Billiards and Tilting Characters for ${\rm SL}_{3}$

Billiards and Tilting Characters for $\boldsymbol{{\rm SL}_{3}}$††This paper
is a contribution to the Special Issue on the Representation Theory of the
Symmetric Groups and Related Topics. The full collection is available at
https://www.emis.de/journals/SIGMA/symmetric-groups-2018.html

George LUSZTIG † and Geordie WILLIAMSON ‡

G. Lusztig and G. Williamson

† Massachusetts Institute of Technology, Cambridge, MA, USA<EMAIL_ADDRESS>http://www-math.mit.edu/~gyuri/

‡ Sydney University, Sydney, NSW, Australia<EMAIL_ADDRESS>http://www.maths.usyd.edu.au/u/geordie/

Received July 18, 2017, in final form February 16, 2018; Published online
February 21, 2018

We formulate a conjecture for the second generation characters of
indecomposable tilting modules for ${\rm SL}_{3}$. This gives many new
conjectural decomposition numbers for symmetric groups. Our conjecture can be
interpreted as saying that these characters are governed by a discrete
dynamical system (“billiards bouncing in alcoves”). The conjecture implies
that decomposition numbers for symmetric groups display (at least) exponential
growth.

tilting modules; billiards; $p$-canonical basis; symmetric group

20C20; 17B10; 20C30

##  1 Introduction

We formulate a conjecture for the second generation characters of
indecomposable tilting modules for $\textrm{SL}_{3}$ in characteristic $p>2$.
These conjectures resulted from our attempts to understand data [25] obtained
following a 9 month calculation in magma on a supercomputer at the MPIM in
Bonn. These results go far beyond existing calculations and are obtained using
a new algorithm [26]. The algorithm relies in an essential way on ideas of
Libedinsky, Riche and the second author (see [15, 21]). The behaviour we
observe appears highly non-trivial, which suggests that proving anything might
be difficult.

On the next page the reader will find a picture. This picture was obtained by
analyzing the output of computer calculations for $p=5$. Exactly how this
picture is used to produce (second generation) tilting characters will be
explained in the final section. Before reading the rest of the paper the
reader is invited to consider this picture and try to discern any patterns.
This paper is an attempt to explain this picture, as well as similar pictures
for $p=3$ and $7$.

Figure 1: Second generation pattern for $p=5$ up to $i=81$.

##  2 Generational philosophy

Let $G$ denote a split simple and simply connected algebraic group over a
field $\Bbbk$ of characteristic $p$. We fix a Borel subgroup and maximal torus
$T\subset B\subset G$. We will try to follow the notation of [24]. In
particular:

$\mathscr{X}$, $\mathscr{X}_{+}$: | weights, dominant weights;
---|---
$\Phi$, $\Phi_{+}$: | roots, positive roots;
$\Phi^{\vee}$, $\Phi_{+}^{\vee}$: | coroots, positive coroots;
$\Sigma$, $\Sigma^{\vee}$: | simple roots, simple coroots;
$\mathscr{X}_{<p}$: |  $p$-restricted weights;
$\rho$, $\alpha_{0}^{\vee}$: | half-sum of $\Phi_{+}$, highest short coroot;
${W_{f}}$, $\mathcal{W}$: | finite Weyl group, affine Weyl group;
$S_{f}$, $\mathcal{S}$: | finite simple reflections, affine simple reflections;
${}^{f}\mathcal{W}$: | minimal coset representatives for ${W_{f}}\setminus\mathcal{W}$;
$\bullet_{p}$, $h$: |  $p$-dilated dot action of $\mathcal{W}$ on $\mathscr{X}$, Coxeter number.

For every dominant weight $\lambda\in\mathscr{X}_{+}$, let $L_{\lambda}$
(resp. $\Delta_{\lambda}$, resp. $T_{\lambda}$) denote the simple (resp.
standard, resp. indecomposable tilting) module with highest weight $\lambda$.

###  2.1 Generations for simple characters

For fixed $p$ and any highest weight $\lambda\in\mathscr{X}_{+}$ the first
author has recently defined characters [17]

$\displaystyle\chi_{\lambda}^{0},\chi_{\lambda}^{1},\chi_{\lambda}^{2},\dots,\chi_{\lambda}^{\infty}\in({\mathbb{Z}}\mathscr{X})^{W_{f}}$

with the following properties:

  1. 1)

$\chi_{\lambda}^{0}$ is the character of the simple highest weight module in
characteristic $0$ (i.e., $\chi_{\lambda}^{0}$ is given by Weyl’s character
formula);

  2. 2)

$\chi_{\lambda}^{1}$ is the character of the simple highest weight module for
the quantum group at a $p^{\rm th}$-root of unity;

  3. 3)

$\chi_{\lambda}^{n}$ is obtained from $\chi_{\lambda}^{n-1}$ by a formula
involving Kazhdan–Lusztig polynomials;

  4. 4)

if $\lambda=\lambda_{0}+p\lambda_{1}+\dots+p^{n}\lambda_{n}$ with
$\lambda_{i}\in\mathscr{X}_{<p}$ and
$\langle\alpha_{0}^{\vee},\lambda_{n}+\rho\rangle\leq p$ (a generalised
Jantzen condition) then
$\chi_{\lambda}^{n}=\chi_{\lambda}^{n+1}=\dots=\chi_{\lambda}^{\infty}$;

  5. 5)

if $p$ is large then $\chi_{\lambda}^{\infty}$ is the character of
$L_{\lambda}$.

######  Remark 2.1.

Recall that the Steinberg tensor product theorem has “one step” for the
quantum group at a root of unity, and “infinitely many steps” for $G$. The
characters $\chi_{\lambda}^{n}$ can be thought of as the simple characters for
an object with an “$n$ step” Steinberg tensor product theorem. It is not known
whether such an object exists. For $\mathfrak{sl}_{2}$ such an object (for any
$n$) has recently been proposed by Angiono [6]. It seems likely that one can
combine recent work of Elias [8] with work of Riche and the second author [21]
to construct such an object in type $A$ for $n=2$.

###  2.2 Generations for tilting characters

As for simple characters, there should exist approximations to tilting
characters. That is, for any $p$ and dominant weight $\lambda\in\mathscr{X}$
there should exist characters

$\displaystyle\theta_{\lambda}^{0},\theta_{\lambda}^{1},\theta_{\lambda}^{2},\dots,\theta_{\lambda}^{\infty}\in({\mathbb{Z}}\mathscr{X})^{W_{f}}$

with the following properties:

  1. 1)

$\theta_{\lambda}^{0}$ is the character of the simple highest weight module in
characteristic $0$ (i.e., $\theta_{\lambda}^{0}$ is given by Weyl’s character
formula);

  2. 2)

$\theta_{\lambda}^{1}$ is the character of the indecomposable tilting module
for the quantum group at a $p^{\rm th}$-root of unity;

  3. 3)

$\theta_{\lambda}^{n}$ is a positive linear combination of the
$\theta_{\mu}^{n-1}$’s for all $n\geq 1$;

  4. 4)

if $\langle\alpha_{0}^{\vee},\lambda+\rho\rangle\leq p^{n+1}$ then
$\theta_{\lambda}^{n}=\theta_{\lambda}^{n+1}=\dots=\theta_{\lambda}^{\infty}$;

  5. 5)

if $p$ is large then $\theta_{\lambda}^{\infty}$ is the character of
$T_{\lambda}$.

######  Remark 2.2.

Parts (4) and (5) for $n=1$ imply Andersen’s conjecture [5], which is still
open (even for $p$ large).

######  Remark 2.3.

The characters $\theta_{\lambda}^{n}$ are defined for a subset of
$\mathscr{X}_{+}$ (roughly, those for which one can apply the tilting tensor
product theorem) in [18].

###  2.3 Our conjecture at $\boldsymbol{v=1}$

Let us assume that $p\geq h$, and consider the $p$-dilated action of the
affine Weyl group from above. For any choice of “generation parameters” $0\leq
m\leq n\leq\infty$ we can write

$\displaystyle\theta^{n}_{x\bullet_{p}0}=\sum_{y\in
W}d_{y,x}^{m,n}\cdot\theta_{y\bullet_{p}0}^{m}$ (2.1)

for some $d_{y,x}^{m,n}\in{\mathbb{Z}}_{\geq 0}$. (The fact that we can form
such expressions follows from the linkage principle and our assumptions
above.) The question of determining characters of indecomposable tilting
characters is equivalent to determining the coefficients $d_{y,x}^{\infty,0}$
for all $y,x\in{}^{f}W$.

######  Remark 2.4.

A philosophy underlying the current paper is that it might be easier to
calculate the $d_{y,x}^{n+1,n}$ for all $n$, rather than calculate the
$d_{y,x}^{\infty,0}$ directly. This is the case for simple characters, as
explained in Section 2.1.

The following is known about these coefficients:

  1. 1)

when $G=\textrm{GL}_{n}$, the $d_{y,x}^{\infty,0}$ are equal to decomposition
numbers for symmetric groups, by work of Donkin [7] and Erdmann [10];

  2. 2)

the $d_{y,x}^{1,0}$ are given as the value at 1 of certain parabolic
Kazhdan–Lusztig polynomials, by work of Soergel [22, 23];

  3. 3)

the $d_{y,x}^{\infty,1}$ are equal to coefficients of James’ “adjustment
matrix” [11].

######  Remark 2.5.

Aside from the above, very little is known. The papers [12, 20] contain some
interesting calculations for $\textrm{SL}_{3}$

In this paper we formulate a conjecture for ($v$-analogues of the)
coefficients $d_{y,x}^{2,1}$, when $G=\textrm{SL}_{3}$. (Note that in the case
of $G=\textrm{SL}_{3}$ the coefficients $d_{y,x}^{1,0}$ given by Soergel’s
algorithm are easily calculated, and may be described by closed formulas.) By
property (4) of the previous section, this provides a conjecture for many
$d_{y,x}^{\infty,1}$, and hence for many coefficients of the adjustment matrix
(and hence decomposition numbers) for three row partitions.

###  2.4 Generations for the $\boldsymbol{p}$-canonical basis

Let $\mathrm{H}$ denote the Iwahori–Hecke algebra of the affine Weyl group
$\mathcal{W}$ over $\mathbb{Z}\big{[}v^{\pm 1}\big{]}$. Consider the anti-
spherical module (see, e.g., [21, 24]):

$\displaystyle\mathrm{AS}_{v}:={\mathrm{sgn}}_{v}\otimes_{\mathrm{H}_{f}}\mathrm{H}=\bigoplus_{x\in{}^{f}\mathcal{W}}{\mathbb{Z}}\big{[}v^{\pm
1}\big{]}n_{x}.$

The anti-spherical module has a canonical basis
$\\{\underline{n}_{x}\,|\,x\in{}^{f}\mathcal{W}\\}$ and a $p$-canonical basis
$\\{{}^{p}\underline{n}_{x}\,|\,x\in{}^{f}\mathcal{W}\\}$ [14, 21].

In [21] (see also [24, Section 2.8]) it is conjectured that if we write

$\displaystyle{}^{p}\underline{n}_{x}=\sum{}^{p}{m_{y,x}}n_{y},$

then the values at 1 of the coefficients ${}^{p}{m_{y,x}}$ express the
characters of tilting modules in terms of Weyl characters.111See [21,
Conjecture 1.7] for a precise formulation. Note that this conjecture is
expected to hold without any restrictions on $p$. Let us give a precise
version of this conjecture for $p\geq h$: for all $x,y\in{}^{f}\mathcal{W}$ we
have (in the notation of Section 2.3)

$\displaystyle d_{y,x}^{\infty,0}={}^{p}{m_{y,x}}(1).$ (2.2)

(Recall that, by the translation principle and still under our assumption
$p\geq h$, knowledge of the left hand side of (2.2) for all
$x,y\in{}^{f}\mathcal{W}$ implies knowledge of the characters of all tilting
modules for $G$.)

######  Remark 2.6.

In [21] this conjecture is proved for $\textrm{GL}_{n}$ and $p>h$. In [9] this
conjecture is proved for $\textrm{GL}_{n}$ for general $p$. In [1, 2, 4] this
conjecture is proved in all types for $p>h$.

The presence of $v$ is the shadow of a non-trivial grading on the category of
tilting modules.222This grading is conjectural in general. In [1, 2, 4, 9, 21]
its existence is established in many cases. It is natural to expect that the
above approximations to tilting characters can also be made to respect this
grading. That is, we expect that for all $p>h$ and $x\in{}^{f}\mathcal{W}$,
there exist elements

$\displaystyle{}^{p}\underline{n}_{x}^{1},{}^{p}\underline{n}_{x}^{2},\dots,{}^{p}\underline{n}_{x}^{\infty}\in\mathrm{AS}_{v}$

such that:

  1. 1)

${}^{p}\underline{n}_{x}^{1}=\underline{n}_{x}$;

  2. 2)

${}^{p}\underline{n}_{x}^{n}$ is a ${\mathbb{Z}}_{\geq 0}[v^{\pm 1}]$-linear
combination of the ${}^{p}\underline{n}_{y}^{n-1}$’s for all $n\geq 1$;

  3. 3)

if $\langle\alpha_{0}^{\vee},x\bullet_{p}0+\rho\rangle\leq p^{n+1}$ then
${}^{p}\underline{n}_{x}^{n}={}^{p}\underline{n}_{x}^{n+1}=\dots={}^{p}\underline{n}_{x}^{\infty}$;

  4. 4)

if $p$ is large then
${}^{p}\underline{n}_{x}^{\infty}={}^{p}\underline{n}_{x}$.

##  3 The new algorithm

In this section we attempt to give the reader some idea of how we perform the
calculations which led to our conjecture. The second author hopes to give more
details in [26]. As explained in the previous section, the main theorems of
[9, 21] (see also [1, 2]) imply that it is enough to calculate the
$p$-canonical basis $\\{{}^{p}\underline{n}_{x}\,|\,x\in{}^{f}W\\}$ in the
anti-spherical module $\mathrm{AS}_{v}$.

Let ${\mathcal{H}}$ denote the diagrammatic Hecke category associated to the
affine Cartan matrix of type $\widetilde{A}_{n}$. Recall that ${\mathcal{H}}$
is built starting from a “realisation”

$\displaystyle\big{(}\\{\alpha_{s}^{\vee}\\}_{s\in
S}\subset{\mathfrak{h}},\\{\alpha_{s}\\}_{s\in
S}\subset{\mathfrak{h}}^{*}\big{)},$

where ${\mathfrak{h}}$ is a free and finite rank ${\mathbb{Z}}$-module,
${\mathfrak{h}}^{*}$ is its dual, and the usual formulas define a
representation of $W$. It will be important below that our realisation is
chosen so that the simple roots $\\{\alpha_{s}\\}_{s\in
S}\subset{\mathfrak{h}}^{*}$ are linearly independent.

The Hecke category is defined over ${\mathbb{Z}}$. After fixing a prime $p$,
extension of scalars yields the Hecke category
${\mathcal{H}}_{{\mathbb{Z}}_{p}}$ defined over the $p$-adic integers. Let
$\mathcal{AS}_{{\mathbb{Z}}_{p}}$ denote the anti-spherical category over
${\mathbb{Z}}_{p}$ considered in [15, 21]. It is a right
${\mathcal{H}}_{{\mathbb{Z}}_{p}}$-module category and one has a canonical
identificiation of right
$[{\mathcal{H}}_{{\mathbb{Z}}_{p}}]_{\oplus}=H$-modules:

$\displaystyle\mathrm{AS}_{v}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}[\mathcal{AS}_{{\mathbb{Z}}_{p}}]_{\oplus},$

where $[-]_{\oplus}$ denotes the split Grothendieck group of an additive
category (see [15, 21] for more details). Under this isomorphism the classes
of the indecomposable self-dual objects yields the $p$-canonical basis in
$\mathrm{AS}_{v}$.

The anti-spherical category $\mathcal{AS}_{{\mathbb{Z}}_{p}}$ is defined by
explicit generators and relations. Thus the question of determining the
characters of its indecomposable objects (and hence the $p$-canonical basis)
is a concrete question of finding idempotents in certain finite rank
${\mathbb{Z}}_{p}$-algebras. However these diagrammatic calculations are
prohibitively difficult in all but the simplest cases.

Instead we exploit the philosophy of localisation. To
$\mathcal{AS}_{{\mathbb{Z}}_{p}}$ are associated progressively simpler
categories:

$\displaystyle\mathcal{AS}_{{\mathbb{Z}}_{p}}\rightsquigarrow\mathcal{AS}_{{\mathbb{Q}}_{p}}\rightsquigarrow
Q\otimes_{R}\mathcal{AS}_{{\mathbb{Q}}_{p}},$

where $\rightsquigarrow$ denotes some form of localisation. The first
localisation $\mathcal{AS}_{{\mathbb{Q}}_{p}}$ is obtained from
$\mathcal{AS}_{{\mathbb{Z}}_{p}}$ by inverting $p$. The second is the main
object of study of [15]: the polynomial ring
$R={\mathbb{Q}}_{p}[\widetilde{\alpha}]$ ($\widetilde{\alpha}$ denotes the
affine simple root) acts on the left on all hom spaces in
$\mathcal{AS}_{{\mathbb{Q}}_{p}}$, and after inverting $\widetilde{\alpha}$
one obtains $Q\otimes_{R}\mathcal{AS}_{{\mathbb{Q}}_{p}}$, where $Q$ denotes
${\mathbb{Q}}_{p}(\widetilde{\alpha})$. Somewhat surprisingly, this category
is semi-simple [15]. (This is analogous to deformed category $\mathcal{O}$,
which is semi-simple for generic parameters. A big difference in the current
setting is that the deformation ring $R$ is always one dimensional.)

The algorithm we used to calculate the $p$-canonical basis now proceeds in two
steps:

  1. 1.

Firstly, $\mathcal{AS}_{{\mathbb{Q}}_{p}}$ is described as a quiver with
relations, and the action of the Hecke category
${\mathcal{H}}_{{\mathbb{Z}}_{p}}$ on $\mathcal{AS}_{{\mathbb{Q}}_{p}}$ is
described explicitly, in terms of the quiver. This is already a very non-
trivial task, and is only feasible for $\textrm{SL}_{2}$, $\textrm{SL}_{3}$
and perhaps $\textrm{SL}_{4}$. It is possible in these cases thanks to the
localisation $Q\otimes_{R}\mathcal{AS}_{{\mathbb{Q}}_{p}}$ (where any
calculation can be reduced to a calculation in matrices with entries in $Q$),
the fact that the Kazhdan–Lusztig conjectures hold in
${\mathcal{A}}_{{\mathbb{Q}}_{p}}$ (thus one has graded dimensions for hom
spaces and knows how the generators of ${\mathcal{H}}_{{\mathbb{Z}}_{p}}$ act
on the Grothendieck group etc.), and the fact that the Kazhdan–Lusztig theory
of these anti-spherical modules in low rank is not too complicated
(Kazhdan–Lusztig polynomials can be written down explicitly).

  2. 2.

Secondly, via the above localisations one can describe
$\mathcal{AS}_{{\mathbb{Z}}_{p}}$ as a ${\mathbb{Z}}_{p}$-lattice inside
$\mathcal{AS}_{{\mathbb{Q}}_{p}}$. Moreover, this lattice is the smallest
lattice which contains the generating object and is stable under the action of
the Hecke category ${\mathcal{H}}_{{\mathbb{Z}}_{p}}$. One may describe this
lattice explicitly and inductively via the
${\mathcal{H}}_{{\mathbb{Z}}_{p}}$-action, using the philosophy of the light
leaf basis. Details of how this is done in this setting will be contained in
[26].

######  Remark 3.1.

Recently, L.T. Jensen has done calculations which omit the first localisation
and calculate the $p$-canonical basis using only the localisation
$Q\otimes_{R}\mathcal{AS}_{{\mathbb{Q}}_{p}}$. This is a much simpler
approach, and appears (much to the second author’s surprise!) not to be any
slower than the approach described above. It may well be that Jensen’s
modification of the algorithm ends up being the more effective.

##  4 Billiards

For the rest of the paper we fix $G=\textrm{SL}_{3}$,
$\mathscr{X}={\mathbb{Z}}\varpi_{1}\oplus{\mathbb{Z}}\varpi_{2}$,
$\Sigma=\\{\alpha_{1},\alpha_{2}\\}$ etc.

Fix $\ell\geq 1$. (For applications to representation theory $\ell$ will be
prime, however for the moment $\ell$ can be any positive integer. Soon we will
assume $\ell\geq 3$.) Elements of the set

$\displaystyle{\mathcal{M}}:={\mathbb{Z}}_{\geq
0}\times\big{\\{}v^{k}\,|\,k\in{\mathbb{Z}}\big{\\}}$

will be called _labels_. Labels will often be denoted by an integer followed
by a bracketed power of $v$; e.g., $51(v^{7})$.

A _labelled point_ is an element of $\mathscr{X}\times{\mathcal{M}}$. Labelled
points will usually be denoted $(\mu,m)$ with $\mu\in\mathscr{X}$ and
$m\in{\mathcal{M}}$ or $(\mu,n,v^{k})$ with $\mu\in\mathscr{X}$ and
$n\in{\mathbb{Z}}_{\geq 0}$.

Our goal in this section is to describe an algorithm which produces a multiset
(i.e., set with multiplicities) of labelled points via an inductive procedure.
Throughout this section we work entirely with multisets. All operations
(union, difference, …) are to be understood in the context of multisets.

###  4.1 Overview of the algorithm

We view the dominant weights $\mathscr{X}_{+}$ as the vertices of a directed
graph $\Gamma$ with edges $\lambda\rightarrow\lambda+\gamma$ if
$\lambda,\lambda+\gamma\in\mathscr{X}_{+}$ and
$\gamma\in\\{\varpi_{1},\varpi_{2}-\varpi_{1},-\varpi_{2}\\}$:

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Our algorithm consists of three steps. We begin with the labelled point
$\big{(}0,0,v^{0}\big{)}$ (our initial “seed”). Each step in our algorithm
enlarges our multiset in a new direction, starting with seeds generated in the
previous step. The first step extends our set along a wall of the dominant
chamber in the direction of $\varpi_{1}$ to produce a set $X$. The second step
extends our set along the walls of the $\ell$-alcoves to produce a multiset
$Y$. Finally, the third step extends our multiset within the interior of each
$\ell$-alcove to produce a multiset $Z$. In the third step, certain labelled
points arising in the second step (seeds) gives rise to two spirals inside the
interior of two adjacent alcoves which move like a billiard. Finally we
consider

$\displaystyle\widetilde{Z}:=Z\setminus X,$

which is the object of our conjecture.

###  4.2 Step 1: the wall of the dominant chamber

Consider the full subgraph of $\mathscr{X}_{+}$ consisting of multiples of
$\varpi_{1}$:

$0$$\varpi_{1}$

Consider a labelled point $m=\big{(}\mu,m\big{(}v^{k}\big{)}\big{)}$ such that
$\mu$ belongs to this full subgraph. A  _small step_ produces the labelled
point $\big{(}\mu^{\prime},(m+2)\big{(}v^{k}\big{)}\big{)}$ where
$\mu\rightarrow\mu^{\prime}$ is the unique edge with source $\mu$. Consider
the set $X$ obtained by repeatedly taking small steps beginning with the
labelled point $\big{(}0,0\big{(}v^{0}\big{)}\big{)}$. In other words, $X$ is
the set

$\displaystyle
X=\big{\\{}\big{(}k\omega_{1},2k\big{(}v^{0}\big{)}\big{)}\,|\,k\in{\mathbb{Z}}_{\geq
0}\big{\\}}.$ (4.1)

We designate the labelled points of the form $(\ell k,2k\ell(v^{0}))$ with
$k>0$ as _seeds_.

######  Remark 4.1.

Of course we could have defined $X$ via (4.1) directly. We prefer the
inductive definition, as it is closer in spirit to the more complicated
definitions which will occur in the next two steps.

###  4.3 Step 2: dynamics on the walls

From now on assume that $\ell\geq 3$. Consider the subgraph of $\Gamma$ with
vertices $\lambda\in\mathscr{X}_{+}$ such that
$\langle\lambda,\alpha^{\vee}\rangle\in\ell{\mathbb{Z}}$ for some
$\alpha\in\Phi_{+}$ and edges $(\lambda,\lambda^{\prime})$ such that
$\langle\lambda,\alpha^{\vee}\rangle=\langle\lambda^{\prime},\alpha^{\vee}\rangle\in\ell{\mathbb{Z}}$
for some $\alpha\in\Phi_{+}$. Let $\Gamma_{\textrm{wall}}$ denote the graph
obtained by removing the gray edges and vertices as in Fig. 2.

Figure 2: The directed graph $\Gamma_{\textrm{wall}}$.

Points in $\mathscr{X}_{+}$ or $\Gamma_{\textrm{wall}}$ such that
$\langle\lambda,\alpha^{\vee}\rangle\in\ell{\mathbb{Z}}$ for all
$\alpha\in\Phi_{+}$ are called _corner points_. A point $\mu$ in
$\mathscr{X}_{+}$ of $\Gamma_{\textrm{wall}}$ is an _almost corner_ if there
exists a corner point $c$ and an arrow $c\rightarrow\mu$ in $\mathscr{X}_{+}$.

Fix a labelled point $\big{(}\mu,n\big{(}v^{k}\big{)}\big{)}$ with
$\mu\in\Gamma_{\textrm{wall}}$. We assume that $\mu$ is such that there is a
unique edge with source $\mu$. From $\big{(}\mu,n\big{(}v^{k}\big{)}\big{)}$
one may obtain new labelled points as follows:

  1. 1.

A _rest_ produces the labelled point
$\big{(}\mu,(n+3)\big{(}v^{k+1}\big{)}\big{)}$.

  2. 2.

A _small step_ produces (as above) the labelled point
$\big{(}\mu^{\prime},(n+2)\big{(}v^{k}\big{)}\big{)}$, where
$\mu\rightarrow\mu^{\prime}$ is the unique edge in $\Gamma_{\textrm{wall}}$
with source $\mu$.

  3. 3.

A _giant leap_ produces either one or two new labelled points, and is only
possible if $\mu$ is not a corner or almost corner point. Define $d$ to be the
direction of the unique arrow with source $\mu$. First we proceed $j<\ell-1$
steps in the graph $\Gamma_{\textrm{wall}}$ in direction $d$ until we reach a
corner point. We then proceed for another $\ell-1-j$ steps in all directions
from the corner point which do not agree with the direction $d$. (There are
either one or two such directions.) A giant leap consists of the resulting
points, which are labelled by $(n+2\ell+1)\big{(}v^{k+1}\big{)}$ (see Fig. 3).

$\mu$$n(v^{k})$$\mu^{\prime\prime}$$(n+2\ell+1)(v^{k+1})$$\mu^{\prime}$$(n+2\ell+1)(v^{k+1})$or$\mu$$n(v^{k})$$\mu^{\prime}$$(n+2\ell+1)(v^{k+1})$
Figure 3: Performing a giant leap with $\ell=5$. (Squares denote corner
points.)

Now suppose that we are given a labelled point $q=(\mu,n,v^{k})$ as above
(i.e., such that there is a unique edge in $\Gamma_{\textrm{wall}}$ with
source $\mu$). We now describe a way of producing a multiset of labelled
points (all with multiplicity 1) beginning with $q$, some of which are
designated as _seeds_ :

  1. 1.

_If $\mu$ is a corner point._ We produce $\ell$ new points as follows: we take
$\ell-1$ small steps to produce labelled points $m=m_{1},\dots,m_{\ell-1}$ and
then rest to produce a new labelled point $m_{\ell}$. The final labelled point
$m_{\ell}$ is designated as a seed. (See Fig. 4.)

  2. 2.

_If $\mu$ is an almost corner point._ We produce $\ell$ new points as follows:
let $m=m_{1}$ and rest once to produce a new labelled point $m_{2}$, now take
$\ell-2$ small steps starting with $m_{2}$ to produce $\ell-2$ labelled points
$m_{3},\dots,m_{\ell-1}$ and finally rest once more to produce a final
labelled point $m_{\ell}$. The final labelled point $m_{\ell}$ is designated
as a seed. (See Fig. 4.)

  3. 3.

_If $\mu$ is neither a corner nor an almost corner point._ We take a giant
leap to produce one or two new labelled points, each of which are designated
seeds.

We now iterate this process as follows. Starting with $q$ we produce a
sequence of multisets $Q_{1},Q_{2},Q_{3},\dots$, where $Q_{1}$ (resp. $Q_{i}$
for $i>1$) is obtained by applying the above procedure to $q$ (resp. to each
seed in $Q_{i-1}$). We say that the union (as multisets)

$\displaystyle Q=\bigcup_{i\geq 0}Q_{i}$

is the result of _applying dynamics on the walls to the seed $q$_.

$\underline{10(v^{0})}$$12(v^{0})$$14(v^{0})$$16(v^{0})$$18(v^{0})$$\underline{21(v^{1})}$

$\underline{54(v^{4})}$$57(v^{5})$$59(v^{5})$$61(v^{5})$$63(v^{5})$$\underline{66(v^{6})}$

Figure 4: An illustration of cases (1) and (2) when $\ell=5$. Seeds are
underlined.

######  Example 4.2.

Fig. 5 gives an example with $\ell=5$. Our initial seed is the unique point
with label $10\big{(}v^{0}\big{)}$. We illustrate a few iterations of the
above algorithm. Seeds are underlined. Note that one more iteration of the
algorithm will produce the labelled point
$\big{(}3\varpi_{1}+7\varpi_{2},88\big{(}v^{8}\big{)}\big{)}$ with
multiplicity 2.

$\underline{32(v^{2})}$$\underline{77(v^{7})}$$\underline{54(v^{4})}$$57(v^{5})$$59(v^{5})$$61(v^{5})$$63(v^{5})$$\underline{66(v^{6})}$$\underline{10(v^{0})}$$12(v^{0})$$14(v^{0})$$16(v^{0})$$18(v^{0})$$\underline{21(v^{1})}$$\underline{54(v^{4})}$$57(v^{5})$$59(v^{5})$$61(v^{5})$$63(v^{5})$$\underline{66(v^{6})}$$\underline{43(v^{3})}$$\underline{77(v^{7})}$$\underline{43(v^{3})}$$\underline{77(v^{7})}$
Figure 5: Dynamics on the walls with $\ell=5$.

Now consider our set $X$ from the previous step. We apply dynamicson walls to
each seed in $X$ (i.e., each labelled point of the form
$\big{(}k\ell\varpi_{1},2k\ell\big{(}v^{0}\big{)}\big{)}$ with $k>0$) to
generate a multiset $Y_{k}$. We now define a new multiset

$\displaystyle Y:=X\cup\bigcup_{k>0}Y_{k}.$

We also remember (for the purposes of the next step) which elements of $Y$
were designated seeds.

######  Remark 4.3.

Consider dynamics on the wall restricted to the full subgraph displayed in
Fig. 6 with seed $q=\big{(}\mu,n\big{(}v^{k}\big{)}\big{)}$. Then $4$
iterations of the above algorithm yields the following points and labels (see
Fig. 6):

$\displaystyle u,u^{\prime}\quad\text{with
label}\quad(n+11)\big{(}v^{k+1}\big{)},$ $\displaystyle
u_{i},u_{i}^{\prime}\quad\text{with
label}\quad(n+14+2(i-1))\big{(}v^{k+2}\big{)}\quad\text{for }{i=1,2,3,4},$
$\displaystyle v,v^{\prime}\quad\text{with
label}\quad(n+23)\big{(}v^{k+3}\big{)},$ $\displaystyle
w,w^{\prime}\quad\text{with label}\quad(n+34)\big{(}v^{k+4}\big{)},$

and then $\mu$ with label $(n+45)\big{(}v^{k+5}\big{)}$ and multiplicity 2.
(The point is that there are two directed paths leading from $\mu$ back to
itself, which causes the multiplicity to double.) Repeating this algorithm
$4i$ times leads to $q_{i}:=\big{(}\mu,n+45i\big{(}v^{k+5i}\big{)}\big{)}$
with multiplicity $2^{i}$. From this observation one deduces easily that for
$\ell=5$ the set $Y$ contains labelled points whose multiplicity grows
exponentially in $n$. Similar considerations show that the same statement
about $Y$ is true for any $\ell$.

$q$$w$$u$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$v$$w^{\prime}$$u^{\prime}$$u_{1}^{\prime}$$u_{2}^{\prime}$$u_{3}^{\prime}$$u_{4}^{\prime}$$v^{\prime}$
Figure 6: Subgraph demonstrating exponential growth of multiplicities for
$\ell=5$.

###  4.4 Step 3: billiards in an alcove

We now describe an algorithm which produces for each seed in $Y$ a multiset of
labelled points. We need a little more notation. We consider the dominant
weights $\mathscr{X}_{+}$ as embedded in the vector space

$\displaystyle\mathscr{X}_{\mathbb{R}}:=\mathscr{X}\otimes_{{\mathbb{Z}}}{\mathbb{R}},$

which we regard as a Euclidean space for some ${W_{f}}$-invariant bilinear
form. An _$\ell$ -alcove_ is a connected component of the complement

$\displaystyle\mathscr{X}_{{\mathbb{R}}}\setminus\bigcup_{\alpha\in\Phi_{+}^{\vee}}\big{\\{}\lambda\,|\,\langle\alpha^{\vee},\lambda\rangle\in\ell{\mathbb{Z}}\big{\\}}.$

An _$\ell$ -regular point_ is a point which belongs to some $\ell$-alcove. An
$\ell$-alcove is _dominant_ if it is contained in the real cone generated by
$\mathscr{X}_{+}$.

Now consider a seed $q=(\mu,m)\in Y$. We want to associate a multiset of
labelled points $\Lambda_{q}$ to $q$. If $\mu$ is a corner or almost corner,
we set $\Lambda_{q}:=\varnothing$.

From now on we assume that $q$ is neither a corner or almost corner. In this
case $\mu$ belongs to the closure of two dominant $\ell$-alcoves, $A^{\prime}$
and $A^{\prime\prime}$. We perform an identical procedure for both alcoves, so
fix one such alcove and call it $A$. Let $\Delta$ denote the full subgraph of
$\Gamma$ consisting of points belonging to $\overline{A}$. Then $\Delta$ has
one of the following forms:

$\ell+1$points

Points of $\Delta$ belong to $\overline{A}$ but not to $A$ are called _wall
points_. Note that our starting point $\mu$ is a wall point.

_Variables and their initialisations:_ Let
$(\mu_{\textrm{start}},m_{\textrm{start}}):=(\mu,m)$ and let
$d_{\textrm{start}}$ denote the unique edge of $\Delta$ with source $\mu$ and
target an interior point of $\Delta$. To begin with, set
$\Lambda:=\\{(\mu_{\textrm{start}},m_{\textrm{start}})\\}$. In the algorithm
we need the following variables, which are initialised as follows:

$\displaystyle\textrm{CurrentPoint}:=\mu_{\textrm{start}},\qquad$
$\displaystyle\textrm{CurrentLabel}:=m_{\textrm{start}},$
$\displaystyle\textrm{CurrentDirection}:=d_{\textrm{start}},\qquad$
$\displaystyle\textrm{CurrentSign}:=1.$

_The loop:_ The multiset of labelled points is obtained by repeating the
following ad infinitum. Let $\mu_{\textrm{new}}$ denote the point obtained by
moving from CurrentPoint one step in CurrentDirection. Write CurrentLabel as
$n\big{(}v^{k}\big{)}$. Two possibilities may occur:

  1. 1.

$\mu_{\textrm{new}}$ is $\ell$-regular:

$\displaystyle\textrm{CurrentLabel}:=(n+2)\big{(}v^{k}\big{)},\qquad\textrm{CurrentPoint}:=\mu_{\textrm{new}},$
$\displaystyle\text{add $(\textrm{CurrentPoint},\textrm{CurrentLabel})$ to
$\Lambda$}.$

(The variables CurrentDirection and CurrentSign remain unchanged.)

  2. 2.

CurrentPoint is $\ell$-regular and $\mu_{\textrm{new}}$ is a wall point:

$\displaystyle\textrm{CurrentLabel}:=(n+3)\big{(}v^{k+\textrm{CurrentSign}}\big{)},\qquad\\!\\!$
$\displaystyle\textrm{CurrentDirection}:=d_{\textrm{new}},$
$\displaystyle\textrm{CurrentSign}:=-\textrm{CurrentSign},\qquad\\!\\!$
$\displaystyle\text{add $(\textrm{CurrentPoint},\textrm{CurrentLabel})$ to
$\Lambda$}.$

Here $d_{\textrm{new}}$ denotes the vector obtained by reflecting
CurrentDirection in the wall on which $\mu_{\textrm{new}}$ lies. (The variable
CurrentPoint remains unchanged.)

######  Example 4.4.

We illustrate the first few steps of the algorithm in some examples. In each
case the starting point $\mu_{\textrm{start}}$ and its label are underlined.
The successive values of CurrentLabel and CurrentPoint are obtained by
following the arrows.

  1. 1.

A reasonably generic example with $\ell=11$:

$(*)$$16(v)$$18(v)$$20(v)$$22(v)$$24(v)$$26(v)$$28(v)$$31(v^{2})$$33(v^{2})$$35(v^{2})$$38(v)$$40(v)$$42(v)$$44(v)$$46(v)$$48(v)$$50(v)$$53(v^{2})$$55(v^{2})$$57(v^{2})$$60(v)$$62(v)$$64(v)$$66(v)$$68(v)$$70(v)$$72(v)$$75(v^{2})$$77(v^{2})$$\dots$$\underline{14(v)}$

If we focus on the point marked with an asterix (*) we obtain the labels:

$\displaystyle\big{\\{}16(v),79\big{(}v^{2}\big{)},82(v),145\big{(}v^{2}\big{)},148(v),211\big{(}v^{2}\big{)},214(v),277\big{(}v^{2}\big{)},280(v),\dots\big{\\}}.$

  2. 2.

An interesting example with $\ell=5$ (we display both alcoves):

$*$$*$$*$$*$$*$$*$$*$$*$

$\dots$

$*$$*$$*$$*$$*$$\underline{21(v)}$$*$$*$$*$$*$$*$$\dots$$*$$*$$*$$*$$*$$*$$*$$*$

In either alcove the sequence of labels obtained by following the arrows
beginning at $21(v)$ is as follows:

$\displaystyle
23(v),25(v),27(v),30{\mathfrak{i}}(v^{2}\big{)},33(v),35(v),37(v),40\big{(}v^{2}\big{)},$
$\displaystyle\qquad 43(v),45(v),47(v),50\big{(}v^{2}\big{)},53(v),\dots\,.$

  3. 3.

Another example with $\ell=5$ (again we display both alcoves):

$*$

…

$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$$*$

…

$\underline{77(v^{7})}$

In either alcove the sequence beginning at $77\big{(}v^{7}\big{)}$ is

$\displaystyle
79\big{(}v^{7}\big{)},81\big{(}v^{7}\big{)},84\big{(}v^{8}\big{)},86\big{(}v^{8}\big{)},89\big{(}v^{7}\big{)},91\big{(}v^{7}\big{)},\dots\,.$

  4. 4.

Finally, an example with $\ell=3$ (again we display both alcoves):

$\underline{13(v)}$$*$$*$$*$$*$$\dots$$*$$*$$*$$*$

$\dots$

The sequence at either middle vertex is: $15(v)$, $18(v^{2})$, $21(v)$,
$24(v^{2})$, $27(v)$, $\dots$.

Now, with $q=(\mu,m)$ as above (so that $\mu$ is neither a corner or almost
corner) we apply the above algorithm to both alcoves which contain $\mu$ in
their closure to produce multisets $\Lambda^{\prime}$ and
$\Lambda^{\prime\prime}$. We then remove $q$ from both, and define
$\Lambda_{q}$ to be the union of the resulting multisets. (Thus $\Lambda_{q}$
contains infinitely many points from both alcoves, but does not contain $q$
itself.)

The above algorithm produces, for each each seed $q\in Y$, a multiset
$\Lambda_{q}$. We define $Z$ to be the union

$\displaystyle Z:=Y\cup\bigcup_{\text{seeds }q\in Y}\Lambda_{q}.$

Finally, we define $\widetilde{Z}:=Z\setminus X$.

######  Example 4.5.

In Fig. 1 we display $\widetilde{Z}$ for $\ell=5$ and all labelled points
$\big{(}\mu,n\big{(}v^{k}\big{)}\big{)}$ with $n\leq 82$. In the notation of
Section 6, the labelled points corresponding to $\mu\in\mathscr{X}_{+}$ are
displayed in the smaller of the two alcoves contained in $B_{\mu}$. The reader
is referred to [25] for further examples.

######  Remark 4.6.

For all $\big{(}\mu,n\big{(}v^{k}\big{)}\big{)}\in\widetilde{Z}$ it is easy to
see that
$\mu\in\mathscr{X}_{++}:={\mathbb{Z}}_{>0}\varpi_{1}\oplus{\mathbb{Z}}_{>0}\varpi_{2}$.

##  5 An altenative construction

In this section we outline an alternative construction of the multiset
$\widetilde{Z}$, which is more natural in some respects.

In the previous section we regarded the dominant weights $\mathscr{X}_{+}$ as
the vertices of a directed graph. An important difference in the current
construction is that now we consider _all_ weights $\mathscr{X}$. That is, we
consider the elements of $\mathscr{X}$ as the vertices of a directed graph
with edges $\lambda\rightarrow\lambda+\gamma$ if
$\lambda,\lambda+\gamma\in\mathscr{X}_{+}$ and
$\gamma\in\\{\varpi_{1},\varpi_{2}-\varpi_{1},-\varpi_{2}\\}$. We write
$\mu\rightarrow\mu^{\prime}$ to indicate that there is a directed edge from
$\mu$ to $\mu^{\prime}$.

We use similar terminology to earlier: the notions of a _wall point_ , _corner
point_ , _almost corner_ and _$\ell$ -regular point_ extend in an obvious way
to $\mathscr{X}$. Note that almost corners are necessarily wall points.

###  5.1 Trees

For us a _rooted tree_ is what many call an “arborescence”: a directed graph
with a distinguished vertex (the _source_) such that there is a unique
directed path from the source to any other vertex.333This is equivalent in an
obvious way to the usual notion of rooted tree.

Given two rooted trees $A$ and $B$, denote by $A*B$ the rooted tree obtained
by adjoining one copy of $B$ to each sink in $A$ at the source of $B$. That
is, if $t_{1},\dots,t_{k}$ denote the sinks of $A$ and we denote by
$b_{1},\dots,b_{k}$ the sources in $B\sqcup B\sqcup\dots\sqcup B$ ($k$
factors), then

$\displaystyle A*B:=(A\sqcup B\sqcup B\sqcup\cdots\sqcup B)/(t_{i}\sim
b_{i}).$

This operation is associative. The image of $A$ (resp. a copy of $B$) under
this quotient map will be called a _component_ of type $A$ (resp. $B$).

Recall that $\ell\geq 3$ is fixed. Consider the two rooted trees:

$I:=$……………$J:=$012$\ell-1$$\ell$$\ell+1$

The integers indicate the distance from the unique source in each graph.

######  Example 5.1.

We illustrate the operation $*$ with our graphs $I$ and $J$. For $\ell=3$, the
graph $I*J*J$ looks as follows:

$\dots$$\dots$$\dots$$\dots$$\dots$$\dots$

Now define:

$\displaystyle
J_{\infty}:=\big{(}I*J^{*\ell-1}\big{)}^{*\infty}:=\big{(}I*J^{*\ell-1}\big{)}*\big{(}I*J^{*\ell-1}\big{)}*\big{(}I*J^{*\ell-1}\big{)}*\cdots.$

Recall the notion of component from above. We say that $v\in J_{\infty}$ is of
_wall type_ if it lies in a component of type $I$ or is the source in a
component of type $J$. We say that $v$ is of _almost corner type_ if it is the
source in a component of type $I$. (Thus if $v\in J_{\infty}$ is of almost
corner type then it is also of wall type.)

######  Example 5.2.

We continue Example 5.1 with $\ell=3$. The points of wall type (resp. of
almost corner type) in $J_{\infty}$ are depicted as open (resp. filled)
circles:

……$\dots$……$\dots$……$\dots$……$\dots$$\dots$$\dots$

###  5.2 The map

Fix an almost corner $\lambda$. We will construct a map

$\displaystyle\Phi=\Phi_{\lambda}\colon\ J_{\infty}\rightarrow\mathscr{X}.$

It will have the following two properties:

  1. 1.

If $v\rightarrow v^{\prime}$ in $J_{\infty}$ then either
$\Phi(v)=\Phi(v^{\prime})$ or $\Phi(v)\rightarrow\Phi(v^{\prime})$ in
$\mathscr{X}$.

  2. 2.

For $v\in J_{\infty}$, $\Phi(v)$ is a wall point (resp. an almost corner) if
and only if $v$ is of wall type (resp. of almost corner type).

Because $J_{\infty}$ is the union of components of types $I$ or $J$ it will
suffice to define the image of $\Phi$ on the source of $J_{\infty}$, and then
define it on each component of type $I$ or $J$ inductively. This is what we do
now:

_Rule $0$:_ Let $v_{0}$ denote the source of $J_{\infty}$. Set
$\Phi(v_{0})=\lambda$ (our fixed almost corner).

_Rule $1$:_ Let $I^{\prime}\subset J_{\infty}$ denote a component of type $I$
and let $v_{0}^{\prime}$ denote the source of $I^{\prime}$. Suppose that
$\mu:=\Phi(v_{0}^{\prime})$ is defined, but that $\Phi$ is not defined on the
rest of $I^{\prime}$. We define $\Phi$ on $I^{\prime}$ as follows:

$\displaystyle\mu_{1}\rightarrow\mu_{1}\rightarrow\mu_{2}\rightarrow\dots\rightarrow\mu_{\ell-2}\rightarrow\mu_{\ell-1}\rightarrow\mu_{\ell-1}.$

Here $\mu=\mu_{1},\mu_{2},\dots,\mu_{\ell}$ are uniquely determined by the
requirement that $\mu=\mu_{1}$ and
$\mu_{1}\rightarrow\mu_{2}\rightarrow\cdots\rightarrow\mu_{\ell-1}$ are all
wall points. (The fact that $\mu_{1},\dots,\mu_{\ell-1}$ are well defined is a
consequence of property (2) above: $\Phi(v_{0}^{\prime})$ is an almost
corner.)

_Rule $2$:_ Let $J^{\prime}\subset J_{\infty}$ denote a component of type $J$
and let $v_{0}^{\prime}$ denote the source of $J^{\prime}$. Suppose that
$\mu:=\Phi(v_{0}^{\prime})$ is defined, but that $\Phi$ is not defined on all
of $J^{\prime}$.

By property (2) above $\mu$ is a wall-point but is not an almost corner. Hence
there exist two edges $d^{\prime},d^{\prime\prime}$ with source $\mu$ and
image an $\ell$-regular point. For $d\in\\{d^{\prime},d^{\prime\prime}\\}$,
define a sequence $\\{(\mu_{i},d_{i})\\}_{i\geq 0}$ as follows:

  1. 1)

$(\mu_{0},d_{0}):=(\mu,d)$;

  2. 2)

if $\mu_{i}+d_{i}$ is an interior point, define
$(\mu_{i+1},d_{i+1}):=(\mu_{i}+d_{i},d_{i})$;

  3. 3)

if $\mu_{i}+d_{i}$ is a wall point, define
$(\mu_{i+1},d_{i+1}):=(\mu_{i},r(d_{i}))$, where $r$ denotes the reflection in
the (unique) wall containing $\mu_{i}+d_{i}$.

Denote by $\\{(\mu^{\prime}_{i},d^{\prime}_{i})\\}_{i\geq 0}$ (resp.
$\\{(\mu^{\prime\prime}_{i},d^{\prime\prime}_{i}))\\}_{i\geq 0})$ the
sequences associated to $d=d^{\prime}$ (resp. $d=d^{\prime\prime}$).

We define $\Phi$ on $J^{\prime}$ as follows:

$\mu^{\prime}_{0}=\mu^{\prime\prime}_{0}$$\mu^{\prime}_{1}$$\mu^{\prime\prime}_{1}$$\mu^{\prime}_{2}$$\mu^{\prime\prime}_{2}$……$\mu^{\prime}_{\ell-1}$$\mu^{\prime\prime}_{\ell-1}$$\mu^{\prime}_{\ell}$$\mu^{\prime\prime}_{\ell}$$\mu^{\prime}_{\ell+1}$$\mu^{\prime\prime}_{\ell+1}$……$\mu^{\prime}_{\ell-1}+d^{\prime}_{\ell-1}$$\mu^{\prime\prime}_{\ell-1}+d^{\prime\prime}_{\ell-1}$

(Note that $\mu^{\prime}_{\ell-1}+d^{\prime}_{\ell-1}$ and
$\mu^{\prime\prime}_{\ell-1}+d^{\prime\prime}_{\ell-1}$ are wall points.)

In order for the above definition to be well defined the we should check that
properties (1) and (2) are satisfied at each step. properties (1) and (2) for
wall points are immediate from the definitions. Property (2) for almost corner
points follows from the following observation: suppose that $J^{\prime}\subset
J_{\infty}$ is a component of type $J$, let $v_{0}^{\prime}$ denote its
source, and let $v_{1}^{\prime}$, $v_{1}^{\prime\prime}$ denote the two sinks
(so that $v_{1}^{\prime}$ and $v_{1}^{\prime\prime}$ are of wall type in
$J_{\infty}$). Then if $\Phi(v_{0}^{\prime})$ is of distance $k\leq\ell-1$
from a corner point, then $\Phi(v_{1}^{\prime})$ and
$\Phi(v_{1}^{\prime\prime})$ are of distance $k-1$ from a corner point.

######  Example 5.3.

We illustrate $\Phi$ on a component of type $J$ with $\ell=5$ (the source is
marked with a circle):

$*$

…

$*$$*$$*$$*$$*$$*$**$*$$*$$*$$*$$*$$*$$*$

…

Note the similarity to the previous section. The only difference is the arrow
joining an $\ell$-regular point to a wall point.

###  5.3 Extending $\boldsymbol{\Phi}$

Let $v_{0}$ denote the source of $J_{\infty}$. For any choice of label
$n\big{(}v^{k_{0}}\big{)}$, we inductively extend $\Phi$ to produce a map

$\displaystyle\widetilde{\Phi}=\widetilde{\Phi}_{\lambda,n(v^{k_{0}})}\colon\
J_{\infty}\rightarrow\mathscr{X}\times{\mathcal{M}}$

such that
$\widetilde{\Phi}(v_{0})=\big{(}\lambda,n\big{(}v^{k_{0}}\big{)}\big{)}$. We
proceed as follows: Suppose that $v\rightarrow v^{\prime}$ in $J_{\infty}$ and
that $\widetilde{\Phi}$ is defined on $v$ but not on $v^{\prime}$, and let
$\widetilde{\Phi}(v)=\big{(}\Phi(v),m\big{(}v^{k}\big{)}\big{)}$. Then
$\widetilde{\Phi}(v^{\prime})=\big{(}\Phi(v^{\prime}),m^{\prime}\big{(}v^{k^{\prime}}\big{)}\big{)}$
with

$\displaystyle m^{\prime}:=\begin{cases}m+2&\text{if
$\Phi(v)\neq\Phi(v^{\prime})$,}\\\ m+3&\text{if
$\Phi(v)=\Phi(v^{\prime})$}\end{cases}$

and

$\displaystyle k^{\prime}:=\begin{cases}k&\text{if
$\Phi(v)\neq\Phi(v^{\prime})$,}\\\ k+1&\begin{array}[]{@{}l@{}}\text{if
$\Phi(v)=\Phi(v^{\prime})$ and}\\\ \text{$v\rightarrow v^{\prime}$ belongs to
a component of type $I$},\end{array}\\\ k\pm
1&\begin{array}[]{@{}l@{}}\text{if $\Phi(v)=\Phi(v^{\prime})$ and}\\\
\text{$v\rightarrow v^{\prime}$ belongs to a component of type
$J$}.\end{array}\end{cases}$

The sign ambiguity in the final case is resolved as follows: suppose that
$J^{\prime}\subset J_{\infty}$ is a component of type $J$, $v_{0}^{\prime}$ is
its initial vertex, and
$\widetilde{\Phi}(v_{0}^{\prime})=\big{(}\mu,m\big{(}v^{k}\big{)}\big{)}$.
Then $\widetilde{\Phi}$ takes values in $\mathscr{X}\times{\mathbb{Z}}_{\geq
0}\times\big{\\{}v^{k},v^{k+1}\big{\\}}$ on $J^{\prime}$.

###  5.4 Pruning

Consider our map

$\displaystyle\Phi=\Phi_{\lambda}\colon\ J_{\infty}\rightarrow\mathscr{X}$

(which depended on the choice of a fixed almost corner $\lambda$). We now
“prune” our tree $J_{\infty}$ to produce a new tree $J_{\lambda}$. Consider
the set

$\displaystyle K_{\lambda}:=\left\\{v\in
J_{\infty}\,\middle|\,\begin{array}[]{@{}c@{}}\text{there exists
}v_{1},\dots,v_{m}\text{ such that }\Phi(v_{1})\notin\mathscr{X}_{++}\\\
\text{ and }v_{1}\rightarrow v_{2}\rightarrow\dots\rightarrow
v_{m}=v\end{array}\right\\}$

and define $J_{\lambda}$ to be the full subgraph with vertices
$J_{\infty}\setminus K_{\lambda}$. (That is, we remove all branches from
$J_{\infty}$ that contain elements which are mapped to weights which are not
strictly dominant.) The restriction of $\Phi$ to $J_{\lambda}$ defines a map

$\displaystyle\Phi\colon\ J_{\lambda}\rightarrow\mathscr{X}_{++}.$

Restricting the extension $\widetilde{\Phi}$ (which depended on an additional
choice of label $n\big{(}v^{k}\big{)}$) of $\Phi$ to $J_{\infty,++}$ yields a
map:

$\displaystyle\widetilde{\Phi}=\widetilde{\Phi}_{\lambda,n(v^{k})}\colon\
J_{\lambda}\rightarrow\mathscr{X}_{++}\times{\mathcal{M}}.$

###  5.5 The alternative construction

For any positive integer $m$ let

$\displaystyle\lambda_{m}:=(\ell-1)m\varpi_{1}+\varpi_{2}\qquad\text{and}\qquad
g_{m}:=\big{(}\lambda_{m},(2m\ell-1)\big{(}v^{-1}\big{)}\big{)}.$

The previous constructions provide us with a rooted tree $J_{\lambda_{m}}$ and
a map

$\displaystyle\widetilde{\Phi}_{g_{m}}\colon\
J_{\lambda_{m}}\rightarrow\mathscr{X}_{++}\times{\mathcal{M}}.$

Consider the union of multisets

$\displaystyle Z^{\prime}:=\bigcup_{m\geq
1}\big{\\{}\widetilde{\Phi}_{g_{m}}(v)\,|\,v\in J_{\lambda_{m}}\big{\\}}.$

The following lemma implies that the above algorithm provides an alternative
construction of $\widetilde{Z}$.

######  Lemma 5.4.

We have $\widetilde{Z}=Z^{\prime}\setminus\\{g_{m}\,|\,m\in{\mathbb{Z}}_{\geq
0}\\}$.

###### Proof (sketch).

Fix $m\\!\geq\\!0$ and consider
$\widetilde{\Phi}\\!:=\\!\widetilde{\Phi}_{g_{m}}$ as defined above. Also, set
$q{=}\big{(}\ell m\varpi_{1},0\big{(}v^{0}\big{)}\big{)}$ and consider the set
$\Lambda_{q}$ defined as above. We claim that we have

$\displaystyle\Lambda_{q}\setminus\\{q\\}=\big{\\{}\widetilde{\Phi}_{g_{m}}(v)\,|\,v\in
J_{\ell m\varpi_{1}}\big{\\}}\setminus\\{g_{m}\\},$ (5.1)

which implies the lemma. The equality (5.1) follows from the following local
considerations:

  1. 1.

The restriction of $\widetilde{\Phi}$ to the initial component of type $I$
takes the same values as part (1) of “Dynamics on walls”, except for the first
two values ($g_{m}$ and $q$ respectively) which are removed in (5.1).

  2. 2.

The restriction of $\widetilde{\Phi}$ to any component of type $I$ other than
the initial segment takes the same values as those produced by part (2) of
“Dynamics on the walls”.

  3. 3.

If $v$ (resp. $v^{\prime}$, $v^{\prime\prime}$) denotes the source (resp.
sinks) of a component of type $J$ then $\widetilde{\Phi}(v^{\prime})$ and
$\widetilde{\Phi}(v^{\prime\prime})$ are obtained from $\widetilde{\Phi}(v)$
by a giant leap, i.e., part (3) of “Dynamics on the walls”.

  4. 4.

For each component of type $J$ in $J_{\infty}$, if we consider the full
subgraph

…………$J_{\textrm{int}}:=$

then the restriction of $\widetilde{\Phi}$ to each branch of
$J_{\textrm{int}}$ produces the same set as the algorithm in Section 4.4
(“Billiards in an alcove”). ∎

##  6 The conjecture

Recall the notion of an $\ell$-alcove in $\mathscr{X}_{\mathbb{R}}$ from
above. An _alcove_ is a $1$-alcove. The alcove
${\mathcal{A}}_{0}:=\\{\lambda\,|\,0<\langle\alpha^{\vee},\lambda\rangle<1\text{
for all }\alpha^{\vee}\in\Phi_{+}^{\vee}\\}$ is the _fundamental alcove_. The
map $x\mapsto x{\mathcal{A}}_{0}$ gives a bijection between the affine Weyl
group $\mathcal{W}$ and the set of alcoves. It restricts to a bijection
between ${}^{f}\mathcal{W}$ and the dominant alcoves.

Given $\mu\in\mathscr{X}$ the open box

$\displaystyle
B_{\mu}:=\big{\\{}\lambda\in\mathscr{X}_{\mathbb{R}}\,|\,\langle\alpha^{\vee},\mu\rangle<\langle\alpha^{\vee},\lambda\rangle<\langle\alpha^{\vee},\mu\rangle+1\text{
for all $\alpha^{\vee}\in\Sigma^{\vee}$}\big{\\}}$

contains exactly 2 alcoves. In this way we obtain a map
$\mathcal{W}\rightarrow\mathscr{X}$ by sending $x\in\mathcal{W}$ to the unique
$\mu\in\mathscr{X}$ such that $B_{\mu}$ contains $x{\mathcal{A}}_{0}$. It
restricts to a map

$\displaystyle\kappa\colon\ {}^{f}\mathcal{W}\rightarrow\mathscr{X}_{+}.$

Let $s_{0}$ denote the simple affine reflection. Consider the elements

$\displaystyle x_{0}:=\operatorname{id},\ x_{1}=s_{0},\ x_{2}=s_{0}s_{1},\
x_{3}=s_{0}s_{1}s_{2},\ x_{4}:=s_{0}s_{1}s_{2}s_{0},\ \dots$

of $\mathcal{W}$. (These are the alcoves along one edge of the dominant cone.)
We have $\kappa(x_{2i})=\kappa(x_{2i+1})=i\varpi_{1}$ for all $i\geq 0$.

We now describe a sort of inverse to $\kappa$. Recall that
$\mathscr{X}_{++}={\mathbb{Z}}_{>0}\varpi_{1}\oplus{\mathbb{Z}}_{>0}\varpi_{2}$
denotes the strictly dominant weights. Consider $\mu\in\mathscr{X}_{++}$ and
let $x,x^{\prime}$ denote the two elements of ${}^{f}\mathcal{W}$ indexing
alcoves contained in $B_{\mu}$. If
${\mathcal{R}}(z)=\\{s\in\mathcal{S}\,|\,zs<z\\}$ denotes the right descent
set then it is easy to see that we have

$\displaystyle{\mathcal{R}}(x)\sqcup{\mathcal{R}}(x^{\prime})=\\{s_{0},s_{1},s_{2}\\}=\mathcal{S}\quad\text{(disjoint
union)}.$

It follows that for any pair $\mu\in\mathscr{X}_{++},s\in\mathcal{S}$ there is
a unique element $x_{\mu}^{s}\in{}^{f}\mathcal{W}$ such that
$s\in{\mathcal{R}}(x_{\mu}^{s})$ and $\mu=\kappa(x_{\mu}^{s})$.

Recall our prime $p$ from above, and consider the multiset $\widetilde{Z}$
constructed in the previous section with $\ell=p$. We now describe how to use
$\widetilde{Z}$ to define new elements in the anti-spherical module
$\mathrm{AS}_{v}$. Consider the ${\mathbb{Z}}$-linear map
$\varphi\colon\mathbb{Z}[v]\rightarrow\mathbb{Z}\big{[}v^{\pm 1}\big{]}$ given
by $v^{0}\mapsto 1$ and $v^{i}\mapsto v^{i}+v^{-i}$ for $i>0$. Set
${}^{p}\zeta_{0}:=\underline{n}_{x_{0}}$. For any $i>0$ let $s\in\mathcal{S}$
denote the unique element of ${\mathcal{R}}(x_{i})$ and consider the element:

$\displaystyle{}^{p}\zeta_{i}:=\underline{n}_{x_{i}}+\sum_{(\mu,n(v^{k}))\in\widetilde{Z};\atop
n\in\\{i,i-1,i-2\\}}\varphi\big{(}v^{k}\big{)}\underline{n}_{x_{\mu}^{s}}.$

######  Conjecture 6.1.

We have:

  1. $1)$

${}^{p}\zeta_{i}={}^{p}\underline{n}_{x_{i}}$ for $0\leq i<2p(p+1)$;

  2. $2)$

${}^{p}\zeta_{i}={}^{p}\underline{n}^{2}_{x_{i}}$ for all $0\leq i$.

######  Remark 6.2.

Some remarks on the conjecture:

  1. 1.

For an example of $\widetilde{Z}$ the reader is referred to Example 4.5.

  2. 2.

It is a nice exercise to compare our conjecture to the results of J.G. Jensen
[12] and Parker [20]. For example, in Fig. 1, the results of Jensen and Parker
are explained by the unique alcoves labelled $12(1)$, $14(1)$, $16(1)$,
$18(1)$ and $21(v)$.

  3. 3.

Our conjecture (in particular the definition of the set $\widetilde{Z}$) does
not seem to make sense for $p=2$.

  4. 4.

We have verified part (1) of the conjecture for $p=3,5$ and in many cases for
$p=7$ by computer. (In fact these calculations led to the conjecture.)

  5. 5.

To determine the $p$-canonical basis in $\mathrm{AS}_{v}$ it is enough to know
the elements ${}^{p}\underline{n}_{x_{i}}$ for all $i>0$. (After exploiting
the automorphism $s_{0}\mapsto s_{0}$, $s_{1}\mapsto s_{2}$, $s_{2}\mapsto
s_{1}$ this can be deduced from a $v$-analogue of the fact that one can apply
the tilting tensor product theorem to determine all tilting characters,
provided one knows the tilting characters along the walls and in the
$(p-1)\rho$-shift of the fundamental box, see [24, Section 1.6]. One can check
by hand that one has

$\displaystyle{}^{p}\underline{n}_{x}=\underline{n}_{x}\qquad\text{for all
$x\in\\{\operatorname{id},s_{0},s_{0}s_{1}s_{2}s_{1},s_{0}s_{1}s_{2}s_{1}s_{0}\\}$}$

and so the only remaining cases are ${}^{p}\underline{n}_{x_{i}}$ for $i>1$.)

  6. 6.

Similarly, to know all elements ${}^{p}\underline{n}^{2}_{x}$ for
$x\in{}^{f}\mathcal{W}$ it should be enough to know
${}^{p}\underline{n}^{2}_{x_{i}}$ for all $i$. Thus our conjecture gives a
formula for the ${}^{p}\underline{n}_{x}^{2}$ (which currently have no other
rigorous definition444However see the last sentence of Remark 2.1.).

  7. 7.

Recall that the exists a bijection between two-sided cells in the affine Weyl
group and nilpotent orbits in the dual group [16]. Moreover, every two sided
cell intersects ${}^{f}\mathcal{W}$ in a left cell (the canonical left cell)
[19]. The “difficult” elements $x_{i}$ for $i>0$ all lie in the left cell
corresponding to the minimal nilpotent orbit of $\textrm{SL}_{3}$. This
suggests that the problem of determining tilting characters should be related
to the geometry of nilpotent orbits. Related results (connecting the
$p$-canonical basis to coherent sheaves on the Springer resolution) may be
found in [3, 4].

  8. 8.

After taking Remark 4.3 into account, our conjecture implies that the
multiplicities of Weyl modules in the tilting module $T_{3pk\varpi_{1}}$ grow
(at least) exponentially in $k$. Thus our conjecture implies that
decomposition numbers for symmetric groups $S_{n}$ grow exponentially in $n$
(see [7, 10] for the connection between tilting module characters and
decomposition numbers), and that the dimension of the tilting module
$T_{k\varpi_{1}}$ grows exponentially in $k$. Neither of these statements is
true for $\textrm{SL}_{2}$ or for the quantum group of $\textrm{SL}_{3}$ at a
$p^{\rm th}$ root of unity.

  9. 9.

The powers of $v$ which occur after $n$ iterations of the algorithm of Section
4.3 are all at least $n$. (By contrast, the algorithm Section 4.4 only ever
changes the power of $v$ by $\pm 1$.) This implies (assuming our conjecture)
that ${}^{p}\underline{n}_{x_{i}}$ involves arbitrarily high powers of $v$ as
$i$ grows. This in turn implies that certain structure constants for the
action of $\underline{h}_{s_{i}}$ on the $p$-canonical basis involve
arbitrarily high powers of $v$. One can use this observation (and [21, Section
1.4]) to conclude that the analogue of the $a$-function for the $p$-canonical
basis for the Hecke algebra $\mathrm{H}$ of $\mathcal{W}$ is unbounded.

  10. 10.

Fix $m=\big{(}\mu,n\big{(}v^{k}\big{)}\big{)}\in\widetilde{Z}$ and let $x$,
$x^{\prime}$ index the two alcoves contained in $B_{\mu}$, chosen such that
${\mathcal{R}}(x_{n})\subset{\mathcal{R}}(x)$. The formula for
${}^{p}\zeta_{i}$ above implies that $m$ contributes
$\varphi\big{(}v^{k}\big{)}$ to the coefficient of $\underline{n}_{x}$ (resp.
$\underline{n}_{x^{\prime}}$, $\underline{n}_{x}$) in ${}^{p}\zeta_{n}$ (resp.
${}^{p}\zeta_{n+1}$, ${}^{p}\zeta_{n+2}$). A key step in arriving at the
multiset $\widetilde{Z}$ is to observe that the elements of the $p$-canonical
basis $\\{{}^{p}\underline{n}_{x_{i}}\\}$ may be decomposed into such
“triples”. After a first version of this paper was written, L.T. Jensen [13]
has proved that such a decomposition is always possible, as a consequence of
more general results on the $p$-canonical basis and the star operations of
Kazhdan and the first author.

Finally, let us explain how to go from the $p$-canonical basis to a picture
similar to that at the beginning of this paper (which is conjecturally
described by $\widetilde{Z}$). It was this procedure (combined with heuristics
as to what constitutes generation $2$) that led us to our conjecture.

Fix an alcove $y{\mathcal{A}}_{0}$ (for $y\in{}^{f}W$) which is not on a wall.
For all $i\geq 0$ such that ${}^{p}n_{y,x_{i}}\neq 0$, we consider the
polynomial obtained from ${}^{p}n_{y,x_{i}}$ by discarding negative powers of
$v$, and write $i(f)$ in the alcove $y{\mathcal{A}}_{0}$. This produces a
diagram, in which the strictly dominant alcoves are decorated by symbols of
the form $i(f)$.555For an example of such a diagram for $p=5$ see
http://www.maths.usyd.edu.au/u/geordie/pCanA2/p5pretriples.pdf. Now, for all
$\mu\in\mathscr{X}_{++}$ we replace each “triple” of the form

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by

$i(f)$

This is what is depicted (for a second generation version) in Fig. 1.

### Acknowledgements

We would like to thank the anonymous referees for their comments.

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