will meet a law of small primes () repeatedly. 
By this we mean the appearance of exceptional relations in cases of -adic expansions with digits 
0, 1, -2, or -1. 
These relations are exceptional in the sense that they contrast with the relations shown above, which describe 
the behavior of generic -adic expansions for large primes . 
Nevertheless, exceptional relations are relevant for all primes, and for =2 only exceptional relations apply.



 §.§ A word about the proof of theorem:main


The basis for our work is the classical fact that the Temperley–Lieb algebra controls the 
finite-dimensional representation theory of . 
The second main ingredient is an explicit description 
of -Jones–Wenzl projectors <cit.>, 
which are characteristic  analogs of the classical Jones–Wenzl projectors, 
that diagrammatically encode the projections 

    (1)^⊗(v-1)→(v-1)→(1)^⊗(v-1).


The bulk of this paper is devoted to a careful study 
of morphisms between -Jones–Wenzl projectors over  
and the linear relations between them. 
This work was supported by extensive computer experimentation using 
Mathematica and SageMATH.



 §.§ Relations to other work


To the best of our knowledge, the quiver underlying the tilting category is new:
We study  as a finitely presented category. So our main concern are the 
relations among composites of generating morphisms, 
rather than just the combinatorics of objects or the 
dimensions of morphism spaces, which appear in the classical literature. 

We would like to acknowledge and reinforce that the  representation theory is, 
of course, well-understood on the level of the modules, 
see <cit.>, 
<cit.>, <cit.>,
<cit.>, <cit.> and <cit.>. 
Further, various other quivers associated to  are known, describing rational modules <cit.> or the
extension algebra for simple <cit.> or Weyl modules 
<cit.>.



 §.§ A graded extension and translation functors


It is possible to give a similar quiver description of  as a positively graded module category of 
the diagrammatic Soergel category  for the Weyl group of type , acting by translation functors. 
The first step in such an extension uses the quantum Satake equivalence (at q=1) <cit.> to
connect the Temperley–Lieb diagrammatic calculus to . 
In fact,  faithfully describes the degree zero part of the antispherical module category for .
The second step uses ideas from <cit.> to relate  and the principal block _0⊂
as long as >2. Along this route,  also inherits a grading from .

In this case, the algebra  essentially describes the degree zero part 
of the principal block _0, 
while the positive degree part is generated by additional degree
1 arrows v→ v+1 and v+1→ v, 
which interact non-trivially with other paths. 
Note another fractal-like structure:  describes , but also the 
degree zero part of _0⊂.
We will not pursue this extension in the present paper.



 §.§ Characteristic zero and higher rank cases


Throughout we could allow the case of characteristic zero, for which  is semisimple. 
In a more interesting variant, one replaces  by its quantum group analog at a complex root of unity, 
using the Jones–Wenzl projectors from <cit.>. The role of  
is then played by the zigzag algebra with vertex set [0] 
and a starting condition, and we would recover a result of <cit.>. 
In this sense we think of  as a positive characteristic version of the zigzag algebra.

We also like to highlight that, to the best of our knowledge, a quiver 
underlying tilting modules for higher rank groups is still unknown, even 
for the quantum group analog in characteristic zero, <cit.> 
for some first steps in this direction. 

We expect the diagrammatic methods used in this paper to 
generalize to SL_N() and GL_N(). This would involve developing characteristic  analogs 
of so-called clasps, living in the corresponding web calculus, 
see <cit.> or <cit.>, 
defined over .



Acknowledgments. D.T. likes to thank Henning Haahr Andersen,
Ben Elias, Hankyung Ko, Catharina Stroppel and Geordie Williamson for teaching
him everything he knows about tilting modules (which is basically nothing, but
that is his fault alone). P.W. would like to thank James Borger, Ben Elias,
Alexandra Grant, Anthony Licata and Scott Morrison for useful discussions.
Special thanks to Henning Haahr Andersen, Jieru Zhu and an anonymous referee for valuable
comments on a draft of this paper, to Nicolas Libedinsky for sharing a draft of
the lovely paper <cit.>, which started this project,
and Noah Snyder for pointing out a missing argument in the proof of
proposition:TLtilt.

Parts of this research were conducted while D.T. visited the Australian National University, 
supported by the Mathematical Sciences Research Visitor Program (MSRVP), 
and while P.W. visited Universität Zürich. Their hospitality and support
are gratefully acknowledged. P.W. was supported by Australian Research Council grants `Braid groups and higher representation theory'
DP140103821 and `Low dimensional categories' DP160103479



§ THE TEMPERLEY–LIEB CALCULUS
 

Let C=(C,,_C,) be a pivotal 
category with (strict) monoidal composition , unit _C, 
and duality . 
We usually write FG:=FG for the composition of morphisms.
We read string diagrams for morphisms in C from bottom to top and left to right, 
    (G)(F) 
    =
    [anchorbase,scale=.22,tinynodes]
    [thin,black,fill=mygray,fill opacity=.35] (-5.5,2) rectangle (-2,.5);
    [thin,black,fill=mygray,fill opacity=.35] (-.5,3) rectangle (3,4.5);
    
    [thick,densely dotted] (-5.5,2.5) node[left,yshift=-2pt] 
    to (3.5,2.5) node[right,yshift=-2pt];
    [thick,densely dotted] (-1.25,5) 
    node[above,yshift=-2pt] to (-1.25,0) node[below];
    
    [usual] (-5,2) to (-5,3.75) node[right,xshift=1pt]… to (-5,5);
    [usual] (-2.5,2) to (-2.5,5);
    [usual] (0,0) to (0,1.25) node[right,xshift=1pt]… to (0,3);
    [usual] (2.5,0) to (2.5,3);
    
    [usual] (-5,0) to (-5,.25) node[right,xshift=1pt]… to (-5,.5);
    [usual] (-2.5,0) to (-2.5,.5);
    [usual] (2.5,4.5) to (2.5,5);
    [usual] (0,4.5) to (0,4.75) node[right,xshift=1pt]… to (0,5);
    
     at (-3.75,1.0) F;
     at (1.25,3.5) G;
    
    =
    [anchorbase,scale=.22,tinynodes]
    [thin,black,fill=mygray,fill opacity=.35] (-5.5,1.75) rectangle (-2,3.25);
    [thin,black,fill=mygray,fill opacity=.35] (-.5,1.75) rectangle (3,3.25);
    
    [usual] (-5,3.25) to (-5,4.125) node[right,xshift=1pt]… to (-5,5);
    [usual] (-2.5,3.25) to (-2.5,5);
    [usual] (0,0) to (0,.625) node[right,xshift=1pt]… to (0,1.75);
    [usual] (2.5,0) to (2.5,1.75);
    
    [usual] (-5,0) to (-5,.625) node[right,xshift=1pt]… to (-5,1.75);
    [usual] (-2.5,0) to (-2.5,1.75);
    [usual] (2.5,3.25) to (2.5,5);
    [usual] (0,3.25) to (0,4.125) node[right,xshift=1pt]… to (0,5);
    
     at (-3.75,2.25) F;
     at (1.25,2.25) G;
    
    =
    [anchorbase,scale=.22,tinynodes]
    [thin,black,fill=mygray,fill opacity=.35] (5.5,2) rectangle (2,.5);
    [thin,black,fill=mygray,fill opacity=.35] (.5,3) rectangle (-3,4.5);
    
    [thick, densely dotted] (5.5,2.5) 
    node[right,yshift=-2pt] to (-3.5,2.5) node[left,yshift=-2pt];
    [thick, densely dotted] (1.25,5) 
    node[above,yshift=-2pt] to (1.25,0) node[below];
    
    [usual] (2.5,2) to (2.5,3.75) node[right,xshift=1pt]… to (2.5,5);
    [usual] (5,2) to (5,5);
    [usual] (0,0) to (0,3);
    [usual] (-2.5,0) to (-2.5,1.25) node[right,xshift=1pt]… to (-2.5,3);
    
    [usual] (2.5,0) to (2.5,.25) node[right,xshift=1pt]… to (2.5,.5);
    [usual] (5,0) to (5,.5);
    [usual] (-2.5,4.5) to (-2.5,4.75) node[right,xshift=1pt]… to (-2.5,5);
    [usual] (0,4.5) to (0,5);
    
     at (3.75,1.0) G;
     at (-1.25,3.5) F;
    
    =(F)(G).
 
The duality maps are pictured as cup and cap string diagrams, subject to the expected string-straightening relations. 
The pivotal structure additionally allows the rotation of string diagrams 
and guarantees that planar-isotopic diagrams represent the same morphism.

Let  be any commutative and unital ring.
(For us  will usually be  or ⊂, 
the prime field of . However, it also makes sense to 
formulate everything for _ and _.)

The category [] (see <cit.>) can be described as the pivotal
-linear category with objects indexed by m∈[0], and with morphisms
from m to n being -linear combinations of unoriented string diagrams
drawn in a horizontal strip ×[0,1] between m marked points on the
lower boundary ×{0} and n marked points on the upper boundary
×{1}, considered up to planar isotopy relative to the boundary and
the relation that a circle evaluates to -2. The composition and tensor product
operations are as described above.

Particular cases of the isotopy and circle relations are

    [anchorbase,scale=.25,tinynodes]
    [usual] (0,-2) to (0,0) to[out=90,in=180] (1,1) to[out=0,in=90] (2,0);
    [usual] (2,0) to[out=270,in=180] (3,-1) to[out=0,in=270] (4,0) to (4,2);
    
    =
    [anchorbase,scale=.25,tinynodes]
    [usual] (0,-2) to (0,2);
     ,  [anchorbase,scale=.25,tinynodes]
    [usual] (0,-2) to (0,0) to[out=90,in=0] (-1,1) to[out=180,in=90] (-2,0);
    [usual] (-2,0) to[out=270,in=0] (-3,-1) to[out=180,in=270] (-4,0) to (-4,2);
    
    =
    [anchorbase,scale=.25,tinynodes]
    [usual] (0,-2) to (0,2);
     ,  [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) to[out=90,in=180] (1,1) to[out=0,in=90] (2,0);
    [usual] (0,0) to[out=270,in=180] (1,-1) to[out=0,in=270] (2,0);
    
    =-2.


In the following we will use 
labeled strands as shorthand notation for bundles of parallel strands:

    [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) to (0,1.5) node[right,xshift=-2pt]m to (0,3);
    
     at (1,-1) a;
     at (1,4) a;
    
    :=
    [m]
    =
    [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) to (0,3);
    [usual] (2,0) to (2,3);
    
     at (1,1.5) ...;
    
     at (1,-1) a;
     at (1,4) m strands;
    
    ,  [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) to[out=90,in=180] (1,1) node[above,yshift=-2pt]m to[out=0,in=90] (2,0);
    
     at (1,-1) a;
     at (1,4) a;
    
    :=
    [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) to[out=90,in=180] (1,1) to[out=0,in=90] (2,0);
    [usual] (-2,0) to[out=90,in=180] (1,3) to[out=0,in=90] (4,0);
    
     at (-1,.5) ...;
     at (3,.5) ...;
    
     at (1,-1) a;
     at (1,4) m caps;
    
    ,  [anchorbase,scale=.25,tinynodes]
    [usual] (0,3) to[out=270,in=180] (1,2) node[below]m to[out=0,in=270] (2,3);
    
     at (1,-1) a;
     at (1,4) a;
    
    :=
    [anchorbase,scale=.25,tinynodes]
    [usual] (0,3) to[out=270,in=180] (1,2) to[out=0,in=270] (2,3);
    [usual] (-2,3) to[out=270,in=180] (1,0) to[out=0,in=270] (4,3);
    
     at (-1,2.5) ...;
     at (3,2.5) ...;
    
     at (1,-1) a;
     at (1,4) m cups;
    
    .

We even omit these numbers or the lines altogether if no confusion can arise.

The category [] furthermore admits a contravariant, -linear 
involution which reflects string diagrams in a horizontal line. 
Several arguments in the following will use this up-down symmetry. However, we will usually not have a left-right symmetry.

Recall that
_[](m,n) is a free -module 
with a basis  given by crossingless matchings.
The through-degree _i of _i∈ is the number of strands connecting the bottom to the top. 
More generally, the through-degree of a general morphism 
F=∑__i∈_i_i 
is defined via F:=max{_i|_i≠ 0}.
Note that FG≤min(F,G), and thus, 
td_i(m,n):={F∈_[](m,n)|F≤ i} form 
a sequence of nested (-)ideals in [].

Instead of m, the number of strands, let us now use v=m+1∈, which will be crucial number for everything that follows.


For v∈ the JW projectors ∈_[](v-1,v-1) are the morphisms, which are
recursively defined by

    [0]
    :=∅  ,
      [1]:=
    [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) to (0,2);
     
    ,  
    :=
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) ;
    
    :=
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) [v-2];
    
    [usual] (-2,0) to (-2,2);
    
    +
    v-2v-1·[anchorbase,scale=.25,tinynodes]
    [JW] (-2.5,1) rectangle (2.5,3);
     at (0,1.9) [v-2];
    
    [JW] (-.5,-1) rectangle (2.5,1);
     at (1,-.1) [v-3];
    
    [usual] (-1.5,1) to[out=270, in=0] (-2.25,.25) to[out=180, in=270] (-3,1) to (-3,3);
    [usual] (-1.5,-1) to[out=90, in=0] (-2.25,-.25) to[out=180, in=90] (-3,-1) to (-3,-3);
    
    [JW] (-2.5,-1) rectangle (2.5,-3);
     at (0,-2.1) [v-2];
       if v>2,

where we use a box with v-1 
bottom and top strands to indicate .


(See <cit.>.) We have
[()]= and =v-1. Furthermore, the following
properties hold, which are best expressed diagrammatically.

.33

    [anchorbase,scale=.25,tinynodes]
    [JW] (-.6,0) rectangle (1.6,2);
    [usual] (-1,0) to (-1,2);
    [usual] (2,0) to (2,2);
     at (.5,.9) [w-1];
    
    [JW] (-1.5,-2) rectangle (2.5,0);
     at (0.5,-1.1) ;
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) ;
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-.6,0) rectangle (1.6,-2);
    [usual] (-1,0) to (-1,-2);
    [usual] (2,0) to (2,-2);
     at (.5,-1.1) [w-1];
    
    [JW] (-1.5,0) rectangle (2.5,2);
     at (0.5,0.9) ;
    
     at (0,2.5) a;
     at (0,-2.5) a;
    ,


.27

    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.2,-1) rectangle (1.2,1);
     at (0,-.1) ;
    
    [usual] (-.5,1) to[out=90,in=180] (0,1.5) node[above,yshift=-2pt]k to[out=0,in=90] (.5,1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =
    0
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.2,1) rectangle (1.2,-1);
     at (0,-.1) ;
    
    [usual] (-.5,-1) to[out=270,in=180] (0,-1.5) node[below,yshift=-1pt]k to[out=0,in=270] (.5,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    ,


.33

    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,-1) rectangle (1,1);
     at (-.25,-.1) ;
    
    [usual] (-1,1) to[out=90,in=0] (-1.5,1.5) to[out=180,in=90] 
    (-2,1) to (-2,0) node[left,xshift=2pt]k to (-2,-1) 
    to[out=270,in=180] (-1.5,-1.5) to[out=0,in=270] (-1,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =(-1)^kvv-k·[anchorbase,scale=.25,tinynodes]
    [JW] (-1.8,-1) rectangle (1.8,1);
     at (0,-.1) [v-1-k];
    
     at (0,2.5) a;
     at (0,-2.5) a;
    .



Here 1≤ w≤ v and the projector of thickness w-1 in
(<ref>) can be at any place. Similarly, the cap or cup in
(<ref>) can be at any place and of any thickness.




 §.§ Characteristic p notions
 

As already suggested by the recursion (<ref>), the 
JW projectors have rational coefficients with respect to  and 
typically cannot be defined in []. To formalize this,
consider the -adic valuation →∪{∞}, 
defined for n∈ as (n)=max{m∈[0]|^m|n} 
(including (0)=∞) and for c=rs∈ as (c):=(r)-(s).


For a non-zero F=∑__i∈_i_i∈[] we let 
(F):=min_i{(_i)}.
We call such a morphism -admissible if
(F)≥ 0.


To highlight morphisms that might not be -admissible, 
we use .̃ as in (<ref>).
Note that F=∑__i∈_i_i∈[] 
is -admissible if and only if every coefficient
_i can be presented as a reduced 
fraction rs with |̸s. 
In this case, F represents 
an element F of [], which is zero if and only if (F)>0. 
If we write F=F_0+F_>0 
with (F_0)=0 and (F_>0)>0, then F=F_0.


We have ([v-1])=0 for v≤, which corresponds to the fact that 
the characteristic zero Weyl module (v-1)=(v-1) stays simple when reduced modulo . 
However, for v>, one typically has ([v-1])<0, 
and in such cases the projectors [v-1] cannot be defined in []. 


However, there are alternative idempotents [v-1]∈[] satisfying 
([v-1])≥ 0 and we will consider their specializations 
[v-1]:=[v-1]∈[]. To this end, recall that 
we write v=a_j,…,a_0=∑_ia_ip^i 
for the -adic expansion of v∈
with digits 0≤ a_i< and a_j≠ 0. (More generally, we also write
b_j,…,b_0:=∑_ib_ip^i for any b_i∈.) 


If v=a_j,…,a_0∈ has only 
a single non-zero digit, then v is called an eve. The set of eves is 
denoted by . 
If v∉, then the mother  of v is 
obtained by setting the rightmost non-zero digit 
of v to zero. 
We will also consider the set 
:={,v2:=[],…} 
of (matrilineal) ancestors of v, 
whose size  is called the generation of v.


Note that =∅ if and only if v∈, and for 
v∉ we write (v) for its eve.

 
For v=a_j,…,a_0, the 
support ⊂ is the set of the 2^ integers of the form
w=a_j,± a_j-1,…,± a_0. The 
integers v[i]=a_j,…,a_i+1,-a_i,a_i-1,…,a_0 
for a_i≠ 0 form the fundamental support ⊂ of v.



Let =3. Then v=23=2,1,23 has [23]=2, 
and [23]=21=2,1,03 and 232=(23)=18=2,0,03. Hence, the ancestry chart of 23 is

    [23]=
    [anchorbase,scale=.25,tinynodes]
     at (10,0) 18∈;
     at (10,2.5) [21]=1;
     at (10,5) [23]=2;
    
    [thin,directed=1] (-4,0) node[left,xshift=.2pt]232 to (-1,0);
    [thin,directed=1] (-4,4) node[left,xshift=.2pt][23] to (-1,4) to (.25,3.3);
    
    [ultra thick,mygray,densely dotted] (0,0) to (-3,2.5);
    [ultra thick,mygray,densely dotted] (0,0) to (-1,2.5);
    [ultra thick,spinach] (0,0) to (1,2.5);
    [ultra thick,mygray,densely dotted] (0,0) to (3,2.5);
    [ultra thick,mygray,densely dotted] (1,2.5) to (0,5);
    [ultra thick,spinach] (1,2.5) to (2,5);
    [ultra thick,mygray,densely dotted] (3,2.5) to (4,5);
    [ultra thick,mygray,densely dotted] (3,2.5) to (6,5);
    
    [spinach,fill=white] (0,.2) circle (.8cm);
    [mygray,fill=mygray!25] (-3,2.7) circle (.8cm);
    [mygray,fill=mygray!25] (-1,2.7) circle (.8cm);
    [spinach,fill=white] (1,2.7) circle (.8cm);
    [mygray,fill=mygray!25] (3,2.7) circle (.8cm);
    [mygray,fill=mygray!25] (0,5.2) circle (.8cm);
    [spinach,fill=white] (2,5.2) circle (.8cm);
    [mygray,fill=mygray!25] (4,5.2) circle (.8cm);
    [mygray,fill=mygray!25] (6,5.2) circle (.8cm);
    
    [tomato] at (0,0) 18;
    [mygray] at (-3,2.5) 19;
    [mygray] at (-1,2.5) 20;
    [spinach] at (1,2.5) 21;
    [mygray] at (3,2.5) 24;
    [mygray] at (0,5) 22;
    [spinach] at (2,5) 23;
    [mygray] at (4,5) 25;
    [mygray] at (6,5) 26;
     .

Moreover, [23]={23=2,1,23,19=
2,1,-23,17=2,-1,23,13=2,-1,-23} and 
[23]={19,17}.
In pictures,

    [anchorbase,scale=.25,tinynodes]
    [thick,spinach] (2,0) to [out=45,in=135] (10,0);
    [thick,spinach] (10,0) to [out=45,in=180] (12,1.15) to [out=0,in=135] (14,0);
    [thick,spinach] (14,0) to [out=45,in=135] (22,0);
    [thick,spinach] (22,0) to [out=45,in=180] (24,1.15) to [out=0,in=135] (26,0);
    [thick,spinach] (26,0) to [out=45,in=135] (34,0);
    [thick,spinach] (34,0) to [out=45,in=180] (36,1.15) to [out=0,in=135] (38,0);
    [thick,spinach] (38,0) to [out=45,in=135] (46,0);
    [thick,myorange,densely dashed] (26,0) to [out=45,in=135] (46,0);
    [thick,myorange,densely dashed] (2,0) to [out=45,in=135] (34,0);
    [thick,myorange,densely dashed] (10,0) to [out=45,in=135] (26,0);
    
    [thick,spinach] (34,0) to [out=45,in=135] (46,0);
    [thick,spinach] (10,0) to [out=45,in=135] (34,0);
    [thick,spinach] (14,0) to [out=45,in=135] (26,0);
    
    [thin,myorange,odirected=1] (36,-2.25) to (26,-2.25) to (26,-.75);
    
    [thick,myorange,odirected=1] (40,-2.75) node[below][23] to 
    (40,-2.25) to (46,-2.25) to (46,-.75);
    [thick,myorange,odirected=1] (38,-2.25) to (38,-1);
    [thick,myorange,odirected=1] (40,-2.25) to (34,-2.25) to (34,-1);
    [thin] (34,-2.75) node[below][23] to (34,-2.25);
    [thin] (34,-2.25) to (38,-2.25);
    [thin,directed=1] (38,-2.25) to (38,-.75);
    [thin,directed=1] (34,-2.25) to (34,-.75);
    
    [spinach,fill=white] (2,.2) circle (.8cm);
    [mygray,fill=mygray!25] (4,.2) circle (.8cm);
    [mygray,fill=mygray!25] (6,.2) circle (.8cm);
    [mygray,fill=mygray!25] (8,.2) circle (.8cm);
    [spinach,fill=white] (10,.2) circle (.8cm);
    [mygray,fill=mygray!25] (12,.2) circle (.8cm);
    [spinach,fill=white] (14,.2) circle (.8cm);
    [mygray,fill=mygray!25] (16,.2) circle (.8cm);
    [mygray,fill=mygray!25] (18,.2) circle (.8cm);
    [mygray,fill=mygray!25] (20,.2) circle (.8cm);
    [spinach,fill=white] (22,.2) circle (.8cm);
    [mygray,fill=mygray!25] (24,.2) circle (.8cm);
    [spinach,fill=corn!25] (26,.2) circle (.8cm);
    [mygray,fill=mygray!25] (28,.2) circle (.8cm);
    [mygray,fill=mygray!25] (30,.2) circle (.8cm);
    [mygray,fill=mygray!25] (32,.2) circle (.8cm);
    [spinach,fill=corn!75] (34,.2) circle (.8cm);
    [mygray,fill=mygray!25] (36,.2) circle (.8cm);
    [spinach,fill=corn!75] (38,.2) circle (.8cm);
    [mygray,fill=mygray!25] (40,.2) circle (.8cm);
    [mygray,fill=mygray!25] (42,.2) circle (.8cm);
    [mygray,fill=mygray!25] (44,.2) circle (.8cm);
    [spinach,fill=corn!75] (46,.2) circle (.8cm);
    
    [tomato] at (2,0) 1;
    [tomato] at (4,0) 2;
    [tomato] at (6,0) 3;
    [mygray] at (8,0) 4;
    [spinach] at (10,0) 5;
    [tomato] at (12,0) 6;
    [spinach] at (14,0) 7;
    [mygray] at (16,0) 8;
    [tomato] at (18,0) 9;
    [mygray] at (20,0) 10;
    [spinach] at (22,0) 11;
    [mygray] at (24,0) 12;
    [spinach] at (26,0) 13;
    [mygray] at (28,0) 14;
    [mygray] at (30,0) 15;
    [mygray] at (32,0) 16;
    [spinach] at (34,0) 17;
    [tomato] at (36,0) 18;
    [spinach] at (38,0) 19;
    [mygray] at (40,0) 20;
    [mygray] at (42,0) 21;
    [mygray] at (44,0) 22;
    [spinach] at (46,0) 23;
    
    ,
     [anchorbase,scale=.25,tinynodes]
     at (0,7) [23];
    [thin] (-3,1.35) to (3,1.35);
    [thin] (-3,4.05) to (3,4.05);
    
    [ultra thick,spinach] (0,0) to (-2.5,2.5);
    [ultra thick,spinach] (0,0) to (2.5,2.5);
    [ultra thick,spinach] (0,5) to (-2.5,2.5);
    [ultra thick,spinach] (0,5) to (2.5,2.5);
    
    [spinach,fill=corn!25] (0,.2) circle (.8cm);
    [spinach,fill=corn!75] (-2.5,2.7) circle (.8cm);
    [spinach,fill=corn!75] (2.5,2.7) circle (.8cm);
    [spinach,fill=corn!75] (0,5.2) circle (.8cm);
    
    [spinach] at (0,0) 13;
    [spinach] at (-2.5,2.5) 17;
    [spinach] at (2.5,2.5) 19;
    [spinach] at (0,5) 23;
    
    ,

where we have highlighted in yellow the support of 23. The solid green arcs indicate successive 
inclusions in fundamental supports, and dashed orange arcs indicate successive inclusions in non-fundamental supports, all starting from 23.


To account for   we need the following admissibility conditions.


Let S⊂[0] be a finite set. 
We consider partitions S=_iS_i of S into subsets S_i 
of consecutive integers, which we call stretches 
(in the -adic expansion of v). For the purpose of this definition, we fix the coarsest such partition. 

The set S is called down-admissible for v=a_j,…,a_0 if:





  (d1)  a_min(S_i)≠ 0 for every i, and


  (d2)  if s∈ S and a_s+1=0, then s+1∈ S.

If S⊂[0] is down-admissible for v=a_j,…,a_0, then we define

    v[S]:=a_j,
    ϵ_j-1a_j-1,…,ϵ_0 a_0,  ϵ_k
    =
    
    1    if k∉ S,
    
    
    -1    if k∈ S.


Conversely, S is 
up-admissible for v=a_j,…,a_0 if the following conditions are satisfied:





  (u1)  a_min(S_i)≠ 0 for every i, and


  (u2)  if s∈ S and a_s+1=-1, then s+1∈ S.


If S⊂[0] is up-admissible for v=a_j,…,a_0, then we define

    v(S):=
    a_r(S)^',…,a_0^',
      
    a_k^'= 
    
    a_k   if  k∉ S, k-1∉ S,
    
    
    a_k+2    if  k∉ S, k-1∈ S,
    
    
    -a_k   if  k∈ S,

where we extend the digits of v by a_h=0 for h>j if necessary.  

If S is up-admissible, then we denote by ⊂[0] the
down-admissible hull of S, the smallest down-admissible
set containing S, if it exists.  


 Let =7. The set S={5,4,3|0}
(here and in the following, we use vertical bars to highlight the coarsest
partition into stretches) is down-admissible but not up-admissible for
v=4,5,0,2,0,6,17. On the other hand, S^'={5,4,3|1,0} is
up-admissible, but not down-admissible for v, and we get

    v[5,4,3|0]=4,5,0,2,0,6,1,7=4,-5,0,-2,0,6,-17
    ,
    
    
    v(5,4,3|1,0)=4,5,0,2,,0,6,17=6,-5,0,-2,2,-6,-17.

Here we have underlined the stretches of digits in S 
and S^'. Furthermore, [S^']={5,4,3,2,1,0}.



We think of the operations v↦ v(S) and v↦ v[S] as reflecting v down and 
up along S, respectively. The admissibility restrictions ensure 
that the down-admissible sets S are in bijection with the elements 
v[S]∈ and that reflecting down and up are inverse operations as 
we will see in lemma:stretch-lemma. Explicitly, for =3 and S={1,0} one gets

    13(1,0)=1,1,13=3,-1,-13=23
    ,
      
    23[1,0]=2,1,23=2,-1,-23=13.
 
See also (<ref>).



For two non-empty sets A,B⊂[0] we define

    (A,B)=min(|a-b|| a∈ A,b∈ B).

We say A and B are distant if (A,B)>1, adjacent if (A,B)=1,
or overlapping if (A,B)=0.
 

If S and S^' are down- or up-admissible for v and S∩ S^'=∅, 
then S∪ S^' will also be down- or up-admissible, respectively. Conversely, 
if S is down- or up-admissible for v and S^'⊂ S, then S^' need not be down- or up-admissible for v. 

For down- or up-admissible sets S, a central object in the following will be
the finest partition into down- or up-admissible subsets
S=_k=0^r(S)S_k (the number r(S)+1 is the size of this
partition), which we order by size of their elements S_k>S_k-1. Note
that the elements of S_k are necessarily consecutive integers, and that this
partition is typically finer than the partition considered in
definition:adm. We call the S_k minimal down- or
up-admissible stretches of v, respectively. It is easy to check that

    v[S]=v[S_r(S)]⋯[S_0],
      
    v(S)=v(S_0)⋯(S_r(S)),

for down- or up-admissible S, respectively.

 
For the set S={5,4,3|0}
(partitioned into stretches by the bar) and v as in
example:ad-sets the finest down-admissible partition is
S={5|4,3|0}= S_2∪ S_1∪ S_0 where
v[S_0],v[S_1],v[S_2]∈. More generally, the down-admissible sets
S with v[S]∈ are exactly the minimal down-admissible stretches for
v.


If S^' is also down- or up-admissible and distant from S, (S,S^')>1, then we have:

    v[S][S^']=v[S^'][S], 
      
    v(S)(S^')=v(S^')(S), 
       
    v(S)[S^']=v[S^'](S).


If S and T are subsets of [0], we write T>S to indicate the
requirement that every element in T be strictly greater than every element in
S. We have the following equivalences of admissibilities.


Consider stretches S^'>S with (S,S^')=1.





  *  S is down-admissible 
for v and S^' is down-admissible for v[S] if and only 
if S^' is down-admissible for v and S is 
up-admissible for v[S^']. In this case we have v[S][S^']=v[S^'](S).


  *  S^' is up-admissible 
for v and S is up-admissible for v(S^') if and only 
if S is down-admissible for v and S^' is 
up-admissible for v[S]. In this case we have v(S^')(S)=v[S](S^').
 



We prove (a). For this 
we write v=a_j,…, a_0, 
S={s,s+1,…,s^'-1} and S^'={s^',s^'+1,…,t-1}.

S is down-admissible for v if and only if a_s≠ 0 and a_s^'≠ 0, and we get

    v[S]=a_j,…, a_t,a_t-1,
    …,a_s^'+1,a_s^'-1,-a_s^'-1-1,…,-a_s,a_s-1,…,a_0.

Now S^' is down-admissible for v[S] if and only if a_s^'≠ 1 and a_t≠ 0, and we get

    v[S][S^']=a_j,…, a_t-1,-a_t-1-1,…,
    -a_s^'+1,-a_s^'-1-1,…,-a_s,a_s-1,…,a_0.

Conversely, S^' is down-admissible for v if and only if a_s^'≠ 0 and a_t≠ 0, and we get

    v[S^']=a_j,…, a_t-1,-a_t-1-1,…,
    -a_s^',a_s^'-1,…,a_s,a_s-1,…,a_0.

Now S is up-admissible for S^' if and only if a_s≠ 0 and a_s^'≠ 1. 
This shows the equivalence of admissibilities. Furthermore, by reflecting 
v[S^'] up along S, it is easy to see v[S^'](S)=v[S][S^']. The case of (b) is analogous. 
 


Let v∈ and S⊂[0] finite.





  *  If S is up-admissible for v, then S is down-admissible for w=v(S) and v=w[S].


  *  If S is down-admissible for v, then S is up-admissible for u=v[S] and v=u(S).




 
Let v=a_j,…,a_0. 
By (<ref>) it suffices to consider 
the case where S={s,s+1,…,s^'-1} 
is a single stretch. For lemma:stretch-lemma.(a), 
suppose that S is up-admissible for v, a_s≠ 0 and a_s^'≠-1. We get

    w=v(S)
       =…,a_s^'+1,a_s^'+2,
    -a_s^'-1,…,-a_s+1,-a_s,a_s-1,…,a_0
       =…,a_s^'+1,a_s^'+1,
    -a_s^'-1-1,…,-a_s+1-1,-a_s,a_s-1,…,a_0.

Since a_s^'+1≠ 0 and 
-a_s≠ 0, S is down-admissible for w and we have: 

    v(S)[S]=…,a_s^'+1,
    a_s^'+1,a_s^'-1-+1,
    …,a_s+1-+1,a_s-,a_s-1,…,a_0=v.

The proof of (b) is completely analogous.




 §.§ Bookkeeping for caps and cups


For this section, we fix v=a_j,…,a_0.

 
For 0≤ i≤ j we consider 
w=a_j,…,a_i+1,-a_i,0,…,0-1 
and x=a_i-1,…,a_0
to define (down and up) diagrams in [] via

    i
    :=
    [x+w]i
    :=
    [anchorbase,scale=.25,tinynodes]
    [usual] (-2,0) to (-2,3) node[above,yshift=-2pt]x;
    [usual] (2.5,0) to[out=90,in=0] 
    (.5,1.25)node[above,yshift=-1pt]a_i^i to[out=180,in=90] (-1.5,0);
    [usual] (3,0) to (3,3) node[above,yshift=-2pt]w;
    
     at (0,-.25) .;
     at (0,3.25) .;
    
    ,  i
    := 
    i[x+w]
    := 
    [anchorbase,scale=.25,tinynodes]	
    [usual] (-2,3) to (-2,0) node[below]x;
    [usual] (2.5,3) to[out=270,in=0] 
    (.5,1.75) node[below,yshift=-2pt]a_i^i to[out=180,in=270] (-1.5,3);
    [usual] (3,3) to (3,0) node[below]w;
    
     at (0,-.25) .;
     at (0,3.25) .;
    
    .

This includes the case of a_i=0, for which we have
i=i=. Note that we use symbols such
as  to indicate the source or target of these morphisms.

Now, suppose that S={s_k>⋯>s_1>s_0} 
and S^'={s_l^'>⋯>s_1^'>s_0^'} are down-, 
respectively, up-admissible for v. Then we set

    S
    :=
    [v[S]-1]S
    :=
    [v[S]-1]s_0⋯s_k
    ,
    S^'
    :=
    [v(S)-1]S^'
    :=
    [v(S)-1]s_l^'⋯s_0^'.



In (<ref>) and in the following we use the usual notation of 
idempotented algebras to drop some of the involved idempotents.
Further, the different orderings of the factors in 
S and S^' ensure that stretches of 
consecutive integers in S and S^' give rise to nested 
caps and cups, respectively.


For S^'>S with (S^',S)=1 the following hold.





  *  S^' is down-admissible 
for v and S is down-admissible for v[S^'] if and only 
if S and S∪ S^' are down-admissible for v. In this case we have SS^'
=S∪ S^'.


  *  S is up-admissible 
for v and S^' is up-admissible for v(S) if and 
only if S^' and S^'∪ S are up-admissible 
for v. In this case we have S^'S=S^'∪ S.


  *  If S^' is 
up-admissible for v and S is down-admissible for v(S^'), then S^'∪ S 
is up-admissible for v. In this case we have SS^'=S^'∪ S.


  *  If S is 
up-admissible for v and S^' is down-admissible for v(S), then
S∪ S^' is down-admissible 
for v. In this case we have S^'S=S∪ S^'.
 


 
The claims about admissibility are not hard to prove and follow, , 
as in the proof of lemma:new-admissible given above. 
Finally, the equalities as SS^'=S∪ S^' are isotopies.



Using the same notation as in definition:cup-cap-operators, 
we define diagrams in []

    i
    :=
    [x+w]i
    :=
    [anchorbase,scale=.25,tinynodes]
    [JW] (0,0) rectangle (4,1);
     at (0,1.85) a_i^i;	
    [usual] (-2,0) to (-2,3) node[above,yshift=-2pt]x;
    [usual] (1.5,1) to(1.5,2) to[out=90,in=0] (0,3) to[out=180,in=90]  (-1.5,2) to (-1.5,0);
    [usual] (2,1) to (2,3)node[above,yshift=-2pt]w;
    
     at (0,-.25) .;
     at (0,3.25) .;
    
    ,  i
    :=
    i[x+w]
    :=
    [anchorbase,scale=.25,tinynodes]
    [JW] (0,2) rectangle (4,3);
     at (0,.9) a_i^i;
    [usual] (-2,3) to (-2,0) node[below,yshift=-2pt]x;
    [usual] (1.5,2) to (1.5,1) to[out=270,in=0] (0,0) to[out=180,in=270] (-1.5,1) to (-1.5,3);
    [usual] (2,2) to (2,0)node[below,yshift=-2pt]w;
    
     at (0,-.25) .;
     at (0,3.25) .;
    
    .

The boxes represent JW projectors of the size implicit in the diagram, namely w+a_i^i =v-a_i^i-x.



Suppose that S={s_k>⋯>s_1>s_0} is down-admissible for v 
and S^'={s_l^'>⋯>s_1^'>s_0^'} is 
up-admissible for v. Then we define trapezes and standard loops

    [S]   :=
    S:=[v[S]-1]s_0⋯s_k 
    ,
    [S^']   :=
    S^':=[v(S^')-1]s_l^'⋯s_0^'[v-1],
    
    [S]   :=Sv-1:=S[v[S]-1]S.


Note that the diagrams defined in definition:jw-cupscaps2 are not left-right symmetric. 


For v=a,b,c we have: 

    [∅]
    =
    [anchorbase,scale=.25,tinynodes]	
    [JW] (-1.5,1) rectangle (1.5,2);
    [JW] (-1.5,-1) rectangle (1.5,-2);
    
    [usual] (0,-1) to (0,1);
    
     at (0,-2.3) a;
    
    ,  [{0}]
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,1) rectangle (1.5,2);
    [JW] (-1.5,-1) rectangle (1.5,-2);
    
    [usual] (0,-1) to (0,1);
    [usual] (-1,-1) to[out=90,in=0] (-1.5,-.5) 
    to[out=180,in=90] (-2,-1) to (-2,-2) node[below,yshift=0pt]c;
    
     at (0,-2.1) a;
    
    ,  [{1}]
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,1) rectangle (1.5,2);
    [JW] (-1.5,-1) rectangle (1.5,-2);
    
    [usual] (0,-1) to (0,1);
    [usual] (-1,-1) to[out=90,in=0] (-1.5,-.5) 
    to[out=180,in=90] (-2,-1) to (-2,-2) node[below,yshift=0pt]b;
    [usual] (-1,1) to[out=270,in=90] (-3,-.5) to (-3,-2) node[below,yshift=0pt]c;
    
     at (0,-2.1) a;
    
    ,   [{1,0}]
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,1) rectangle (1.5,2);
    [JW] (-1.5,-1) rectangle (1.5,-2);
    
    [usual] (1,-1) to (1,1);
    [usual] (-1,-1) to[out=90,in=0] (-1.5,-.5) 
    to[out=180,in=90] (-2,-1) to (-2,-2) node[below,yshift=0pt]b;
    [usual] (0,-1) to[out=90,in=0] (-1.5,0) 
    to[out=180,in=90] (-3,-1) to (-3,-2) node[below,yshift=0pt]c;
    
     at (0,-2.1) a;
    
    .



We record that S=v[S]-1, 
S=v-1, and Sv-1=v[S]-1.



 §.§ The p-Jones–Wenzl projectors


For v∈ and s∈[0] let
vs denote the youngest
ancestor of v whose sth 
digit is zero. (By convention, v-1=v.) 
For each down-admissible S for v we let 

    λ_v,S
    :=∏_s∈ S 
    (-1)^a_s^svs-1[S]vs[S]∈.

Note that (λ_v,S)=-|S|.


Let v=1,2,6,4,0,6,67 and S={5|3,2,1,0}. 
Then we have ∑_s∈ S a_s=18, so the overall sign is positive. 
The relevant reflected ancestors in the telescoping product (<ref>) are
v-1[S]=1,-2,6,-4,0,-6,-67, v3[S]=1,-2,6,0,0,0,07, 
v4[S]=1,-2,0,0,0,0,07, and v5[S]=1,0,0,0,0,0,07. So we get

    λ_v,S=
    1,-2,0,0,0,0,071,-2,6,-4,0,-6,-671,0,0,0,0,0,071,-2,6,0,0,0,07
    =
    485105689087
    ,
      [7](λ_v,S)=-5.



The following is immediate from (<ref>).


If S^'>S are down-admissible for v, then
λ_v,S∪ S^'=λ_v[S^'],Sλ_v,S^'.


As we will see below, the following definition is a reformulation of <cit.>.


For v-1∈[0] the rational JW projector [v-1]∈_[](v-1,v-1) is defined to be

    [anchorbase,scale=.25,tinynodes]
    [pQJW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) [v-1];
    
    :=
    [v-1]
    :=
    ∑_v[S]∈[v] λ_v,S Sv-1
    =
    ∑_v[S]∈[v] λ_v,S·[S].
  


 
For v=a,b,c we have

    [anchorbase,scale=.25,tinynodes]
    [usual,white] (0,0) to (0,2) node[above,yshift=-2pt]2;
    [usual,white] (0,-6) to (0,-8) node[below]2;
    
    [pQJW] (-1.5,-4) rectangle (1.5,-2);
     at (0,-3.1) [v-1];
    
    =
    [anchorbase,scale=.25,tinynodes]
    
    [JW] (-1.5,-1) rectangle (1.5,-2);
    [JW] (-1.5,-5) rectangle (1.5,-4);
    [JW] (-1.5,-8) rectangle (1.5,-7);
    [usual] (-1,-2) to (-1,-4);
    [usual] (0,-2) to (0,-4);
    [usual] (1,-2) to (1,-4);
    
    [usual] (-1,-5) to (-1,-7)(0,-5) to (0,-7)(1,-5) to (1,-7);
    
    +(-1)^ca,b,-ca,b,0·[anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,-1) rectangle (1.5,-2);
    [JW] (-1.5,-5) rectangle (1.5,-4);
    [JW] (-1.5,-8) rectangle (1.5,-7);
    [usual] (0,-2) to (0,-4);
    [usual] (1,-2) to (1,-4);
    [usual] (-1,-2) to[out=270,in=0] (-1.5,-2.5) 
    to[out=180,in=270] (-2,-2) to (-2,-1) node[above,yshift=-2pt]c;
    
    [usual] (0,-7) to (0,-5);
    [usual] (1,-7) to (1,-5);
    [usual] (-1,-7) to[out=90,in=0] (-1.5,-6.5) 
    to[out=180,in=90] (-2,-7) to (-2,-8) node[below]c;
    
    +(-1)^ba,-b,0a,0,0·[anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,-1) rectangle (1.5,-2);
    [JW] (-1.5,-5) rectangle (1.5,-4);
    [JW] (-1.5,-8) rectangle (1.5,-7);
    [usual] (0,-2) to (0,-4);
    [usual] (1,-2) to (1,-4);
    [usual] (-1,-2) to[out=270,in=0] (-1.5,-2.5) 
    to[out=180,in=270] (-2,-2) to (-2,-1) node[above,yshift=-2pt]b;
    [usual] (-1,-4) to[out=90,in=270] (-3,-2) to (-3,-1) node[above,yshift=-2pt]c;
    
    [usual] (0,-7) to (0,-5);			
    [usual] (1,-7) to (1,-5);
    [usual] (-1,-7) to[out=90,in=0] (-1.5,-6.5) 
    to[out=180,in=90] (-2,-7) to (-2,-8) node[below]b;
    [usual] (-1,-5) to[out=270,in=90] (-3,-7) 
    to (-3,-8) node[below]c;
    
    +(-1)^b+ca,-b,-ca,0,0·[anchorbase,scale=.25,tinynodes]	
    [JW] (-1.5,-1) rectangle (1.5,-2);
    [JW] (-1.5,-5) rectangle (1.5,-4);
    [JW] (-1.5,-8) rectangle (1.5,-7);
    [usual] (1,-2) to (1,-4);
    [usual] (-1,-2) to[out=270,in=0] (-1.5,-2.5) 
    to[out=180,in=270] (-2,-2) to (-2,-1) node[above,yshift=-2pt]b;
    [usual] (0,-2) to[out=270,in=0] (-1.5,-3.5) 
    to[out=180,in=270] (-3,-2) to (-3,-1) node[above,yshift=-2pt]c;
    
    [usual] (1,-7) to (1,-5);
    [usual] (-1,-7) to[out=90,in=0] (-1.5,-6.5) 
    to[out=180,in=90] (-2,-7) to (-2,-8) node[below]b;
    [usual] (0,-7) to[out=90,in=0] (-1.5,-5.5) 
    to[out=180,in=90] (-3,-7) to (-3,-8) node[below]c;
    
    .




The elements [v-1] agree with the 
ones defined in <cit.>.
(Note however that we have mirrored their definition.)



Careful inspection of the recursive definition in <cit.>. 
More precisely, in our notation their recursion works as follows. 
If v∈, then [v-1]=[v-1]. Otherwise,

    [anchorbase,scale=.25,tinynodes]
    [pQJW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) [v-1];
    
    =
    ∑_[v][S]∈[[v]] λ_[v],S (
    [anchorbase,tinynodes]
    1.65.3[S]0.2[JW] (-.3,-.2) rectangle (1.65,.2);
    1.65.3[S]0-.2
     at (.675,-.05) [v[S]-1];
    
    [usual] (-.15,.2) to (-.15,.5) node[above,yshift=-2pt]a_s^s;
    [usual] (-.15,-.2) to (-.15,-.5) node[below,yshift=-1pt]a_s^s;
    
    +(-1)^a_s^sv[S][s][S]·[anchorbase,tinynodes]
    1.65.3[S]0.2[JW] (.45,-.2) rectangle (1.65,.2);
    1.65.3[S]0-.2
     at (1.05,-.05) [v[S][s]-1];
    
    [usual] (.3,.2) to[out=270,in=0] (.075,.075) 
    to[out=180,in=270] (-.15,.2) to (-.15,.5) node[above,yshift=-2pt]a_s^s;
    [usual] (.3,-.2) to[out=90,in=0] (.075,-.075) 
    to[out=180,in=90] (-.15,-.2) to (-.15,-.5) node[below,yshift=-1pt]a_s^s;
    ),

where a_s is the first non-zero digit of v.


By lemma:pjw-well-defined, we can 
refer to results of <cit.> without further notice.

(<cit.>.)
For any v∈ we have ([v-1])≥ 0.



We define the JW projectors [v-1]:=[v-1]∈_[](v-1).


In illustrations we distinguish the three types of JW projectors via 

    [v-1]=
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) [v-1];
    
    ,  [v-1]=
    [anchorbase,scale=.25,tinynodes]
    [pQJW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) [v-1];
    
    ,  [v-1]=
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,0) rectangle (1.5,2);
     at (0,.9) [v-1];
    
    ,

called JW, rational JW and JW projectors, respectively.


Note that these projectors behave quite differently, for the projectors 
as in example:the-half-projectors we have

    [anchorbase,scale=.25,tinynodes]
    [JW] (-1,-1) rectangle (2,1);
     at (.5,-.1) [22];
    
    [usual] (-.5,1) to[out=90,in=180] (0,1.5) node[above,yshift=-2pt]5 to[out=0,in=90] (.5,1);
    [usual] (1.5,1) to (1.5,2);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =0,
      [anchorbase,scale=.25,tinynodes]
    [pJW] (-1,-1) rectangle (2,1);
     at (.5,-.1) [22];
    
    [usual] (-.5,1) to[out=90,in=180] (0,1.5) node[above,yshift=-2pt]5 to[out=0,in=90] (.5,1);
    [usual] (1.5,1) to (1.5,2);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1,-1) rectangle (2,1);
     at (.5,-.1) [17];
    
    [usual] (-.5,1) to[out=90,in=0] (-1,1.5) node[above,yshift=-2pt]5 to[out=180,in=90] (-1.5,1) to (-1.5,-1);
    [usual] (1.5,1) to (1.5,2);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    .



 
We have a pivotal, -linear functor 

    []→,
      (v-1)=(1)^⊗(v-1),

which sends the idempotent [v-1] to the projection 
(1)^⊗(v-1)→(v-1)→(1)^⊗(v-1). This functor induces an equivalence of 
pivotal -linear categories upon additive Karoubi completion.  



By proposition:pjw-well-defined and the construction of [], 
the only non-trivial statement is the fully-faithfulness of .
This is known; 
however, for completeness, let us give a short 
(but not new, <cit.> 
or <cit.>) argument for this. 
First, the same statement over  is a classical result and dates back to 
work of Rumer–Teller–Weyl. This implies that hom spaces on both sides have the same dimension over . 
The point is now the flatness of both sides. Precisely, the standard basis  works for [], 
showing that the dimensions of hom spaces in [] are independent 
of the characteristic. The same is true in the image of : 
The module (1) is a tilting module regardless of the characteristic, and the same 
thus holds for (1)^⊗(v-1). This implies that the hom spaces in 
([]) are also independent of the characteristic. Finally, one can check that () remains linear independent, and
the claim follows since all involved dimensions are finite and the same on both sides.




§ THE QUIVER ALGEBRA




 §.§ Generators and relations


In order to prove theorem:main
we have to give a 
presentation of the algebra

    :=⊕_v,w∈ _[](
    [v-1],[w-1]
    )
 
by generators and relations. To this end, we first introduce notation for certain elements.


Let S and S^' be down- and up-admissible for v, respectively. Then we define

    S[v-1]
       :=[v[S]-1]S[v-1]
    ,
    
    S^'[v-1]
       :=[v(S^')-1]S^'[v-1]
    ,
    
    Sv-1[v-1]
       :=[v-1]SS[v-1].

We call the latter the loop on v-1 down through v[S]-1.


We will consider the morphisms S[v-1] and 
S[v-1] as generators for , but restrict to the 
cases when S and S^' are minimal admissible stretches of 
consecutive integers. Then these morphisms can be pictured as

    S[v-1]=
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,.5) rectangle (-2,-.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
     at (0,-.2) [v-1];
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (.5,.5);
    
    [usual] (-1,.5) to (-1,1.5);
    [usual] (1,.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    ,  S^'[v-1]=
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,.5) rectangle (-2,-.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
     at (0,-.2) [v-1];
    
    [usual] (-.5,1.5) to[out=270,in=180] (0,1) to[out=0,in=270] (.5,1.5);
    
    [usual] (-1,.5) to (-1,1.5);
    [usual] (1,.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    ,  Sv-1[v-1]=
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-2,.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
    [pJW] (2,3.5) rectangle (-2,4.5);
     at (0,-.2) [v-1];
     at (0,3.8) [v-1];
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (.5,.5);
    [usual] (-.5,3.5) to[out=270,in=180] (0,3) to[out=0,in=270] (.5,3.5);
    
    [usual] (-1,.5) to (-1,1.5);
    [usual] (1,.5) to (1,1.5);
    [usual] (-1,2.5) to (-1,3.5);
    [usual] (1,2.5) to (1,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    .


We define two functions ,→ 
(where we again see ) via

    (a)=
     
    (-1)^a2a   if  1≤ a≤-2,
    
    
    0 
       if a=0 or a=-1,
      (a) =
    
    -(a+1a)
       if  1≤ a≤-1,
    
    
    -2    if a=0.

Note that (-1)=(-1)=0 and (a)=(-a-1)^-1 for a≠ 0,-1. 
Then for each finite S⊂[0] we define scaling 
operators ,,∈ on v=a_j,…,a_0 as

    [v-1] 
    =
    (a_max(S)+1)[v-1], 
      [v-1] 
    = 
    (a_max(S)+1)[v-1],
    
    [v-1] 
    =
    (a_max(S)+1-1)[v-1].

These are not considered as generators of , but as mere 
bookkeeping devices for the appearing scalars.

(Generators and relations.)
The algebra  is generated by [v-1] for v∈, 
and elements S[v-1] and S^'[v-1], 
where S and S^' denote minimal down- and up-admissible stretches for v, respectively.
These generators are subject to the following complete set of relations.





  *  Idempotents.

    [v-1][w-1]=δ_v,w[v-1],
      [v[S]-1]S[v-1]
    =
    S[v-1],
      [v(S^')-1]S^'[v-1]
    =
    S^'[v-1].



  *  Containment.
If S^'⊂ S, then we have

    S^'S[v-1]=0,
      SS^'[v-1]=0.



  *  Far-commutativity.
If (S,S^')>1, then 

    SS^'[v-1]=S^'S[v-1],
      SS^'[v-1]=S^'S[v-1],
      SS^'[v-1]=S^'S[v-1].



  *  Adjacency relations.
If (S,S^')=1 and S^'>S, then

    S^'S[v-1]=S∪S^'[v-1],
      SS^'[v-1]=S^'∪S[v-1],
    
    S^'S[v-1]=SS^'[S][v-1],
      SS^'[v-1]=[S]S^'S[v-1].



  *  Overlap relations.
If S^'≥ S with S^'∩ S={s} and S^'⊄S, then we have

    S^'S[v-1]={s}SS^'∖{s}[v-1],
      SS^'[v-1]=S^'∖{s}S{s}[v-1].



  *  Zigzag.

    SS[v-1]=[S][S][S][v-1] 
    +T[S][S]T[S][v-1].

Here, if the down-admissible hull [S], or 
the smallest minimal down-admissible stretch T with T>[S] does not exist, then the involved symbols are 
zero by definition.

 
(Basis) The elements of the form

    [w-1]S_i_l^'⋯S_i_0^'S_i_0⋯S_i_k[v-1],

with S_i_l^'>⋯>S_i_0^', and S_i_0<⋯<S_i_k, 
form a basis for [w-1][v-1]. 

(Complete) Any word 
[w-1]F[v-1] in the generators 
of  can be rewritten as a linear
combination of basis elements from (Basis) using only the relations (1)–(6).



The algebra  is a path algebra of an underlying quiver as follows. The idempotents [v-1] correspond to vertices of a quiver, call these v-1. The elements S[v-1] and 
S^'[v-1] correspond to arrows starting at the vertex v-1, and either pointing to downwards or upwards (which is leftwards respectively rightwards in <ref>) to v[S]-1 or v(S)-1.



In the special case of v=w,
theorem:main-tl-section.(Basis) says that loops form a basis of
the endomorphism spaces. Furthermore, we will see in lemma:dualnumbers
that all loops are products of loops
Sv-1[v-1] for S minimal down-admissible.



In theorem:main-tl-section.(4) and (6), the right-hand sides of the shown relations feature 
morphisms indexed by admissible subsets that are not necessarily minimal. 
We shall see in lemma:loops-1 that 
such morphisms decompose into products of generators

    S[v-1]
    :=
    S_i_1⋯S_i_k[v-1],
      S^'[v-1]
    :=
    S^'_i_l⋯S^'_i_1[v-1],

where the products are taken over the minimal down- respectively up-admissible stretches S_i_j and 
S^'_i_j, 
such that S=_j S_i_j and S^'=_j S^'_i_j, with S_i_1<⋯<S_i_k and 
S^'_i_l>⋯>S^'_i_1. 

In theorem:main-tl-section we use (<ref>) as a shorthand notation, 
but	one could also take S[v-1] and S^'[v-1] 
for (not necessarily minimal) admissible S and S^' as 
generators for . 
This requires listing the additional relations

    S[v-1]
    =
    S_1S_2[v-1],
      S^'[v-1]
    =
    S^'_2S^'_1[v-1],

for down-admissible S_1<S_2 with S=S_1∪ S_2 and up-admissible S^'_2>S^'_1 
with S^'=S^'_2∪ S^'_1, 
in addition to the relations  
theorem:main-tl-section.(1-6) among minimal generators. 
One advantage of such a presentation is that it exhibits  as 
a quadratic algebra, since relations theorem:main-tl-section.(4-6)
now turn into quadratic relations with respect to the enlarged generating set.


The proof of theorem:main-tl-section will occupy the remainder of this paper.
However, we already note that theorem:main-tl-section.(1) 
holds by the definition of  as the endomorphism algebra of a direct sum. 
Moreover, assuming the relations theorem:main-tl-section.(1-6), we get:

(Completeness—theorem:main-tl-section.(Complete).)
Let [w-1]F[v-1]∈. 
Then there is a finite sequence of relations theorem:main-tl-section.(1-6)
rewriting it as a linear combination of elements of the form theorem:main-tl-section.(Basis).



We can immediately restrict to the case where [w-1]F[v-1] is 
a product of generators of  (rather than a linear combination of such). In order to prove the claim, 
we will show that, if [w-1]F[v-1] is not of the desired form, 
then we can measure its complexity by counting out-of-order pairs of the following forms, 
all other pairs are called in-order.


 


  * S^'S or SS^' for S^'≥ S.


  * SS^'.


A case-by-case check will verify that we can use our relations 
to reduce these to in-order pairs, which then inductively shows the claim. 
For the case-by-case check we write down the 
list of all combinations how stretches S and S^' can 
meet. A priori, there 
are 13 such cases illustrated by

    min(S)<min(S^')     1a) [anchorbase,tinynodes]
     (.75,.1) to (.25,.1);
    [densely dashed] (-.25,-.1) to (-.75,-.1);
    
    ,  1b) [anchorbase,tinynodes]
     (.5,.1) to (0,.1);
    [densely dashed] (0,-.1) to (-.5,-.1);
    
    ,  1c) [anchorbase,tinynodes]
     (.4,.1) to (-.1,.1);
    [densely dashed] (.1,-.1) to (-.4,-.1);
    
    ,  1d) [anchorbase,tinynodes]
     (-.5,.1) to (.5,.1);
    [densely dashed] (0,-.1) to (-.5,-.1);
    
    ,  1e) [anchorbase,tinynodes]
     (-.5,.1) to (.5,.1);
    [densely dashed] (-.25,-.1) to (.25,-.1);
    
    ,
    
    min(S)>min(S^')     2a) [anchorbase,tinynodes]
    [densely dashed] (.75,-.1) to (.25,-.1);
     (-.25,.1) to (-.75,.1);
    
    ,  2b) [anchorbase,tinynodes]
    [densely dashed] (.5,-.1) to (0,-.1);
     (0,.1) to (-.5,.1);
    
    ,  2c) [anchorbase,tinynodes]
    [densely dashed] (.4,-.1) to (-.1,-.1);
     (.1,.1) to (-.4,.1);
    
    ,  2d) [anchorbase,tinynodes]
    [densely dashed] (-.5,-.1) to (.5,-.1);
     (0,.1) to (-.5,.1);
    
    ,  2e) [anchorbase,tinynodes]
    [densely dashed] (-.5,-.1) to (.5,-.1);
     (-.25,.1) to (.25,.1);
    
    ,
    
    min(S)=min(S^')     3a) [anchorbase,tinynodes]
     (0,.1) to (-.5,.1);
    [densely dashed] (0,-.1) to (-1,-.1);
    
    ,  3b) [anchorbase,tinynodes]
     (0,.1) to (-1,.1);
    [densely dashed] (0,-.1) to (-1,-.1);
    
    ,  3c) [anchorbase,tinynodes]
     (0,.1) to (-1,.1);
    [densely dashed] (0,-.1) to (-.5,-.1);
    
    .

where the solid line represents S and the dashed line S^', with smaller entries appearing further to the right. 
Some of the illustrated cases never arise when considering minimal admissible stretches
and the remaining cases are precisely covered by our relations.
Let us do this in detail for the out-of-order pair S^'S. 
First, the cases 2a)–2e) as well as 1e) and 3c) are ruled out by the assumption S^'≥ S. 
The case 1a) is far-commutativity, the case 1b) adjacency, while 1d) and 3b) are covered by containment. 
The relation 3a) does not occur as S^' would not be minimal. 
The remaining case 1c) is only possible if S^'∩ S={min(S^')}, in which case we can 
apply the overlap relation.




 §.§ Basic properties of pJW projectors


We invite the reader to illustrate the statements and proofs of the
next lemmas using the explicit diagrammatic examples of trapezes from
example:trapezes and of JW projectors from
example:the-half-projectors.

(See <cit.>.) 
Suppose that S and S^' are down-admissible for v. Then we have 

    [v[S]-1]SS^'[v[S^']-1] 
    =
    [anchorbase,tinynodes]
    [JW] (.15,.3) rectangle (1.25,.6);
     at (0.7,.4) [v[S]-1];
    [JW] (.15,-.3) rectangle (1.25,-.6);
     at (0.7,.-.5) .875[v[S^']-1];
    1.325.3[S]-.075.31.325.3.9[S^']-.075-.3
    =
    δ_S,S^'λ^-1_v,S^'·[anchorbase,tinynodes]
    [JW] (.15,-.3) rectangle (1.25,.3);
     at (0.7,-.05) [v[S^']-1];
    
    =
    δ_S,S^'λ^-1_v,S^'[v[S^']-1].

Thus, the summands λ_v,SSv-1 
in (<ref>) are orthogonal idempotents in [].


 
Suppose S is down-admissible for v, 
and S^'={s,…,s^'-1} is a minimal 
down-admissible stretch for v. Then we have 

    [anchorbase,tinynodes]
    1.2.6[S]00[usual] (.125,.6) 
    to[out=90,in=180] (0.25,.725) node[above,yshift=-2pt]S^' to[out=0,in=90] (.375,.6);
    
    =	
    
    (-1)^a_s^svs[S]vs-1[S]· [S∖S^']    if s∈ S,s^'∉ S,
    
     [S∪ S^']    if s∉ S,s^'∈ S,
    
    
    0     otherwise.



We will also use the non-zero cases in the form:

    λ_v,S·[anchorbase,tinynodes]
    1.2.6[S]00[usual] (.125,.6) 
    to[out=90,in=180] (0.25,.725) node[above,yshift=-2pt]S^' to[out=0,in=90] (.375,.6);
    
    =
    λ_v[S^'],S∖S^'· [S∖S^']    if s∈ S,s^'∉ S,
    
    λ_v,S· [S∪ S^']    if s∉ S,s^'∈ S.


 
Relation (<ref>) implies that S^'S[v[S]-1]=0 
if either s∈ S, s^'∈ S or s∉ S, s^'∉ S. 
For the other cases, we define S_+={t∈ S| t>s^'} and 
S_-={t∈ S| t<s}. If s∈ S and s^'∉ S, then we use 
far commutativity, relation (<ref>), and S^'=s to compute

    S^'S[v[S]-1]=
    S_+S^'S^'S_-[v[S]-1]    =
    (-1)^a_s^svs[S]vs-1[S]S_+S_-[v[S]-1]
       =
    (-1)^a_s^svs[S]vs-1[S]S∖S^'[v[S]-1].

Similarly, if s∉ S but s^'∈ S, we use far commutativity and 
an isotopy to compute

    S^'S[v[S]-1]=
    S^'S_+∪{s^'}S_-[v[S]-1]=
    S_+∪{s^'}S^'S_-[v[S]-1]=
    S^'∪ S[v[S]-1],

which finishes the proof.



 
Suppose that S^'={s,…,s^'-1} is the smallest minimal down-admissible stretch for v and let S be down-admissible for vs=. Then we have:

    [anchorbase,scale=.3,tinynodes]
    2.40.6-.05cm[S]00[usual] (-.6,-.6) to (-.6,.6) 
    to[out=90,in=180] (0,1.2) node[above,yshift=-2pt]S^' to[out=0,in=90] (.6,.6);
     at (1,1.2) a;
     at (1,-.6) a;
    	
    =
    [S].875[S∪ S^']
    =
    S[v[S∪ S^']-1]S∪ S^'    if s^'∉ S,
    
    .875[S∪ S^'][S]
    =
    S∪ S^'[v[S]-1]S    if s^'∈ S.



 
Similar, but easier than the proof of lemma:capidem.



Let e=(v) and w≤ v=a_j,…,a_0. Then we have

    [anchorbase,scale=.25,tinynodes]
    [pQJW] (-.5,0) rectangle (2.5,2);
    [usual] (-1,0) to (-1,2);
     at (1,.9) [w-1];
    
    [JW] (-1.5,-2) rectangle (2.5,0);
     at (.5,-1.1) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pQJW] (-.5,0) rectangle (2.5,-2);
    [usual] (-1,0) to (-1,-2);
     at (1,-1.1) [w-1];
    
    [JW] (-1.5,0) rectangle (2.5,2);
     at (0.5,.9) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    ,
      [anchorbase,scale=.25,tinynodes]
    [JW] (-.5,0) rectangle (2.5,2);
    [usual] (-1,0) to (-1,2);
     at (1,.9) [e-1];
    
    [pQJW] (-1.5,-2) rectangle (2.5,0);
     at (.5,-1.1) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pQJW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [JW] (-.5,0) rectangle (2.5,-2);
    [usual] (-1,0) to (-1,-2);
     at (1,-1.1) [e-1];
    
    [pQJW] (-1.5,0) rectangle (2.5,2);
     at (0.5,.9) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    .




The first pair of equalities is clear since [w-1] contains 
[w-1] with coefficient 1 and otherwise only cap and cup 
diagrams, which are killed by (<ref>). 
For a down-admissible set S, let i(S)=max{s∈ S| a_s≠ 0}. 
For the second pair of equalities we express [v-1] as

    [v-1]
    =
    [v-1]+
    ∑_i=0^j-1(∑_v[S]∈[v],i=i(S) λ_v,S Sv-1)_:=[v-1](i).
 
It follows from lemma:idemp that the 
summands [v-1](i) are orthogonal idempotents. Note 
that we can write [v-1](i)=iF(v,i)i for 
some morphism F(v,i). In particular [v-1](i) absorbs [vi-1] 
or smaller, and it annihilates all [k] for k>vi-1. In particular, it absorbs [e-1].


We prove now a significant generalization of <cit.> 
and the analog of (<ref>).

(Classical absorbtion.)
Let w≤ v. Then we have

    [anchorbase,scale=.25,tinynodes]
    [pJW] (-.5,0) rectangle (2.5,2);
    [usual] (-1,0) to (-1,2);
     at (1,.9) [w-1];
    
    [pJW] (-1.5,-2) rectangle (2.5,0);
     at (.5,-1.1) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-.5,0) rectangle (2.5,-2);
    [usual] (-1,0) to (-1,-2);
     at (1,-1.1) [w-1];
    
    [pJW] (-1.5,0) rectangle (2.5,2);
     at (0.5,.9) [v-1];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    .
 


 
We distinguish two cases. If w≤ e=(v), then we have 

    [w-1][v-1]=[w-1][e-1][v-1]=
    [e-1][v-1]=[v-1]

and the other equation follows by reflection.

On the other hand, if w≥(v), then 
∩[w]≠∅. Let z=vs=wt 
denote the youngest common ancestor of v and w. It follows 
that u:=vs-1 is the oldest ancestor of v with u≥ w. 
Now, we have [v-1][w-1]=[v-1] and [v-1][w-1](j)=0 for any j, as well as 

    [v-1](i)[w-1]=
    [v-1](i)     if i<s,
    
    
    0     if  i≥ s,
        [v-1](i)[w-1](j)=δ_vi,wj[v-1](i).

The latter is a consequence of lemma:idemp. Moreover, 
for each i≥ s, there exists exactly one j, such that vi=wj. Thus, we have

    [v-1][w-1]
       =[v-1][w-1]
    +∑_i [v-1](i)[w-1]
    +∑_i,j [v-1](i)[w-1](j)
    
       =[v-1]+∑_i<s [v-1](i)
    +∑_i≥ s [v-1](i)
    =[v-1].

The computation for [w-1][v-1] is analogous.



For =3 we have

    [anchorbase,scale=.25,tinynodes]
    [pJW] (-.5,0) rectangle (2.5,2);
    [usual] (-1,0) to (-1,2);
     at (1,.9) [2];
    
    [pJW] (-1.5,-2) rectangle (2.5,0);
     at (.5,-1.1) [3];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) [3];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-.5,0) rectangle (2.5,-2);
    [usual] (-1,0) to (-1,-2);
     at (1,-1.1) [2];
    
    [pJW] (-1.5,0) rectangle (2.5,2);
     at (0.5,.9) [3];
    
     at (1,2.5) a;
     at (1,-2.5) a;
    
    ,  [anchorbase,scale=.25,tinynodes]
    [pJW] (.5,0) rectangle (-2.5,2);
    [usual] (1,0) to (1,2);
     at (-1,.9) [2];
    
    [pJW] (1.5,-2) rectangle (-2.5,0);
     at (-.5,-1.1) [3];
    
     at (-1,2.5) a;
     at (-1,-2.5) a;
    ≠[anchorbase,scale=.25,tinynodes]
    [pJW] (1.5,-1) rectangle (-2.5,1);
     at (-.5,-.1) [3];
    
     at (-1,2.5) a;
     at (-1,-2.5) a;
    ≠[anchorbase,scale=.25,tinynodes]
    [pJW] (.5,0) rectangle (-2.5,-2);
    [usual] (1,0) to (1,-2);
     at (-1,-1.1) [2];
    
    [pJW] (1.5,0) rectangle (-2.5,2);
     at (-.5,.9) [3];
    
     at (-1,2.5) a;
     at (-1,-2.5) a;
    
    .



We also have the following relations with no classical analog.

(Non-classical absorbtion and shortening.)
Let S be a
down-admissible stretch for v. Then we have

    [v[S]-1]S[v-1]
    =
    S[v-1]
       ↭[anchorbase,scale=.25,tinynodes]
    [pJW] (2,0.5) rectangle (-2,-.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
     at (0,-.2) [v-1];
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (.5,.5);
    
    [usual] (-1,.5) to (-1,1.5);
    [usual] (1,.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,0.5) rectangle (-2,-.5);
     at (0,-.2) [v-1];
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (.5,.5);
    
    [usual] (-1,.5) to (-1,2.5);
    [usual] (1,.5) to (1,2.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    ,
    
    [v-1]S[v[S]-1]
    =
    [v-1]S   ↭[anchorbase,scale=.25,tinynodes]
    [pJW] (2,0.5) rectangle (-2,-.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
     at (0,1.8) [v-1];
    
    [usual] (-.5,1.5) to[out=270,in=180] (0,1) to[out=0,in=270] (.5,1.5);
    
    [usual] (-1,.5) to (-1,1.5);
    [usual] (1,.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,1.5) rectangle (-2,2.5);
     at (0,1.8) [v-1];
    
    [usual] (-.5,1.5) to[out=270,in=180] (0,1) to[out=0,in=270] (.5,1.5);
    
    [usual] (-1,-.5) to (-1,1.5);
    [usual] (1,-.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    ,
    
    [v[S]-1]S
    ([v-vs][vs-1])
    =
    S[v-1]
       ↭[anchorbase,scale=.25,tinynodes]
    [pJW] (2,0.5) rectangle (0,-.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
    
    [usual] (-.5,-.5)to (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (.5,.5);
    
    [usual] (-1,-.5) to (-1,1.5);
    [usual] (1,.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,0.5) rectangle (-2,-.5);
     at (0,-.2) [v-1];
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (.5,.5);
    
    [usual] (-1,.5) to (-1,2.5);
    [usual] (1,.5) to (1,2.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    ,
    
    
    ([v-vs][vs-1])S[v[S]-1]
    =[v-1]S   ↭[anchorbase,scale=.25,tinynodes]
    [pJW] (2,0.5) rectangle (-2,-.5);
    [pJW] (2,1.5) rectangle (0,2.5);
    
    [usual] (-.5,2.5) to (-.5,1.5) to[out=270,in=180] (0,1) to[out=0,in=270] (.5,1.5);
    
    [usual] (-1,.5) to (-1,2.5);
    [usual] (1,.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,1.5) rectangle (-2,2.5);
     at (0,1.8) [v-1];
    
    [usual] (-.5,1.5) to[out=270,in=180] (0,1) to[out=0,in=270] (.5,1.5);
    
    [usual] (-1,-.5) to (-1,1.5);
    [usual] (1,-.5) to (1,1.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    .

Here vs denotes the youngest ancestor of v for which the sth digit is zero.


 
If suffices to prove these relations in the case of minimal down-admissible stretches. 
To be consistent with the above notation, let us write 
S^'={s,…,s^'-1} instead of S.


In order to verify the first relation we compute, using (<ref>), that

    S^'[v-1]
       =
    S^'∑_v[S]∈[v] λ_v,SSv-1
       =
    ∑_s∈ S,s^'∉ S λ_v[S^'],S∖S^' S∖S^'[v[S]-1]S
    + 
    ∑_s∉ S,s^'∈ S λ_v,SS∪ S^'[v[S]-1]S.

For s with s∈ S,s^'∉ S, we 
define S_-=S∖ S^'. For S with 
s∉ S,s^'∈ S, we define S_+=S∪ S^'.  
It is easy to verify that the sets S_- and S_+ are 
down-admissible for v[S^'].

Then lemma:idemp implies that each summand in 
(<ref>) is invariant under left multiplication by a unique 
summand in [v[S^']-1]=
∑_v[S^'][X]∈[v[S^']] λ_v[S^'],X Xv[S^']-1,
while it is killed under left multiplication 
by any other summand. 
This proves the first equation; the second absorption equation follows by reflection symmetry.

For later use, note that the relevant summands of 
[v[S^']-1] 
are the λ_v[S^'],XXv[S^']-1 
for which s,s^'∈ X or s,s^'∉ X. 

Now we are ready to prove the projector shortening relations.
We start by expanding

    S^'([v-vs][vs-1])
    =
    S^'∑_vs[S]∈[vs] λ_vs,S ([v-vs]Svs-1).

Note that the down-admissible sets S for 
vs=a_j,…,a_s+1,0,0,…,0 are 
exactly the down-admissible sets of v which are contained 
in {s+1,…,j-1} and that λ_vs,S=λ_v,S.
Recall that, by (<ref>), we have 

    S^'
    ([v-vs]Svs-1)
    =
    S[vs-1[S∪ S^']-1]S∪ S^'    if s^'∉ S,
    
    S∪ S^'[vs-1[S]-1]S    if s^'∈ S.

In the resulting elements we either see S, 
with s,s^'∉ S or S∪ S^' 
with s,s^'∈ S∪ S^'.

Now, if X is down-admissible for v[S^'], we 
compute 

    (
    λ_v[S^'],X Xv[S^']-1
    )
    S^'
    (
    λ_vs,S [v-vs]Svs-1
    )
    
    
    = 
    
    c_1(v,Y) S_-[v[Y]-1]Y    if s∉ 
    X=:S_-, s^'∉ S,  
    X(≥ s^')=S,  
    Y:=S_-∪ S^',
    
    
    c_2(v,Y) S_+[v[Y]-1]Y    if s∈ X=:S_+,  s^'∈ S,  
    X(≥ s^')=S, Y:=S_+∖ S^',
    
    
    0     otherwise,

where the scalars c_1(v,Y) and c_2(v,Y) are computed as follows.

    c_1(v,Y) 
       = 
    λ_v[S^'],X λ_vs,S λ^-1_v[S^']s-1,S
       =
    λ_v[S^'],X λ_vs,S λ^-1_vs,S  
    =
    λ_v[S^'],Y∖S^'.
    
    
    c_2(v,Y)
       =
    λ_v[S^'],X λ_vs,S λ^-1_v[S^']s-1,S∪ S^'
       =
    (-1)^a_s^svs-1[Y]vs[Y]λ_v,Y λ_vs,S 
    (-1)^a_s^svs[Y]vs-1[Y]λ^-1_vs,S=λ_v,Y.

Thus, by (<ref>), we have

    [v[S^']-1]S^'([v-vs][vs-1])
       = 
    ∑_X,S 
    (
    λ_v[S^'],X Xv[S^']-1
    ) 
    S^'
    (
    λ_vs,S [v-vs]Svs-1
    )
       = 
    S^'[v-1].

This establishes the third relation. The analogous relation for cups follow by reflection.


The characteristic  analog of (<ref>) is:

(Partial trace.)





  *  For v∉, a_s being the first non-zero digit of v, we have

    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) [v-1];
    
    [usual] (-1,1) to[out=90,in=0] (-1.5,1.5) to[out=180,in=90] 
    (-2,1) to (-2,0) node[left,xshift=2pt]a_s^s to (-2,-1) 
    to[out=270,in=180] (-1.5,-1.5) to[out=0,in=270] (-1,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =(-1)^a_s^s2·[anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (2.5,1);
     at (0.5,-.1) [-1];
    
     at (0,2.5) a;
     at (0,-2.5) a;
    .

On the other hand, if v∈ and v≥, then the (partial) trace of [v-1] is zero.


  *  Let S be down-admissible for
v and S^' the smallest minimal down-admissible stretch for v. Then
the partial trace on the bundle of strands specified by S^' evaluates as:

    _S^'(Sv-1)
    =
    S∖S^'-1   if S^'⊂ S,
    
    
    (-1)^v-2·S-1   if S^'⊄S.




 
The second claim in proposition:p-properties-trace.(a), concerning the case of v∈, 
follows from [v-1]=[v-1] and (<ref>), which produces a scalar a∈ with (a)>0.
The case v∉ follows immediately by applying (<ref>) 
to the two expressions in the bracket in (<ref>).

In proposition:p-properties-trace.(a) we have already seen the case S=∅, 
so we assume that S≠∅. We then apply the projector shortening 
relations from proposition:absorb, and get the following two cases 
for _S^'(Sv-1), depending on whether S^'⊂ S or S^'⊄S.

    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-2,.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
    [pJW] (2,3.5) rectangle (-2,4.5);
     at (0,-.2) [v-1];
     at (0,3. 8) [v-1];
    
    [usual] (-1.5,.5) to[out=90,in=180] (-1,1) to[out=0,in=90] (-.5,.5);
    [usual] (-1.5,3.5) to[out=270,in=180] (-1,3) to[out=0,in=270] (-.5,3.5);
    
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    [usual] (-1.5,4.5) to[out=90,in=90] (-3,4.5) to (-3,2) to (-3,-.5) to[out=270,in=270] (-1.5,-.5);
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-1,.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
    [pJW] (2,3.5) rectangle (-1,4.5);
    
    [usual] (-1.5,-.5) to (-1.5,.5) to[out=90,in=180] (-1,1) to[out=0,in=90] (-.5,.5);
    [usual] (-1.5,4.5) to (-1.5,3.5) to[out=270,in=180] (-1,3) to[out=0,in=270] (-.5,3.5);
    
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    [usual](-1.5,4.5) to[out=90,in=90] (-3,4.5) to (-3,2) 	to (-3,-.5) to[out=270,in=270] (-1.5,-.5);
    
    = 
    (-1)^v-2· [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-1,.5);
    [pJW] (2,1.5) rectangle (-1,2.5);
    [pJW] (2,3.5) rectangle (-1,4.5);
    
    [usual] (-.5,.5) to (-.5,1.5);
    [usual] (-.5,2.5) to (-.5,3.5);	
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
      if 
    S^'⊂ S
    ,
    
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-2,.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
    [pJW] (2,3.5) rectangle (-2,4.5);
     at (0,-.2) [v-1];
     at (0,3.8) [v-1];
    
    [usual] (-1,.5) to[out=90,in=180] (-.5,1) to[out=0,in=90] (0,.5);
    [usual] (-1,3.5) to[out=270,in=180] (-0.5,3) to[out=0,in=270] (0,3.5);
    
    [usual] (-1.5,.5) to (-1.5,1.5);
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (-1.5,2.5) to (-1.5,3.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    [usual] (-1.5,4.5) to[out=90,in=90] (-3,4.5) to (-3,2)	 to (-3,-.5) to[out=270,in=270] (-1.5,-.5);
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-.5,.5);
    [pJW] (2,1.5) rectangle (-2,2.5);
    [pJW] (2,3.5) rectangle (-.5,4.5);
    
    [usual] (-1,-.5)to (-1,.5) to[out=90,in=180] (-.5,1) to[out=0,in=90] (0,.5);
    [usual] (-1,4.5) to (-1,3.5) to[out=270,in=180] (-.5,3) to[out=0,in=270] (0,3.5);
    
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (-1.5,2.5) to (-1.5,3.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    [usual](-1.5,2.5) to (-1.5,4.5) to[out=90,in=90] (-3,4.5) to (-3,2) to (-3,-.5) to[out=270,in=270] (-1.5,-.5) to  (-1.5,1.5) ;
    
    = 
    (-1)^v-2· [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-.5,.5);
    [pJW] (2,1.5) rectangle (-1,2.5);
    [pJW] (2,3.5) rectangle (-.5,4.5);
    
    [usual] (-1,-.5)to (-1,.5) to[out=90,in=180] (-.5,1) to[out=0,in=90] (0,.5);
    [usual] (-1,4.5) to (-1,3.5) to[out=270,in=180] (-.5,3) to[out=0,in=270] (0,3.5);
    
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    = 
    (-1)^v-2· [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-1.5,.5);
    [pJW] (2,1.5) rectangle (-1.5,2.5);
    [pJW] (2,3.5) rectangle (-1.5,4.5);
    
    [usual] (-1,.5) to[out=90,in=180] (-.5,1) to[out=0,in=90] (0,.5);
    [usual] (-1,3.5) to[out=270,in=180] (-.5,3) to[out=0,in=270] (0,3.5);
    
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.5,2.5) to (1.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
      if 
    S^'⊄S
    .

Here we have used proposition:p-properties-trace for the second equation in the bottom row.



Note that (<ref>) and (<ref>) (for eves) give a recursive way to compute 
traces. For example, for v=a,b,c we have 
=a,b,0, v2=a,0,0, and

    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (2.5,1);
     at (.5,-.1) [v-1];
    
    [usual] (-1,1) to[out=90,in=0] (-1.5,1.5) node[above,yshift=-2pt]{2,1,0} to[out=180,in=90] 
    (-2,1) to (-2,0) to (-2,-1)
    to[out=270,in=180] (-1.5,-1.5) to[out=0,in=270] (-1,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (3,1);
     at (.75,-.1) [v-1];
    
    [usual] (-1,1) to[out=90,in=0] (-1.5,1.5) to[out=180,in=90] 
    (-2,1) to (-2,0) node[left,xshift=2pt]c to (-2,-1)
    to[out=270,in=180] (-1.5,-1.5) to[out=0,in=270] (-1,-1);
    [usual] (0,1) to[out=90,in=0] (-1.5,2.5) to[out=180,in=90] 
    (-3,1) to (-3,0) to (-3,-1) to[out=270,in=180] (-1.5,-2.5) to[out=0,in=270] (0,-1);
    [usual] (1,1) to[out=90,in=0] (-1.5,3.5) to[out=180,in=90] 
    (-4,1) to (-4,0) to (-4,-1) to[out=270,in=180] (-1.5,-3.5) to[out=0,in=270] (1,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =(-1)^c2·[anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (3,1);
     at (.75,-.1) [-1];
    
    [usual] (0,1) to[out=90,in=0] (-1.5,2.5) to[out=180,in=90] 
    (-3,1) to (-3,0) node[right,xshift=-2pt]b to (-3,-1) to[out=270,in=180] (-1.5,-2.5) to[out=0,in=270] (0,-1);
    [usual] (1,1) to[out=90,in=0] (-1.5,3.5) to[out=180,in=90] 
    (-4,1) to (-4,0) to (-4,-1) to[out=270,in=180] (-1.5,-3.5) to[out=0,in=270] (1,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =(-1)^b+c2^2·[anchorbase,scale=.25,tinynodes]
    [pJW] (-1.5,-1) rectangle (3,1);
     at (.75,-.1) [v2-1];
    
    [usual] (1,1) to[out=90,in=0] (-1.5,3.5) to[out=180,in=90] 
    (-4,1) to (-4,0) node[right,xshift=3pt,yshift=-16pt]a^2-1 to (-4,-1) to[out=270,in=180] (-1.5,-3.5) to[out=0,in=270] (1,-1);
    
     at (0,2.5) a;
     at (0,-2.5) a;
    
    =0.

The proposition also implies that the (full) trace of 
the JW projectors are zero unless v<.




 §.§ Morphisms between pJW projectors—the linear structure


First, we state direct consequences of classical absorption, see 
proposition:p-properties, and non-classical absorption, see proposition:absorb.







  *  
If S=_j=0^kS_i_j with S_i_k>⋯>S_i_1>S_i_0, each down-admissible for v,
and S^'=_j=0^lS_i_j^' with S_i_l^'>⋯>S_i_1^'>S_i_0^', 
each up-admissible for v, then 

    S[v-1]=S_i_0⋯S_i_k[v-1],
        S^'[v-1]=S_i_l^'⋯S_i_0^'[v-1].



  *  
Let S and S^' be down- and up-admissible for v, respectively. Then we have

    [v[S]-1]S[v-1]
       =
    S[v-1]
    =[v[S]-1]S([v-v(S)S][vS-1]).
    
    [v(S^')-1]S^'[v-1] 
       =
    [v(S^')-1]S^'
    =([v-v(S^')S^'][v(S^')S^'-1])S^'[v-1].

Here vS denotes the youngest ancestor 
of v for which all digits with indices in S are zero.


Let F be a cap (or cup) configuration, a
Temperley–Lieb diagram consisting only of caps (or cups), with source (target)
v-1. We say that F is ancestor-centered for v, and write
F∈, if each cap or cup has its center immediately to the
right of an ancestor strand of v.


A way to illustrate ancestor-centering is imagine a line with a marker 
⋆ to the right of each 
ancestor strand of v. (There are v-1 strands in total and  many ⋆.)
For example, for =3 and v=13=1,1,13 we have

    [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) node[below]∙ to (0,2);
    [usual] (-1,0) node[below]∙ to (-1,2);
    [usual] (-2,0) node[below]∙ to (-2,2);
    [usual] (-3,0) node[below]∙ to (-3,2);
    [usual] (-4,0) node[below]∙ to (-4,2);
    [usual] (-5,0) node[below]∙ to (-5,2);
    [usual] (-7,0) node[below]∙ to[out=90,in=0] (-7.5,1) to[out=180,in=90] (-8,0) node[below]∙;
    [usual] (-6,0) node[below]∙ to[out=90,in=0] (-7.5,2) to[out=180,in=90] (-9,0) node[below]∙;
    [usual] (-10,0) node[below]∙ to[out=90,in=0] (-10.5,1) to[out=180,in=90] (-11,0) node[below]∙;
    
     at (-7.5,-.55) ⋆;
     at (-10.5,-.55) ⋆;
    ∈,  [anchorbase,scale=.25,tinynodes]
    [usual] (0,0) node[below]∙ to (0,2);
    [usual] (-1,0) node[below]∙ to (-1,2);
    [usual] (-2,0) node[below]∙ to (-2,2);
    [usual] (-3,0) node[below]∙ to (-3,2);
    [usual] (-4,0) node[below]∙ to (-4,2);
    [usual] (-6,0) node[below]∙ to[out=90,in=0] (-6.5,1) to[out=180,in=90] (-7,0) node[below]∙;
    [usual] (-5,0) node[below]∙ to[out=90,in=0] (-6.5,2) to[out=180,in=90] (-8,0) node[below]∙;
    [usual] (-9,0) node[below]∙ to (-9,2);
    [usual] (-10,0) node[below]∙ to[out=90,in=0] (-10.5,1) to[out=180,in=90] (-11,0) node[below]∙;
    
     at (-7.5,-.55) ⋆;
     at (-10.5,-.55) ⋆;
    ∉.

Moreover, we have i,i∈, with mid point right to 
a_j,…,a_i+1,0,…,0∈. 
More generally, the morphisms S and S are ancestor-centered if S is down- or up-admissible, 
respectively.


The following is the analog of (<ref>).

 
For a cap configuration d we have d[v-1]=0
unless d∈. Analogously for cup configurations.


 
By assumption, d contains a cap which is 
not centered around an ancestor of v. By expanding d[v-1] 
along (<ref>), we see that this cap hits a JW projector in every 
summand in (<ref>), and thus annihilates [v-1].








  *  Suppose that S and S^' are down-admissible for w and v, respectively, with w[S]=v[S^'].  Then we have

    [w-1]SS^'[v-1] 
    = 
    S[v[S^']-1]S^' 
    +∑_X,Y 
    c_X,YX[v[Y]-1]Y,

for some coefficients c_X,Y∈, where X and Y are down-admissible for w and v, respectively, and w[X]=v[Y]<v[S^'].
(In other words 
[w-1]SS^'[v-1]-
S[v[S^']-1]S^'∈td_>.)


  *  We have isomorphisms of -vector spaces

    _[]([v-1],[w-1])
    ≅S[v[S]-1]S^'≅[w-1]SS^'[v-1],
 
where (S,S^') ranges over pairs of sets that are down-admissible for w and v, respectively, such that w[S]=v[S^'].
In particular, _[]([v-1])≅Sv-1.



Note that the second isomorphism in (<ref>) is unitriangular by (<ref>). 
We will refer to morphisms of the form S[v[S]-1]S^' as standard morphisms 
and to morphisms of the form [w-1]SS^'[v-1] as -morphisms.

 
The proof of (a) proceeds by iterating lemma:capidem. 
Let S=_iS_i and S^'=_jS^'_j be the partitions 
into minimal admissible stretches of consecutive integers with the 
usual ordering. Then we expand

    S^'[v-1]=S^'_1⋯S^'_l[v-1]
    =∑_X S^'_1⋯S^'_l-1·λ_v,X·S^'_lX[v[X]-1]X
    ∈∑_X⊃S^'_l S^'_1⋯S^'_l-2·λ_v[X∖S^'_l],X∖S^'_l·S^'_l-1R∖S^'_l[v[X]-1]X  
    +td_>,
 
since by (<ref>), we have
λ_v,X·S^'_lX=λ_v,X∖S^'_l·X∖S^'_l
if S^'_l⊂ X, and otherwise max(S^'_l)+1∈ X and
thus v[X]<v[S^']. Here we write td_> for
the ideal of morphisms of smaller through-degree than the leading term. We now iterate this argument to find 

    S^'[v-1] 
    ∈[v[S^']-1]S^'+td_>,
      [w-1]S∈S[w[S]-1]
    +td_>,

which together imply (<ref>).

To see the first isomorphism in (<ref>):
For a given F∈_[](v-1,w-1), we compute

    [w-1]F [v-1] 
       = 
    ∑_S,S^' (
    λ_w,SS[w[S]-1]S
    )
    F
    (
    λ_v,S^'S^'[v[S^']-1]S^'
    )
    
       = 
    ∑_S,S^' δ_w[S],v[S^']S
    (λ_w,Sλ_v,S^'[w[S]-1]
    SFS^'[v[S^']-1]
    )S^'
       = 
    ∑_S,S^' δ_w[S],v[S^']c_X,S,S^'S[v[S^']-1]S^',

where c_X,S,S^'∈. In the last two lines, we have used 
lemma:idemp and the fact the 
JW projectors have no endomorphisms besides scalar multiples of the identity, (<ref>).
Finally, (<ref>) implies then the second isomorphism in (<ref>). 


(theorem:main-tl-section.(Basis).)





  *  Suppose that a -admissible morphism is
expressed as 

    ∑_S,S^' r_S,S^'·[w-1]SS^'[v-1] 
    ∈_[](v-1,w-1),

where r_S,S^'∈ and the sum (S,S^') ranges over pairwise distinct pairs of
sets that are down-admissible for w and v, respectively, such that
w[S]=v[S^']. Then every coefficient r_S,S^' is
-admissible.


  *  We have the -vector space isomorphisms 

    _[]([v-1],[w-1])≅[w-1]SS^'[v-1]
    ,
 
where (S,S^') ranges over the same set as above. In particular 

    _[]([v-1])≅Sv-1|S  down-admissible for  v.
 




 
For the first claim, we proceed by induction on the through-degree.
Note that the through-degree of
[w-1]SS^'[v-1] is w[S]=v[S^']. Let
(S,S^') be the pair labeling the summand with maximal through-degree.
Then r_S,T is -admissible since it is the coefficient of the (maximal
through-degree) basis element
S[v[S^']-1]S^' in (<ref>). Thus,
we can subtract
r_S,S^'·[w-1]SS^'[v-1] to obtain
another -admissible sum, which now has strictly lower
through-degree since
r_S,S^'·[w-1]SS^'[v-1] was the
only summand with this maximal through-degree. If the resulting sum is non-zero, then the
remaining coefficients are now -admissible by the induction hypothesis.
The basis step for the induction concerns the morphism of minimal possible
through-degree, which is -admissible (and thus also its coefficient)
since there are no correction terms in (<ref>).

To see (b), for any given F∈_[](v-1,w-1), we choose a 
lift F̃∈_[](v-1,w-1)⊂_[](v-1,w-1). By (<ref>), the 
-admissible morphism [w-1]F̃ [v-1] can be expanded in the -morphism 
basis over . By (a), all appearing coefficients are -admissible 
and can be specialized to . This results in an expansion of 
[w-1]F [v-1] in terms of the -morphisms over . 
Note that all such morphisms are still linearly independent, since they have distinct through-degrees.




 §.§ Morphisms between pJW projectors—the algebra structure








  *  
The algebra _[]([v-1]) is commutative.


  *  Every Sv-1 is nilpotent.
As a consequence, every element of non-maximal 
through-degree in _[]([v-1]) is nilpotent.




By lemma:admsum.(b), _[]([v-1]) has a basis that is invariant under reflection. 
Thus, for all a,b∈_[]([v-1]) we have [a]=a and [b]=b, and then ab=[a][b]=[(ba)]=ba. 
This implies that _[]([v-1]) is commutative.

To see (b), we shall use induction on Sv-1. 
We work over  and start by expanding 
Sv-1∈Sv-1+td_> into a sum of orthogonal quasi-idempotents and 
noting that Sv-1 has eigenvalue divisible by . If S was maximal, 
then we have (Sv-1)^2=(Sv-1)^2=0 in _[]([v-1]). 
Otherwise, if S≠∅, we conclude (Sv-1)^2<Sv-1. 
By lemma:admsum.(b), (Sv-1)^2 is a linear combination of loops S^'v-1 with 
S^'v-1<Sv-1. Then the induction hypothesis implies that (Sv-1)^2, and thus 
also Sv-1, is nilpotent.


(Containment—theorem:main-tl-section.(2).)
Let S be a stretch that is down- or up-admissible for v and S^'⊂ S 
down-admissible for v[S] or up-admissible for v(S) respectively. Then we have

    S^'S[v-1]=0,
      SS^'[v-1]=0.




Note that by projector absorption, we 
have S^'S[v-1]=S^'S[v-1]. 
This is a cap configuration consisting of a pair of collections of 
concentric caps. The right one is not ancestor-centered and, thus, kills [v-1] by lemma:anccent. 


(Far-commutativity—theorem:main-tl-section.(3).) 
Suppose that S and S^' are down-admissi­ble, T 
and T^' up-admissible 
and (S,S^')>1, (S,T)>1, and (T,T^')>1. 
The following hold.


    SS^'[v-1]=S^'S[v-1],
      ST[v-1]=TS[v-1],
      TT^'[v-1]=T^'T[v-1].



 
These relations follow from projector absorption. 
For example, for the first relation we compute

    SS^'[v-1]
    = 
    S[w]S^'[v-1]
    =
    SS^'[v-1]
    =
    S^'S[v-1]
    =
    S^'[z]S[v-1]
    = 
    S^'S[v-1].

Here we have used an isotopy of caps in the third equality.


(Adjacency relations 1—theorem:main-tl-section.(4).)
If (S,S^')=1 and S^'>S, then
the following equations hold whenever one side, and thus also the other one, is admissible

    S^'S[v-1]=SS^'[v-1],
      SS^'[v-1]=S^'S[v-1].



 
The first relation follows from projector shortening and absorption, as can be best verified graphically, 
    S^'S[v-1]
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2.5,0.5) rectangle (-2,-.5);
    [pJW] (2.5,2.5) rectangle (-2,1.5);
    [pJW] (2.5,4.5) rectangle (-2,3.5);
     at (.25,-.2) v-1;
    
    [usual] (-1.25,1.5) to[out=270,in=180] (-.75,1) to[out=0,in=270] (-.25,1.5);
    
    [usual] (.25,2.5) to[out=90,in=180] (.75,2.75) to[out=0,in=90] (1.25,2.5);
    [usual] (0,2.5) to[out=90,in=180] (.75,3) to[out=0,in=90] (1.5,2.5);
    [usual] (-.25,2.5) to[out=90,in=180] (.75,3.25) to[out=0,in=90] (1.75,2.5);
    
    [usual] (-1.75,.5) to (-1.75,1.5) 
    (-1.75,2.5) to (-1.75,3.5)
    (-1.25,2.5) to (-1.25,3.5) 
    (0,.5) to (0,1.5)
    (.25,.5) to (.25,1.5)
    (1.25,.5) to (1.25,1.5)
    (1.5,.5) to (1.5,1.5)
    (1.75,.5) to (1.75,1.5)
    (2.25,.5) to (2.25,1.5)
    (2.25,2.5) to (2.25,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2.5,0.5) rectangle (-2,-.5);
    [pJW] (2.5,2.5) rectangle (-.75,1.5);
    [pJW] (2.5,4.5) rectangle (-2,3.5);
     at (.25,-.2) v-1;
    
    [usual] (-1.25,3.5)to (-1.25,1.5) to[out=270,in=180] (-.75,1) to[out=0,in=270] (-.25,1.5);
    
    [usual] (.25,2.5) to[out=90,in=180] (.75,2.75) to[out=0,in=90] (1.25,2.5);
    [usual] (0,2.5) to[out=90,in=180] (.75,3) to[out=0,in=90] (1.5,2.5);
    [usual] (-.25,2.5) to[out=90,in=180] (.75,3.25) to[out=0,in=90] (1.75,2.5);
    
    [usual] (-1.75,.5) to (-1.75,3.5) 
    (0,.5) to (0,1.5)
    (.25,.5) to (.25,1.5)
    (1.25,.5) to (1.25,1.5)
    (1.5,.5) to (1.5,1.5)
    (1.75,.5) to (1.75,1.5)
    (2.25,.5) to (2.25,1.5)
    (2.25,2.5) to (2.25,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2.5,0.5) rectangle (-2,-.5);
    [pJW] (2.5,2.5) rectangle (.75,1.5);
    [pJW] (2.5,4.5) rectangle (-2,3.5);
     at (.25,-.2) v-1;
    
    [usual] (-1.25,3.5) to (-1.25,1.5) to[out=270,in=180] (-.75,1) to[out=0,in=270] (-.25,1.5) 
    to (-.25,2.5) to[out=90,in=180] (.75,3.25) to[out=0,in=90] (1.75,2.5);
    
    [usual] (0,.5) to (0,2.5) to[out=90,in=180] (.75,3) to[out=0,in=90] (1.5,2.5);
    [usual] (.25,0.5) to (.25,2.5) to[out=90,in=180] (.75,2.75) to[out=0,in=90] (1.25,2.5);
    
    [usual] (-1.75,.5) to (-1.75,3.5) 
    (1.25,.5) to (1.25,1.5)
    (1.5,.5) to (1.5,1.5)
    (1.75,.5) to (1.75,1.5)
    (2.25,.5) to (2.25,1.5)
    (2.25,2.5) to (2.25,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2.5,0.5) rectangle (-2,-.5);
    [pJW] (2.5,4.5) rectangle (-2,3.5);
     at (.25,-.2) v-1;
    
    [usual] (-1.25,3.5)to[out=270,in=90] (1.75,2) to (1.75,0.5);
    
    [usual] (0,.5) to (0,1.5) to[out=90,in=180] (.75,2) to[out=0,in=90] (1.5,1.5) to (1.5,.5);
    [usual] (.25,.5) to (.25,1.25) to[out=90,in=180] (.75,1.75) to[out=0,in=90] (1.25,1.25) to (1.25,.5);
    
    [usual] (-1.75,.5) to (-1.75,3.5)
    (2.25,.5) to (2.25,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    S∪ S^'[v-1]=SS^'[v-1].

Here we have used projector shortening twice, 
then projector absorption and an isotopy.
The second relation is analogous.


The following four statements will be proved jointly by induction in v. 
The proofs depend on each other in a non-trivial way.

(The endomorphisms.) 
Let v∈ with minimal 
down-admissible stretches S_j,…,S_0. Then we have the algebra isomorphism

    _[]([v-1])
    ≅[][S_jv-1,…,S_0v-1]
    /⟨(S_jv-1)^2,…, (S_0v-1)^2⟩,

and if S is down-admissible for v, then Sv-1 = ∏_k|S_k⊂SS_kv-1. 
Furthermore, if S is down-admissible for v, then we have 

    SSS[v-1]=0, 
      [v-1]SSS=0.



(Adjacency relations 2—theorem:main-tl-section.(4).)
Let S^'>S be 
down-admissible stretches of 
consecutive integers for v with (S,S^')=1. Then we have

    S^'S[v-1]=SS^'[S][v-1], 
      [v-1]SS^'=[v-1][S]S^'S.



(Overlap relations—theorem:main-tl-section.(5).)
Suppose that S is a minimal down-admissible stretch for v and S^'≥ S a 
minimal down-admissible stretch for v[S] with S^'∩ S={s} and S^'⊄S, then we have 

    S^'S[v-1]={s}SS^'∖{s}[v-1],
      [v-1]SS^'=[v-1]S^'∖{s}S{s}.



(Zigzag—theorem:main-tl-section.(6).) 
Suppose that S is an 
up-admissible stretch for v. If S is also down-admissible for v, then we have

    SS[v-1]=SS[S][v-1] 
    +TSST[S][v-1].
 
Here T denotes the smallest minimal down-admissible 
stretch with T>S, provided it exists. If not, then the 
equation holds without the second term on the right-hand side.

Furthermore, if S is not down-admissible for v, then we have

    SS[v-1]= 
    -2 
    [v-1].
 
Here [S] denotes the down-admissible hull of S, if it exists. 
If not, then the right-hand side is defined to be zero. 




§ INDUCTIVE PROOF OF THE RELATIONS


In this section we will use the 
far-commutativity relations from lemma:far-commutation, the containment 
relations from lemma:nilpotent, and the adjacency relations from 
lemma:adjacent-generators-1, sometimes without explicitly mentioning them. lemmaLemmas
Further, we only prove lemma:adjacent-generators-2 and <ref>, and lemmaLemma
(<ref>) for the first shown relations as the other ones are equivalent by reflection.


Throughout this section, unless stated otherwise, we use the convention that S denotes 
either a minimal down- or up-admissible stretch for v, and U>T> 
are the following minimal down-admissible stretches for v. 
To declutter the notation, we will suppress ∪ symbols in many expressions, 
for example STU:=S∪ T∪ U. 
Further, we 
introduce shorthand notation for the states where we have already proven 
the above Lemmas for certain v∈ N.






  * (v) means lemma:real-zigzag holds for all zigzags of 
the form X[w-1]X where w≤ v and X is down-admissible 
for w, except possibly for the case w=v and X=S, the smallest minimal down-admissible stretch for v.


  * (v) means lemma:adjacent-generators-2 on adjacent generators holds for all w≤ v.


  * (v) means lemma:overlap on overlapping generators holds for all w≤ v.


  * (v) means lemma:real-zigzag, holds for all zigzags of the form X[w-1]X where w≤ v.


  * (v) means lemma:dualnumbers, which describes _[]([w-1]), holds for all w≤ v.



Here we would like to draw the readers attention to the fact that the 
relevant quantity for zigzags is not where they start, but how high they reach.


The inductive proof of these conditions will proceed in the order shown. 
As base cases we observe that (v), (v), (v) and (v) are 
all vacuously satisfied for 1≤ v≤. Then, assuming that these conditions 
all hold for v-1, we will first deduce (v), then (v) and 
(v), followed by (v), and finally (v).


(v) follows if we have (v-1).



We need to show that we can resolve all zigzags of the form YY[v[Y]-1] 
where Y denotes a down-admissible stretch for v such 
that Y≠ S, the smallest minimal down-admissible stretch for v. If S⊄Y, 
then this is possible using projector absorption and (v-1). 
In the remaining cases we write Y_+:=Y∖ S and employ the same 
trick, but for Y_+⊅S. If Y_+ is down-admissible for v, we get

    YY[v[Y]-1]
       =SY_+Y_+S[v[Y]-1]
    
       =SY_+Y_+S[Y][v[Y]-1]+  
    STY_+Y_+TS[Y][v[Y]-1]
    
       =Y_+SSY_+[Y][v[Y]-1]
    +TY_+SSY_+T[Y][v[Y]-1]
    
       =YY[Y][v[Y]-1]+
    TYYT[Y][v[Y]-1].

Here T denotes the smallest minimal down-admissible stretch T>Y for 
v[Y], if it exists. We have also underlined the 
locations where relations are applied. If Y_+ is not down-admissible for v, then we instead get

    YY[v[Y]-1]
       =SY_+Y_+S[v[Y]-1]
    =-2S[Y_+][Y_+]S[v[Y]-1]
    
       =-2[Y_+]SS[Y_+][v[Y]-1]
    =-2[Y][Y][v[Y]-1],

or zero, if [Y_+] (and thus [Y]) does not exist.




 §.§ Adjacency relations

Next we focus on establishing (v). These relations are irrelevant for
=2, so we will assume >2 in this subsection. For this we need an
approximate result first.


Suppose that S<T<U are adjacent minimal down-admissible stretches for v. Then we have

    TS[v-1] 
    ∈ST[S][v-1] 
    +V_>U.

Here V_>U=ST[v-1]|∃ t∈ T such that t>U 
is the span of morphisms with T exceeding U. 

Similarly, if the stretches are up-admissible for v, then we have

    ST[v-1]
    ∈[S]TS[v-1]+W_>U.

where W_>U=ST[v-1]|∃ s∈ S such that s>U. 

In either case, if U is a largest down-admissible stretch for v, 
or if no down-admissible stretch exists above T, then the 
relations from lemma:adjacent-generators-2 hold on the nose.


 
Let us write  for the scalar appearing in
[v-1]=[S][v-1]. (The functions , , and
 were defined in subsec:main-TL.) We would like to
identify

    [w-1]TS[v-1]
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2.75,0.5) rectangle (-2.25,-.5);
    [pJW] (2.75,2.5) rectangle (-2.25,1.5);
    [pJW] (2.75,4.5) rectangle (-2.25,3.5);
     at (.25,-.2) [v-1];
    
    [usual] (-1.25,.5) to[out=90,in=180] (-.75,1) to[out=0,in=90] (-.25,.5);
    [usual] (0,.5) to (0,.9) to[out=90,in=270]  (-1.25,1.5); 
    [usual] (.25,2.5) to[out=90,in=180] (.75,3) to[out=0,in=90] (1.25,2.5);
    [usual] (1.5,2.5) to (1.5,2.9) to[out=90,in=270]  (.25,3.5); 
    
    [usual] (-2,.5) to (-2,1.5);
    [usual] (-1.25,2.5) to (-1.25,3.5);
    [usual] (.25,.5) to (.25,1.5);
    [usual] (-2,2.5) to (-2,3.5);
    [usual] (1.25,.5) to (1.25,1.5);
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.75,.5) to (1.75,1.5);
    [usual] (1.75,2.5) to (1.75,3.5);
    [usual] (2.5,.5) to (2.5,1.5);
    [usual] (2.5,2.5) to (2.5,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
      and  · [anchorbase,scale=.25,tinynodes]
    [pJW] (2.5,0.5) rectangle (-2.25,-.5);
    [pJW] (2.5,2.5) rectangle (-2.25,1.5);
    [pJW] (2.5,4.5) rectangle (-2.25,3.5);
     at (.25,-.2) [v-1];
    
    [usual] (-1.25,3.5) to[out=270,in=180] (-.75,3) to[out=0,in=270] (-.25,3.5);
    [usual] (-1.25,2.5) to[out=90,in=270] (0,3.1) to (0,3.5); 
    [usual] (0,.5) to[out=90,in=180] (.5,1) to[out=0,in=90] (1,.5);
    [usual] (1.25,.5) to (1.25,.9) to[out=90,in=270]  (.25,1.5); 
    
    [usual] (-2,.5) to (-2,1.5);
    [usual] (-1.25,.5) to (-1.25,1.5);
    [usual] (.25,2.5) to (.25,3.5);
    [usual] (-2,2.5) to (-2,3.5);
    [usual] (1.5,.5) to (1.5,1.5);
    [usual] (1.5,2.5) to (1.5,3.5);
    [usual] (2.25,.5) to (2.25,1.5);
    [usual] (2.25,2.5) to (2.25,3.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [w-1]ST[v-1].

By projector absorption it suffices to do this in the case when S is the smallest minimal down-admissible stretch of v.
We will start by computing the characteristic zero analogs of both sides. 

Suppose that V⊂[0] is down-admissible for v, then by lemma:capidem we have 

    Sλ_v,V Vv-1=
    λ_v[S],V∖S V∖S[v[V]-1]V   if S⊂ V,T⊄V, 
    
    λ_v,V VS[v[V]-1]V   if S⊄V,T⊂ V,
    
    
    0    otherwise.

After another application of lemma:capidem we get

    TSλ_v,V Vv-1=
    λ_v[S],V∖S VT∖S[v[V]-1]V   if S⊂ V,T⊄V,U⊂ V, 
    
    λ_v,V λ^-1_v[S],VS λ_v[T](S),VS∖T VS∖T[v[V]-1]V   if S⊄V,T⊂ V,U⊄V,
    
    
    0    otherwise.

These possibilities for V index the -morphism basis for _[]([v-1],[w-1]) from lemma:admsum. 
The term of maximal through-degree arises for V=T. 
Now, by the unitriangularity of the basis change between -morphisms and standard morphisms, 
we can read off the coefficient of ST[v-1] 
in the -morphism expansion of TS[v-1] 
as the coefficient of S[v[T]-1]T in (<ref>). 
Writing S={i,i+1,…,i_1-1} and 
T={i_1,…,i_2-1}, we compute this coefficient as

    q   = 
    λ_v,T λ_v[T](S),S λ^-1_v[S],ST   
    =
    λ_v,T λ_v[S],T   
    =(-1)^^i_1a_j,…,a_i_2,-a_i_2-1,…,-a_i_1+1,-a_i_1,0,…,0a_j,…,a_i_2,-a_i_2-1,…,-a_i_1+1,1-a_i_1,0,…,0
       =(-1)^^i_1vS[T]vS[T]+^i_1.

From this, we immediately get q≡(a_i_1-1)=, as desired. 

To finish the proof, we also need to show that UTSU[v-1] (the basis morphism of second 
highest through-degree 
in _[]([v-1],[w-1])) does not occur 
in the -morphism expansion of TS[v-1]. 
Thanks to triangularity of the basis change, this can be verified by computing the coefficient q^' of 
UT[v[SU]-1]SU in the difference 
[w-1]TS[v[T]-1]-q[w-1]ST[v-1], 
and showing that it reduces to zero modulo .

To this end, we again use lemma:capidem to expand 

    [w-1]STλ_v,V Vv-1=
    λ_v,V λ^-1_v[T],VT λ_v[T](S),VT∖S VT ∖S[v[V]-1]V   if S⊂ V,T⊄V,U⊂ V,
    
    λ_v[T],V∖T VS∖T[v[V]-1]V   if S⊄V,T⊂ V,U⊄V,
    
    
    0    otherwise.


Focusing on the case V=S∪ U, we compute the crucial coefficient q^' as

    q^' = 
    λ_v[S],U -  
    λ_v,T λ_v[S],T λ_v[S],SU λ_v[T](S),TU λ_v[T],SUT    = 
    λ_v,U ( 
    1- 
    λ_v,T λ_v[T](S),TU λ_v[S],T λ_v[T],TU)
    
       =
    λ_v,U ( 
    1 - 
    λ_v,T λ_v[SU](T),Tλ_v[S],T λ_v[U](T),T),

where we have used λ_v[S],U =λ_v,U, λ_v[S],SU =λ_v,U λ_v[U],S 
and λ_v[T],SUT =λ_v[T],UT λ_v[U],S in the first line, and in the second line

    λ_v[T](S),TU   =λ_v[T](S),U λ_v[T](S)[U],T=λ_v[T],U λ_v[SU](T),T,
    
    λ_v[T],TU   =λ_v[T],U λ_v[T][U],T=λ_v[T],U λ_v[U](T),T.

Now we note that

    λ_v[SU](T),Tλ_v[U](T),T   =
    a_j,…,a_i_3,-a_i_3-1,…,-a_i_2,0,…,0,a_i_1-1,0,…,0a_j,…,a_i_3,-a_i_3-1,…,-a_i_2,0,…,0,a_i_1,0,…,0 
    =v[U]S-^i_1v[U]S,

and together with (<ref>) we can continue

    (<ref>)   = 
    λ_v,U ( 
    
    (vS[T]+^i_1)v[U]S
    -vS[T](v[U]S-^i_1)(vS[T]+^i_1 )v[U]S)
    
       = 
    λ_v,U ( 
    ^i_1(v[U]S+vS[T])(vS[T]+^i_1)v[U]S).

This is divisible by ^-|U|^|S|^|S|+|T|+|U|^|S|^|S|=^|T| and, thus, 
q^' is zero modulo . This completes the proof of the first claim of the lemma. The second one is analogous.



(v) follows if we have (v-1), (v-1) and (v).  


The proof will be split into two parts. 
First we give a proof that works under a technical 
assumption, which is generically satisfied. 
In the second part, we treat the remaining cases.


By (v-1) and projector absorption we may assume that 
S is a smallest down-admissible stretch. At first, we will also 
assume that S<T are minimal down-admissible stretches for v and 
that T is also down-admissible for v[S]. By lemma:isotopies, 
this implies that S is up-admissible for v and T is down-admissible for v(S). 

We already know that the desired equation holds up to certain potential error terms, 
    TS[v-1] 
    =_1ST[v-1] 
    +∑_X ( 
    c_XXUTSUX 
    +d_XXSTX)
    [v-1],

where the summation runs over down-admissible subsets 
X>U, c_X,d_X∈ and where we write _1:=(a_max(S)+1-1) 
for v=a_j,…,a_0. 
We now multiply this equation with S 
on the left and with T on the right 
and rewrite it using w=v[T] and lemma:adjacent-generators-1 into

    STTS[w-1]=
       _1SSTT[w-1] 
    
       
    +∑_X (
    c_XXUTSw-1 
    +d_XXw-1SSTT)[w-1].

This equation 
can be simplified using (v). In this proof attempt, we only consider the 
generic case where T (and thus also TS) is down-admissible for w. 
So, using (v) for the pair (v,ST) we get:

    STTS[v[T]-1] 
    =_1TSv[T]-1 
    +_1UTSv[T]-1,

where _1:=(b_max(S∪ T)+1) and 
_1:=(b_max(S∪ T)+1) are computed 
from v[T]=b_i,…,b_0. Further, 
using (v) for (v[T](S),S) and (v,T) as well as (v-1) we compute

    SSTT[v[T]-1] 
    = 
    (_2Sv[T]-1+
    _2TSv[T]-1)(_1Tv[T]-1 
    +_1UTv[T]-1)
    
    
    =_2_1TSv[T]-1
    +_2_1UTSv[T]-1,

where _1 and _1 are as above, while 
_2:=(b_max(S)+1)=(-a_max(S)+1). We also have

    _2:=(b_max(S)+1)=(-a_max(S)+1)
    =_1^-1.
 
(Note that _1 is invertible since >2.) Using these two computations and 
(v-1), the equation (<ref>) transforms into

    0=0+∑_X ( 
    (c_X+_2_1d_X)XUTSw-1 
    +_2_1d_XXTSw-1).

Since the -loops Yw-1 form a basis 
of _[]([w-1]) and the scalars _1 and _2 are non-zero by admissibility and >2,
we conclude d_X=0 and then c_X=0. Thus all error terms in (<ref>) vanish. This completes the proof in the case where S and T are minimal.

In the general case, we partition S and S^' into minimal 
down-admissible stretches S_1<⋯<S_k and S^'_1<⋯<S^'_l, 
respectively. Then we have

    S^'S[v-1]
       =
    S^'_1S^'_2⋯S^'_lS_1⋯S_k-1S_k[v-1]
    
       =
    S_1⋯S_k-1S^'_1S_kS^'_2⋯S^'_l[v-1]
    
       =
    S_1⋯S_k-1S_kS^'_1[S_k]S^'_2⋯S^'_l[v-1]
    
       =
    S_k⋯S_1S^'_1⋯S^'_l[S_k][v-1]
    =SS^'[S][v-1].

Here we have first used far-commutativity, then (v-1) on the 
adjacent minimal stretches S_k<S^'_1, and finally lemma:adjacent-generators-1. 
Note also that [S_k]=[S] far-commutes with S^'_2⋯S^'_l.



In the previous proof we made the assumption that T, and thus also T∪ S, is down-admissible for w=v[T]. 
Now suppose this is not the case. 
At first we can proceed in a very similar way as 
in the previous proof. Whenever we use zigzag 
relations, we have to replace T by [T]=T∪ U and set the -term to zero. Hence, we get

    STTS[w-1] 
    =_1UTSw-1,
    
    SSTT[w-1] 
    = 
    (_2Sw-1+
    _2UTSw-1)
    (_1UTw-1)=_2_1UTSw-1,

and the equation (<ref>) transforms into

    0=0+∑_X ( 
    (c_X+_2_1d_X)XUTSw-1).

This implies that the coefficients c_X and d_X are unit 
multiples of each other for every X. Next we will use a 
different strategy to show that d_X=0, which thus implies c_X=0 and finishes the proof. 
The strategy is to multiply both sides of (<ref>) by Uv-1 on 
the right, to equate the first two terms, to kill all terms with coefficients 
c_X, and to preserve all terms with coefficients d_X. 





The first two terms are rewritten as

    TSUv-1   =TUSU[v-1]
    =UTSU[v-1]  
    
    _1STUv-1   =_1SUTU[v-1]=_1USTU[v-1],

which are equal by virtue of (v-1) since v[T](S)[U]<v. 
We also note that the scalar that appears is exactly _1^-1. 
After subtracting these terms from the multiple of (<ref>), we are left with

    0=∑_X ( 
    c_XXUTSUXUv-1   
    +d_XXSTXUv-1).

We first claim that XUTSUXUv-1=0. 
To verify this, we distinguish between the two cases in which X is distant or adjacent to U. 
In the first case, we get

    XUTSUXUv-1  
    =XUTSUUUX[v-1]=0,

since UUUX[v-1]=0 thanks to (v-1) as v[X]<v. 
In the second case, we get

    XUTSUXUv-1  
    =XUTSUXUU[v-1] 
    =XUTSXUUU[v-1]=0,

since U^2[v-1]=0. This proves the claim.

Our second claim is that XSTXUv-1≠ 0 
for every X and that these morphisms are linearly independent. 
Again it matters whether X is distant or adjacent to U. In the first case we get

    XSTXUv-1  
    =XSTUUX[v-1]  
       =XUSTUX[v-1]
    
       ∼XUTSUX[v-1].

Here we have used (v-1) for 
v[U∪  X]<v to proceed to the second 
line. (We use ∼ to indicate unit proportionality.) In the second case we compute

XSTXUv-1  
   =XSTUXU[v-1]       

  
   ∼XSTUUX[v-1]      


   ∼XSTUUX[v-1]
      =XSUTUX[v-1]


          ∼XUTSUX[v-1]

This time we have used (v-1), namely on the ancestor vT<v 
using projector absorption, to get to the second and the fourth line, 
and (v) in the form of a zigzag relation for v[X](U)<v, noting 
that U is down-admissible for v[X], to get to the third line. The 
proportionality constants that appear in these steps are units and 
XUTSUX[v-1] are linearly independent as X varies. 

Finally, the two claims and equation (<ref>) imply that d_X=0 
for every X, and thus also c_X=0, which finishes the proof of (v). 







 §.§ Overlap relations

Next, we focus on establishing (v). We again start with an approximate version.


Suppose that S<T are adjacent minimal down-admissible stretches for v and S^'≥ S 
is a minimal down-admissible stretch for v[S] with S^'∩ S={s} and S^'⊄S, then we have 

    S^'S[v-1]={s}SS^'∖{s}[v-1] 
    +V_>T,
      [v-1]SS^'=[v-1]S^'∖{s}S{s}+W_>T.

Here we use the notations V_>T=XY[v-1]|∃ y∈ Y such that y>T and 
W_>T=[v-1]YX|∃ y∈ Y such that y>T.
In either case, if T is a largest down-admissible stretch for v 
then the relations from lemma:overlap hold on the nose.






We will use the notation w=v[T](R) and 
{s}=S∩ S^', R=S∖{s}, and 
note S^'=T∪{s}. We will also consider the 
minimal down-admissible stretch U>T for v, if it exists. 
For the purpose of this proof it is useful to explicitly write 
down the relevant parts of the continued fraction expansions of v, w and other entities

    v 
       =…,0,a_x,0,…,a_u,0,…,0,1,0,0,…,0,a_r,…,
    
    
    v[S] 
       =…,0,a_x,0,…,a_u,0,…,0,0,-1,-1,…,-1,-a_r,…,
    
    
    v[T] 
       =…,0,a_x,0,…,a_u-1,-1,…,-1,-1,0,0,…,0,a_r,… 
    =w[R],
    
    
    v[T][S] 
       =…,0,a_x,0,…,a_u-1,-1,…,-1,-2,-1,
    -1,…,-1,-a_r,…,
    
    
    v[S][S^']
       =…,0,a_x,0,…,a_u-1,-1,…,
    -1,-1,1,-1,…,-1,-a_r,…=w.

Here we have highlighted the digit in position s in red.

From this description, it is straightforward to see 
that _[]([v-1],[w-1]) is spanned by morphisms of the following four different types

    XRTX[v-1],
      XUS^'SUX[v-1],
      X{s}STX[v-1],
      XUTRUX[v-1],

where X denotes a down-admissible subset for v with X>U, 
which may be empty. The basis elements of highest and second highest 
through-degree among the above are RT[v-1] and 
{s}ST[v-1], and all other basis elements are in the subspace V_>T. 

Our task is to show that RT[v-1] 
appears with coefficient 0 and {s}ST[v-1] 
appears with coefficient 1 if we expand S^'S[v-1] 
in this basis. We again start with a computation in characteristic zero.

Two applications of lemma:capidem establish

    S^'Sλ_v,V Vv-1=
    
    λ_v[S],V∖S VS^'∖S[v[V]-1]V   if S⊂ V,T⊄V,U⊂ V, 
    
    λ_v,V λ^-1_v[S],VS λ_w,VR∖T VR∖T[v[V]-1]V   if S⊄V,T⊂ V,U ⊄V,
    
    
    0    otherwise.

where we have used V∪ S∖ S^'=V∪ R∖ T in the second case.

The coefficient q of the maximal through-degree 
basis element RT[v-1] in 
S^'S[v-1] is equal to the
coefficient shown for R[v[T]-1]T in (<ref>). This is

    q=λ_v,T λ^-1_v[S],TS λ_w,R≡
    (-1)^^s…,0,a_x,0,…,a_u-1,-1,…,-1,-1,0/…,0,a_x,0,…,a_u-1,-1,…,-1,-1,1≡ 0 .

This shows that RT[v-1] appears with coefficient 0 in 
S^'S[v-1]. The term {s}ST[v-1], however, does 
not seem to appear at all in (<ref>). 
Since it is of second highest through-degree in _[]([v-1],[w-1]), 
its coefficient is congruent to the coefficient of 
{s}[v[ST]-1]ST in the -morphism expansion of 
qR[v[T]-1]T.

Using lemma:capidem, it is straightforward to 
compute that R[v[T]-1]T equals R[v[T]-1]T-λ_w,{s}{s}[v[ST]-1]ST 
up to terms of lower through-degree. 
Thus, we compute the coefficient of interest as

    -λ_w,{s} q ≡
    -…,a_u-1,-1,…,-1,-1/…,a_u-1,-1,…,-1,0…,a_u-1,-1,…,-1,0/…,a_u-1,-1,…,-1,1≡
    1 ,

and this finishes the proof.






(v) follows if we have (v-1), (v-1), (v-1), and (v).



As usual, (v-1) and projector absorption 
allows us to restrict to the case when S is the 
smallest minimal down-admissible stretch for v.
By lemma:overlapapprox we then have

    S^'S[v-1] 
       ={s}ST[v-1]
    
       +∑_X≠∅ c_XXRTX[v-1] 
       +∑_X d_XXUS^'SUX[v-1]
    
       +∑_X≠∅ e_XX{s}STX[v-1] 
       +∑_X f_XXUTRUX[v-1].

Here U>T denotes another adjacent minimal down-admissible stretch for v, 
if it exists, and X ranges over down-admissible subsets X>U for v. 
Our task is to show that the scalars c_X,d_X,e_X,f_X∈ are all zero. 

We start by multiplying both sides of (<ref>) by T on the right. After rearranging, we get

    S^'ST[v[T]-1] 
       ={s}STT[v[T]-1]
    
       +∑_X≠∅ c_XXRXTT[v[T]-1]
    
       +∑_X d_XXUS^'STUX[v[T]-1]
    
       +∑_X≠∅ e_XX{s}SXTT[v[T]-1]
    
       +∑_X f_XXUTRTUX[v[T]-1].

The next step is to apply the zigzag relations and for this 
we shall assume that we are in the generic case, where T 
is down-admissible for v[T] (and thus >2). This also implies that S^' is 
down-admissible for v[T](R), and using the zigzag relations provided by (v) for v[T](R) we compute

    S^'ST[v[T]-1]
       =S^'S^'R[v[T]-1]
    
       =(a_u)S^'S^'R[v[T]-1]
    
         +(a_u)US^'S^'UR[v[T]-1]
    
       =(a_u)S^'ST[v[T]-1]
    
         +(a_u)US^'STU[v[T]-1].

Similarly we compute

    {s}STT[v[T]-1]
       ={s}STT[v[T]-1]
    +{s}SUTTU[v[T]-1]
    
       ={s}TST[v[T]-1]
    +{s}UTSTU[v[T]-1]
    
       =(-2)T{s}{s}RT[v[T]-1]
    
         +(-2)UT{s}{s}RTU[v[T]-1]
    
       =S^'{s}RT[v[T]-1]
    +US^'{s}RTU[v[T]-1]
    
       =S^'ST[v[T]-1]
    +US^'STU[v[T]-1],

where we write =(a_u-1) and =(a_u-1), and we 
have used (v) on the pair (v,T) and smaller instances, as well as 
(v-1). Thus, we have equated the first two terms in (<ref>). We also simplify the c_X terms

    XRXTT[v[T]-1] 
       =XRXTT[v[T]-1]+ 
    XRXUTTU[v[T]-1]
    
       =XTRTX[v[T]-1]+ 
    XUTRTUX[v[T]-1],

where we have again used (v) on the pair (v,T), then (v-1), 
and smaller instances of zigzag relations in the case 
when X≠∅ is adjacent to U for the final step. To be explicit, the sequence of transformations is

    XUTTU[v[T]-1]=
    TUXTU[v[T]-1]∼
    TUUTX[v[T]-1]∼UTTUX[v[T]-1].

The simplification of the f_X term proceeds in complete analogy to (<ref>) and we get

    X{s}SXTT[v[T]-1]= 
    XS^'STX[v[T]-1]
    +XUS^'STUX[v[T]-1],

having again used only (v) and (v-1).

Finally, after all these simplifications, (<ref>) gives the following linear system

    0= c_X,  
    0= c_X+ f_X,  
    0=d_X+ e_X,  
    0= e_X,

which, since ≠ 0, implies that all unwanted scalars are zero. 






Now suppose that T is not down-admissible for v[T], 
which happens exactly if a_u=1 in the notation from above. 
In this case we have [T]=T∪ U for v[T] and 
[S^']=S^'∪ U for v[T](R). 

We proceed exactly as above, with the only differences being that 
no  terms arise and =-2.
The linear system resulting from (<ref>) is

    0=-2c_X+f_X,
      
    0=d_X-2e_X.

(Note that if =2, we immediately see d_X=0=f_X.) 
To see that all coefficients are zero, 
we multiply (<ref>) by Uv-1, 
expecting that this should allow us to equate the first two 
terms, kill the d_X and f_X terms, and not hurt the c_X 
and e_X terms. Let us check these assertions in turn.

For the first term we get

    S^'SUU[v-1]=
    S^'USU[v-1]=
    US^'SU[v-1].


For the second term we compute

    {s}STUU[v-1]=
    {s}UTSU[v-1]= 
    US^'SU[v-1],

where the second step works as in (<ref>) and 
requires (v-1) and (v). This equates the first two terms. 

Now we claim that the d_X and f_X terms are killed by the loop along U

    XUS^'SUXUU[v-1]=0
    =XUTRUXUU[v-1].


If X≠∅ is adjacent to U, then both assertions follow from

    UXUU[v-1]=
    (U)^2XU[v-1]=0.

If X is distant from U or empty, then we use far-commutativity to see substrings 
of the form UUUX[v-1]=0 by (v-1).

Now we claim that the c_X and e_X terms survive the multiplication by the loop along U: 

    XRTXUU[v-1]≠ 0
    ≠X{s}STXUU[v-1].

To see this, let us first observe XUU[v-1]=UUX[v-1]. 
This is clear if X is distant from U, and it follows 
from (v-1) and (v), otherwise. Using this observation, we compute

    XRTXUU[v-1]=
    XRTUUX[v-1]
       =
    XUTRUX[v-1]≠ 0
    
    X{s}STXUU=
    X{s}STUUX   =
    X{s}UTSUX
       =XUS^'SUX≠ 0

where the last step works as in 
(<ref>) and requires (v-1) and (v).

After these simplifications, we see that 
(<ref>) multiplied by Uv-1 
shows c_X=0=e_X, which (for >2) in turn implies d_X=0=f_X. This completes the proof of (v). 





Let us also note the following consequence.

 
Suppose that a minimal stretch S is down-admissible for v but not for v[S], 
and suppose the down-admissible hull 
 exists. 
Then (v) implies

    S[v-1]=S[v-1],
      [v-1]S=[v-1]S.




Let s=max(S) and S^'={s}∪∖ S and R=S∖{s}. Then S^' is down-admissible for v[S] and we use (v) to compute

    S[v-1]
    =
    RS^'S[v-1]
    =
    R{s}SS^'∖{s}[v-1]
    =
    {s}R[v-1]
    =
    S[v-1].

The other relation follows by reflection.




 §.§ Zigzag relations


 
The zigzag relations from lemma:real-zigzag hold in generation 2.


 
Suppose that S^' is a down-admissible stretch 
for v such that w=v[S^'] is of generation 2. Then, 
using the projector shortening property from proposition:absorb, we get 

    S^'S^'[w-1]= 
    _(v-w)/2([(v+w)/2-1]).

This partial trace is not covered by proposition:p-properties-trace, 
but since (v+w)/2 is of generation at most 2, it can be straightforwardly computed: 
One first expands [(v+w)/2-1] into a linear combination of standard loops and 
computes their partial traces using (<ref>). The result follows by changing 
back into the loop basis of _[]([w-1]) and reducing the coefficients to . 

The basis change from loops to standard loops for w of generation 2 with minimal down-admissible stretches S<T is

    ∅w-1   =∅w-1
    +(-1)^w-[w]w[S][w]·Sw-1
    +(-1)^[w]-w2[w][T]w2·Tw-1
    +(-1)^w-w2w[ST]w2·STw-1,
    
    Sw-1   =Sw-1
    +(-1)^w-w2([w][T])^2w[T]w2·Tw-1, 
    
    Tw-1   =Tw-1
    +(-1)^w-[w]w[ST][w][T]·STw-1,
    
    STw-1   =STw-1.

The inverse basis change can be readily computed from this. The basis change in generation 1 is easier and left as an exercise for the reader.



(v) follows if we have (v), (v-1), (v) and (v).


The proof again splits into two parts. First we give a 
proof that works under a technical assumption, which is generically 
satisfied. In the second part, we refine this proof to work in all cases.

 
We need to consider the zigzag  S[v-1]S where S is the smallest minimal down-admissible stretch of v.
Let us also assume that we are in the generic case, where S 
is also down-admissible for v[S] (and thus >2), and we denote by T the minimal down-admissible stretch for v[S] that is adjacent and T>S. 

By the unitriangularity of the basis change between the loops basis and the standard 
loops basis for _[]([v[S]-1]) and by the generation 2 case in 
lem:gen2zigzag, we may assume that

    SS[v[S]-1]
    =[S]Sv[S]-1+
    [S]STv[S]-1+
    ∑_U⊄S∪ T x_UUv[S]-1,

with error terms x_UUv[S]-1 with x_U∈. 
Our job is to show that we have x_U=0 for all 
such U. If we multiply (<ref>) 
by Sv[S]-1, then (v-1) implies 0=0+0+∑_Xx_XSXv[S]-1 
and thus x_X=0, where X runs over all U as above, for which S⊄U. 
On the other hand, if we multiply (<ref>) by Tv[T]-1, then we get

    SSTT[v[S]-1]  
    =[S]STv-1[v[S]-1]  
    +∑_X x_XTXv[S]-1,

where now X runs over all remaining U such that T⊄X. Then, by (v), we also get

    SSTT[v[S]-1]  
    =  
    [S]STv[S]-1.
 
This implies x_X=0 for such X. 
The only coefficients 
x_U that are left to be considered are the ones for 
which S∪ T⊂ U. Now we apply the partial 
trace pTr_ST:=pTr_(v[S]-v[S]ST) to both 
sides of (<ref>). For this we 
will use the notation w=v[S]S and u=v[S]ST, and we get

    (-1)^w+1-u2[u-1] 
    = 
    [S](-1)^w-u2
    [u-1]+ 
    [S][u-1] 
    +∑_U≠∅x_UUu-1.

This is because pTr_ST(SS[v-1])=pTr_T([z-1])
where z=vS, which differs from w by 
increasing its first non-zero digit a by one. 
The coefficient of [u-1] on the right-hand side 
is computed as follows

    -(-1)^w-u2  
    (a+1a) 
    +(-1)^w-u2a
    =(-1)^w+1-u2.

After subtracting the multiples of [u-1] from both sides, we conclude x_U=0. 


 
Now suppose that S is smallest minimal down-admissible stretch 
for v, but not down-admissible for v[S]. Then (<ref>) takes the form

    SS[v[S]-1] 
    =-2v[S]-1+
    ∑_U⊄ x_UUv[S]-1.

We first multiply this by v[S]-1 and deduce

    SS[v[S]-1]= 
    SS[v[S]-1]=0

from (v). Since (v[S]-1)^2=0 
by (v-1), we get x_U=0 unless ⊂U. Then a 
partial trace argument as above shows that all remaining x_U are also zero.



(v) follows if we have (v-1), (v), (v), and (v).



We first prove (<ref>). By (v-1) and projector absorption, 
we may assume that S is a smallest minimal down-admissible stretch. 
Suppose first that S is down-admissible for v[S]. Then we have

    SSS[v-1] 
       = 
    SSS[v-1]+TSSTS[v-1]
    
       = 
    0+TSSST[v-1]=0.

where we have used 
(v), (v) and finally (v-1) to 
deduce [v[T](S)-1]SSS=0. 
Now, suppose that S is not down-admissible for v[S]. Then we instead get

    SSS[v-1] 
       = 
    -2S[v-1]
    
       = 
    -2S[v-1]=0.

Here we have used (v) and lemma:nilpotent. 

Next we need to prove that (S_kv-1)^2=0 and 
Xv-1=∏_k| S_k⊂ XS_kv-1. 
The first relation simply follows from (<ref>), 
    (S_kv-1)^2=S_kS_kS_kS_k[v-1]=0.

Next, suppose we already know that Xv-1=∏_S_k⊂ XS_kv-1 
for subsets X that decompose into l-1 minimal stretches, and let us 
now consider X∪ S_i where X>S_i. The result is clear if X and S_i 
are distant, so we will assume that S_i is adjacent to 
X. By projector absorption and (<ref>), we may further assume that S_i=S is the 
smallest minimal down-admissible stretch for v. Then we compute

    Sv-1Xv-1
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (2,-.5) rectangle (-2,.5);
    [pJW] (2,2) rectangle (-2,3);
    [pJW] (2,4.5) rectangle (-2,5.5);
     at (0,-.2) [v-1];
     at (0.75,1.25) …;
     at (0.75,3.75) …;
     at (0,2.3) [v-1];
     at (0,4.8) [v-1];
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (-1.5,3) to[out=90,in=180] (-1,3.5) to[out=0,in=90] (-.5,3);
    [usual] (-1.5,4.5) to[out=270,in=180] (-1,4) to[out=0,in=270] (-.5,4.5);
    
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (1.5,.5) to (1.5,2);
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (0,3) to (0,4.5)(1.5,3) to (1.5,4.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (0,-.5) rectangle (2,.5);
    [pJW] (2,2) rectangle (-2,3);
    [pJW] (2,4.5) rectangle (-1,5.5);
     at (0,2.3) [v-1];
    
    [usual] (-.5,-.5) to (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (-1.5,3) to[out=90,in=180] (-1,3.5) to[out=0,in=90] (-.5,3);
    [usual] (-1.5,5.5) to (-1.5,4.5) to[out=270,in=180] (-1,4) to[out=0,in=270] (-.5,4.5);
    
    [usual] (-1.5,-.5) to (-1.5,2);
    [usual] (1.5,.5) to (1.5,2);
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (0,3) to (0,4.5)(1.5,3) to (1.5,4.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (0,-.5) rectangle (2,.5);
    [pJW] (2,2) rectangle (-2,3);
    [pJW] (2,3.25) rectangle (-0.25,4.25);
    [pJW] (2,4.5) rectangle (-1,5.5);
     at (0.75,1.25) …;
     at (0,2.3) [v-1];
    
    [usual] (-.5,-.5) to (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (-1.5,3) to[out=90,in=180] (-1,3.5) to[out=0,in=90] (-.5,3);
    [usual] (-1.5,5.5) to (-1.5,4.5) to[out=270,in=180] (-1,4) to[out=0,in=270] (-.5,4.5);
    
    [usual] (-1.5,-.5) to (-1.5,2);
    [usual] (1.5,.5) to (1.5,2);
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (0,3) to (0,3.25)(1.5,3) to (1.5,3.25)(0,4.25) to (0,4.5)(1.5,4.25) to (1.5,4.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (0,-.5) rectangle (2,.5);
    [pJW] (2,2) rectangle (-1,3);
    [pJW] (2,3.25) rectangle (-0.25,4.25);
    [pJW] (2,4.5) rectangle (-1,5.5);
    
    [usual] (-.5,-.5) to (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (-1.5,3) to[out=90,in=180] (-1,3.5) to[out=0,in=90] (-.5,3);
    [usual] (-1.5,5.5) to (-1.5,4.5) to[out=270,in=180] (-1,4) to[out=0,in=270] (-.5,4.5);
    
    [usual] (-1.5,-.5) to (-1.5,3);
    [usual] (1.5,.5) to (1.5,2);
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (0,3) to (0,3.25)(1.5,3) to (1.5,3.25)(0,4.25) to (0,4.5)(1.5,4.25) to (1.5,4.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (0,-.5) rectangle (2,.5);
    [pJW] (2,2) rectangle (-1,3);
    [pJW] (2,4.5) rectangle (-1,5.5);
    
    [usual] (-.5,-.5) to (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (-1.5,3) to[out=90,in=180] (-1,3.5) to[out=0,in=90] (-.5,3);
    [usual] (-1.5,5.5) to (-1.5,4.5) to[out=270,in=180] (-1,4) to[out=0,in=270] (-.5,4.5);
    
    [usual] (-1.5,-.5) to (-1.5,3);
    [usual] (1.5,.5) to (1.5,2);
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (0,3) to (0,4.5)(1.5,3) to (1.5,4.5);
     at (1,2.5) a;
     at (1,-.5) a;
    
    =
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1,-.5) rectangle (2,.5);
    [pJW] (2,2) rectangle (-1,3);
    [pJW] (2,4.5) rectangle (-1,5.5);
    
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (-1.5,3) to[out=90,in=180] (-1,3.5) to[out=0,in=90] (-.5,3);
    [usual] (-1.5,5.5) to (-1.5,4.5) to[out=270,in=180] (-1,4) to[out=0,in=270] (-.5,4.5);
    
    [usual] (-1.5,-.5) to (-1.5,3);
    [usual] (1.5,.5) to (1.5,2);
    [usual] (-1.5,.5) to (-1.5,2);
    [usual] (0,3) to (0,4.5)(1.5,3) to (1.5,4.5);
     at (1,2.5) a;
     at (1,-.5) a;
    [anchorbase,scale=.25,tinynodes]
    [pJW] (-1,-.5) rectangle (2,.5);
    [pJW] (2,2) rectangle (-1,3);
    [usual] (-.5,.5) to[out=90,in=180] (0,1) to[out=0,in=90] (0.5,.5);
    [usual] (-.5,2) to[out=270,in=180] (0,1.5) to[out=0,in=270] (.5,2);
    [usual] (1.5,.5) to (1.5,2);
     at (1,2.5) a;
     at (1,-.5) a;
    
    = 
    X-1.

The equation pTr_v-(Sv-1Xv-1)=X-1 and 
proposition:p-properties-trace imply 
Sv-1Xv-1=xXv-1+yXSv-1 where (-1)^v-2x+y=1. 
Equivalently, we can write Xv-1(-x+Sv-1)=yXSv-1. 
Suppose that x≠ 0, then (-x+Sv-1) would be a unit in 
_[]([v-1]), so we can write 

    Xv-1=(-x+Sv-1)^-1yXSv-1.

However, the left-hand side has through-degree v[X], while 
the right-hand side has through-degree at most v[X∪ S]<v[X], 
a contradiction. Thus, we have x=0 and y=1, and 
consequently Sv-1Xv-1=XSv-1.


This completes the proof of theorem:main-tl-section, which 
by proposition:TLtilt, completes the proof of theorem:main.


In addition to the eve base cases 1≤ v ≤ for the induction we have
explicitly seen certain relations in cases of low generation. For example, for
v of generation 1, the description of the endomorphism algebra can be
deduced from the proof of lemma:nil-ploop while the adjacency and
overlap relations hold vacuously. For v of generation 2, we have seen the
adjacency relations in lemma:adjacent-generators-3 and the overlap
relations in lemma:overlapapprox. Finally, zigzag relations for loops
based at w of generation 2 were treated in lem:gen2zigzag.




§ SOME CONCLUSIONS




 §.§ The fractal nature of Z

The quiver underlying  is a graph with countably infinitely many connected components. 
In each connected component there is a unique vertex e-1 with e∈, 
and we denote the vertex set of this component by (e)_.


We have (1)_3={0<4<6<10<12<16<18<22<⋯}, (<ref>).


The decomposition of the quiver implies that the algebra  decomposes as

    = 
    ∐_e∈ _e-1
    ,     _e-1:=⊕_v,w∈(e)_ [w-1][v-1].

Let M_ denote the free monoid on the set L={0,…,-1}.
We represent the elements of M_ by words [b_k,…,b_0] for b_i∈ L 
and the multiplication is given by 

    [b_k,…,b_0]⊙ 
    [a_j,…,a_0]
    :=[a_j,…,a_0,b_k,…,b_0].

(Note that the empty word ∅ is the neutral element.) 
Elements of M_ with differing numbers of leading zeros are considered as distinct, 
and so they should not be interpreted as -adic expansions of natural numbers. 
However,  carries an action of M_ defined by 

    M_×→,  
    [b_k,…,b_0]⊙a_j,…,a_0
    :=a_j,…,a_0,b_k,…,b_0.


The fractal nature, the self-similarity,
of  is now captured by the following proposition.


The monoid M_ acts on  by 
algebra endomorphisms: For each w∈ M_, there is an 
algebra endomorphism ϕ_w∈() acting on idempotents by

    ϕ_w([v-1]) 
    := 
    [w⊙ v-1],

and on arrows by reindexation. Moreover, we have

    ϕ_zϕ_w
    = 
    ϕ_z⊙ w for w,z∈ M_.

Finally, if w=[0,…,0], then ϕ_w maps any _e-1
isomorphically to _w⊙ e-1. 
In this sense,  is generated under the 
action of M_ by the summands _e-1 for 
e∈{1,…,-1}.



This is a direct consequence of theorem:main-tl-section.


We also note that, since  is 
the direct sum of  many copies of _e-1 for 
e∈{1,…,-1}, 
the underlying quiver of  
is a fractal graph in the sense of Ille–Woodrow <cit.>, 
albeit in the trivial sense that any countable graph without edges and more than one vertex can be considered as a fractal factor. 



 §.§ A few words about tilting modules


Let us work over the ground field .
First, recall the category of finite-dimensional modules for  has 
simple (v-1), Weyl (v-1), dual Weyl (v-1) and 
indecomposable tilting modules (v-1) for v∈, the latter being the
indecomposable objects of ,
see <cit.> for a concise summary of 
the main definitions and properties regarding .

Let us now elaborate a bit further on the representation-theoretic implications of corollary:main. 
Almost all of these are, of course, well-understood. 
However, the reader might find it helpful to see how they can be 
derived from our results in the previous sections.

It is well-known that

    =
    ⊕_e∈ _e-1,
      _e-1=
    {(v-1)|
    v-1∈ (e)_},

whose direct summands are called blocks, which are equivalent as additive, -linear 
categories. From our discussion we immediately get the following.


There is an equivalence of additive, -linear categories

    ^'_e-1_e-1,

sending indecomposable tilting modules to indecomposable projectives. 
Moreover, _(X,Y)=0 if 
X∈_e-1 
and Y∈_e^'-1 for e,e^'∈,e≠ e^'.
Finally, there is an isomorphism of algebras _e-1≅_e^'-1 
for all e,e^'∈ with equal non-zero digits.


Directly from theorem:main and corollary:main, combined with 
theorem:main-tl-section and the above.


In fact _e-1≅_e^'-1 
for all e,e^'∈, but such isomorphisms involve non-trivial rescalings in our presentation.

Another consequence we get are the tilting–dual Weyl multiplicities.


We have

    ((v-1):(w-1))
    =
    
    1
       if w∈,
    
    
    0
       else.




Note that the basis theorem:main-tl-section.(Basis) is part of the
family of bases constructed in <cit.>, and the
proposition follows from the construction of these bases in , and our
main statements theorem:main and corollary:main.


Hence, we get the tilting characters 
χ_v-1^=∑_w∈((v-1):(w-1))χ_w-1^,
where the characters χ_w-1^=χ_w-1^ 
are the characteristic zero characters well-known by Weyl's character formula. 
By reciprocity, <cit.>, we also get the Weyl–simple multiplicities. 
To state these explicitly, let us recursively define a set X(v) as follows. For
v≤-1 let X(v)={0}. For v>-1 let

    X(v)=
     X((v-a_0)/)∪(a_0+1+ X((v-a_0-)/))
       if a_0≠-1,
    
     X((v-a_0)/)
       if a_0=-1,

where we again meet .
Then we get

    [(w-1):(v-1)]
    =
    
    1
       if v∈ w-2X(w),
    
    
    0
       else.

Thus, we get χ_w-1^
=∑_v∈[(w-1):(v-1)]χ_v-1^
=∑_v∈ w-X(w)χ_v-1^,
which determines the simple characters by inverting the change of basis matrix.


For =3 we have
χ_22^
=χ_22^+χ_18^+χ_16^+χ_12^, <cit.>.
Moreover, we also get χ_22^=χ_22^+χ_18^+χ_12^+χ_10^.


The final consequence we would like 
to derive in this paper is the following. 


Let I_v={(w-1)| w≥ v}. 
For any thick -ideal I≠ 0 in  there exists 
k∈[0] such that

    I
    =
    I_^k≠2={(v-1)|(_((v-1)))
    ≤ k
    },

with the latter equality holding in case ≠ 2.



We will use that (1) is a -generator of  and
Weyl and dual Weyl modules have classical characters.

Assume that (v-1)∈I for v minimal. Then it is clear by 
proposition:tilting-weyl that I=I_v. Thus, it remains to determine the 
possible minimal v.
To this end, note that 
the decomposition of (1)(w-1)≅(w-1)(1) 
into its indecomposable summands is completely determined by proposition:tilting-weyl 
and the classical SL_2() tensor product combinatorics.
Analyzing now how tensoring with (1) affects the support, 
proposition:tilting-weyl then also implies that v needs to be a prime 
power, and conversely, that a prime power works as a minimal v.

Finally, the last statement follows from proposition:p-properties-trace since 
[e-1]=[e-1]=[e-1] for e∈.


Thus, the thick -ideals in  are
=
I_^0⊃I_^1⊃I_^2⊃I_^3⊃I_^4⊃⋯.


The elements of I_^1 are the so-called negligible modules.


Note that the above implies that  has no projective modules 
since these would form the minimal thick -ideal.



§ TABLE OF NOTATION AND CENTRAL CONCEPTS


In general we use a tilde, f̃, to indicate that we work over , an overline, f, to indicate that we have something that reduces mod  but we want to consider it over , and no extra decoration if we work in [p].



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