Billiards and Tilting Characters for SL 3
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2018), 015, 22 pages Billiards and Tilting Characters for SL George LUSZTIG † and Geordie WILLIAMSON ‡† Massachusetts Institute of Technology, Cambridge, MA, USA
E-mail: [email protected]
URL: ‡ Sydney University, Sydney, NSW, Australia
E-mail: [email protected]
URL:
Received July 18, 2017, in final form February 16, 2018; Published online February 21, 2018https://doi.org/10.3842/SIGMA.2018.015
Abstract.
We formulate a conjecture for the second generation characters of indecompos-able tilting modules for SL . This gives many new conjectural decomposition numbers forsymmetric groups. Our conjecture can be interpreted as saying that these characters aregoverned by a discrete dynamical system (“billiards bouncing in alcoves”). The conjectureimplies that decomposition numbers for symmetric groups display (at least) exponentialgrowth. Key words: tilting modules; billiards; p -canonical basis; symmetric group We formulate a conjecture for the second generation characters of indecomposable tilting modu-les for SL in characteristic p >
2. These conjectures resulted from our attempts to understanddata [25] obtained following a 9 month calculation in magma on a supercomputer at the MPIMin Bonn. These results go far beyond existing calculations and are obtained using a new algo-rithm [26]. The algorithm relies in an essential way on ideas of Libedinsky, Riche and the secondauthor (see [15, 21]). The behaviour we observe appears highly non-trivial, which suggests thatproving anything might be difficult.On the next page the reader will find a picture. This picture was obtained by analyzing theoutput of computer calculations for p = 5. Exactly how this picture is used to produce (secondgeneration) tilting characters will be explained in the final section. Before reading the rest ofthe paper the reader is invited to consider this picture and try to discern any patterns. Thispaper is an attempt to explain this picture, as well as similar pictures for p = 3 and 7. Let G denote a split simple and simply connected algebraic group over a field k of characteristic p .We fix a Borel subgroup and maximal torus T ⊂ B ⊂ G . We will try to follow the notationof [24]. In particular: a r X i v : . [ m a t h . R T ] F e b G. Lusztig and G. Williamson ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ` v ) , ( v ` v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ` v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ` v ) , ( v ` v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ` v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( v ) , ( ) , ( v ) , ( v ) , ( ) , ( v ) , ( ) , ( ) , ( v ) , ( ) , Figure 1.
Second generation pattern for p = 5 up to i = 81. illiards and Tilting Characters for SL X , X + : weights, dominant weights;Φ, Φ + : roots, positive roots;Φ ∨ , Φ ∨ + : coroots, positive coroots;Σ, Σ ∨ : simple roots, simple coroots; X
Recall that the Steinberg tensor product theorem has “one step” for the quantumgroup at a root of unity, and “infinitely many steps” for G . The characters χ nλ can be thought ofas the simple characters for an object with an “ n step” Steinberg tensor product theorem. It isnot known whether such an object exists. For sl such an object (for any n ) has recently beenproposed by Angiono [6]. It seems likely that one can combine recent work of Elias [8] with workof Riche and the second author [21] to construct such an object in type A for n = 2. As for simple characters, there should exist approximations to tilting characters. That is, forany p and dominant weight λ ∈ X there should exist characters θ λ , θ λ , θ λ , . . . , θ ∞ λ ∈ ( Z X ) W f with the following properties:1) θ λ is the character of the simple highest weight module in characteristic 0 (i.e., θ λ is givenby Weyl’s character formula);2) θ λ is the character of the indecomposable tilting module for the quantum group at a p th -root of unity; G. Lusztig and G. Williamson3) θ nλ is a positive linear combination of the θ n − µ ’s for all n ≥ (cid:104) α ∨ , λ + ρ (cid:105) ≤ p n +1 then θ nλ = θ n +1 λ = · · · = θ ∞ λ ;5) if p is large then θ ∞ λ is the character of T λ . 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 θ nλ are defined for a subset of X + (roughly, those for which onecan apply the tilting tensor product theorem) in [18]. v = 1 Let us assume that p ≥ h , and consider the p -dilated action of the affine Weyl group from above.For any choice of “generation parameters” 0 ≤ m ≤ n ≤ ∞ we can write θ nx • p = (cid:88) y ∈ W d m,ny,x · θ my • p (2.1)for some d m,ny,x ∈ Z ≥ . (The fact that we can form such expressions follows from the linkageprinciple and our assumptions above.) The question of determining characters of indecomposabletilting characters is equivalent to determining the coefficients d ∞ , y,x for all y, x ∈ f W . Remark 2.4.
A philosophy underlying the current paper is that it might be easier to calculatethe d n +1 ,ny,x for all n , rather than calculate the d ∞ , y,x directly. This is the case for simple characters,as explained in Section 2.1.The following is known about these coefficients:1) when G = GL n , the d ∞ , y,x are equal to decomposition numbers for symmetric groups, bywork of Donkin [7] and Erdmann [10];2) the d , y,x are given as the value at 1 of certain parabolic Kazhdan–Lusztig polynomials, bywork of Soergel [22, 23];3) the d ∞ , y,x 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 someinteresting calculations for SL In this paper we formulate a conjecture for ( v -analogues of the) coefficients d , y,x , when G = SL . (Note that in the case of G = SL the coefficients d , y,x given by Soergel’s algo-rithm are easily calculated, and may be described by closed formulas.) By property (4) of theprevious section, this provides a conjecture for many d ∞ , y,x , and hence for many coefficients ofthe adjustment matrix (and hence decomposition numbers) for three row partitions. p -canonical basis Let H denote the Iwahori–Hecke algebra of the affine Weyl group W over Z (cid:2) v ± (cid:3) . Consider theanti-spherical module (see, e.g., [21, 24]):AS v := sgn v ⊗ H f H = (cid:77) x ∈ f W Z (cid:2) v ± (cid:3) n x . The anti-spherical module has a canonical basis { n x | x ∈ f W} and a p -canonical basis { p n x | x ∈ f W} [14, 21].illiards and Tilting Characters for SL p n x = (cid:88) 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 termsof Weyl characters. Let us give a precise version of this conjecture for p ≥ h : for all x, y ∈ f W we have (in the notation of Section 2.3) d ∞ , y,x = p m y,x (1) . (2.2)(Recall that, by the translation principle and still under our assumption p ≥ h , knowledge ofthe left hand side of (2.2) for all x, y ∈ f W implies knowledge of the characters of all tiltingmodules for G .) Remark 2.6.
In [21] this conjecture is proved for GL n and p > h . In [9] this conjecture isproved for 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. It is natural to expect that the above approximations to tilting characters can also be made torespect this grading. That is, we expect that for all p > h and x ∈ f W , there exist elements p n x , p n x , . . . , p n ∞ x ∈ AS v such that:1) p n x = n x ;2) p n nx is a Z ≥ [ v ± ]-linear combination of the p n n − y ’s for all n ≥ (cid:104) α ∨ , x • p ρ (cid:105) ≤ p n +1 then p n nx = p n n +1 x = · · · = p n ∞ x ;4) if p is large then p n ∞ x = p n x . In this section we attempt to give the reader some idea of how we perform the calculations whichled to our conjecture. The second author hopes to give more details in [26]. As explained in theprevious section, the main theorems of [9, 21] (see also [1, 2]) imply that it is enough to calculatethe p -canonical basis { p n x | x ∈ f W } in the anti-spherical module AS v .Let H denote the diagrammatic Hecke category associated to the affine Cartan matrix oftype (cid:101) A n . Recall that H is built starting from a “realisation” (cid:0) { α ∨ s } s ∈ S ⊂ h , { α s } s ∈ S ⊂ h ∗ (cid:1) , where h is a free and finite rank Z -module, h ∗ is its dual, and the usual formulas define a rep-resentation of W . It will be important below that our realisation is chosen so that the simpleroots { α s } s ∈ S ⊂ h ∗ are linearly independent.The Hecke category is defined over Z . After fixing a prime p , extension of scalars yieldsthe Hecke category H Z p defined over the p -adic integers. Let AS Z p denote the anti-sphericalcategory over Z p considered in [15, 21]. It is a right H Z p -module category and one has a canonicalidentificiation of right [ H Z p ] ⊕ = H -modules:AS v ∼ → [ AS Z p ] ⊕ , See [21, Conjecture 1.7] for a precise formulation. Note that this conjecture is expected to hold without anyrestrictions on p . This grading is conjectural in general. In [1, 2, 4, 9, 21] its existence is established in many cases.
G. Lusztig and G. Williamsonwhere [ − ] ⊕ denotes the split Grothendieck group of an additive category (see [15, 21] for moredetails). Under this isomorphism the classes of the indecomposable self-dual objects yields the p -canonical basis in AS v .The anti-spherical category AS Z p is defined by explicit generators and relations. Thus thequestion of determining the characters of its indecomposable objects (and hence the p -canonicalbasis) is a concrete question of finding idempotents in certain finite rank Z p -algebras. Howeverthese diagrammatic calculations are prohibitively difficult in all but the simplest cases.Instead we exploit the philosophy of localisation. To AS Z p are associated progressively sim-pler categories: AS Z p (cid:32) AS Q p (cid:32) Q ⊗ R AS Q p , where (cid:32) denotes some form of localisation. The first localisation AS Q p is obtained from AS Z p by inverting p . The second is the main object of study of [15]: the polynomial ring R = Q p [ (cid:101) α ]( (cid:101) α denotes the affine simple root) acts on the left on all hom spaces in AS Q p , and after inverting (cid:101) α one obtains Q ⊗ R AS Q p , where Q denotes Q p ( (cid:101) α ). Somewhat surprisingly, this category issemi-simple [15]. (This is analogous to deformed category O , which is semi-simple for genericparameters. A big difference in the current setting is that the deformation ring R is always onedimensional.)The algorithm we used to calculate the p -canonical basis now proceeds in two steps:1. Firstly, AS Q p is described as a quiver with relations, and the action of the Hecke cate-gory H Z p on AS Q p is described explicitly, in terms of the quiver. This is already a verynon-trivial task, and is only feasible for SL , SL and perhaps SL . It is possible in thesecases thanks to the localisation Q ⊗ R AS Q p (where any calculation can be reduced to a cal-culation in matrices with entries in Q ), the fact that the Kazhdan–Lusztig conjectures holdin A Q p (thus one has graded dimensions for hom spaces and knows how the generatorsof H Z p act on the Grothendieck group etc.), and the fact that the Kazhdan–Lusztig theo-ry of these anti-spherical modules in low rank is not too complicated (Kazhdan–Lusztigpolynomials can be written down explicitly).2. Secondly, via the above localisations one can describe AS Z p as a Z p -lattice inside AS Q p .Moreover, this lattice is the smallest lattice which contains the generating object and isstable under the action of the Hecke category H Z p . One may describe this lattice explicitlyand inductively via the H Z p -action, using the philosophy of the light leaf basis. Details ofhow 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 andcalculate the p -canonical basis using only the localisation Q ⊗ R AS Q p . This is a much simplerapproach, and appears (much to the second author’s surprise!) not to be any slower than theapproach described above. It may well be that Jensen’s modification of the algorithm ends upbeing the more effective. For the rest of the paper we fix G = SL , X = Z (cid:36) ⊕ Z (cid:36) , Σ = { α , α } etc.Fix (cid:96) ≥
1. (For applications to representation theory (cid:96) will be prime, however for themoment (cid:96) can be any positive integer. Soon we will assume (cid:96) ≥ M := Z ≥ × (cid:8) v k | k ∈ Z (cid:9) will be called labels . Labels will often be denoted by an integer followed by a bracketed powerof v ; e.g., 51( v ).illiards and Tilting Characters for SL labelled point is an element of X × M . Labelled points will usually be denoted ( µ, m ) with µ ∈ X and m ∈ M or ( µ, n, v k ) with µ ∈ X and n ∈ Z ≥ .Our goal in this section is to describe an algorithm which produces a multiset (i.e., set withmultiplicities) of labelled points via an inductive procedure. Throughout this section we workentirely with multisets. All operations (union, difference, . . . ) are to be understood in thecontext of multisets. We view the dominant weights X + as the vertices of a directed graph Γ with edges λ → λ + γ if λ, λ + γ ∈ X + and γ ∈ { (cid:36) , (cid:36) − (cid:36) , − (cid:36) } :Γ = . . ....... (cid:36) (cid:36) Our algorithm consists of three steps. We begin with the labelled point (cid:0) , , v (cid:1) (our initial“seed”). Each step in our algorithm enlarges our multiset in a new direction, starting with seedsgenerated in the previous step. The first step extends our set along a wall of the dominantchamber in the direction of (cid:36) to produce a set X . The second step extends our set along thewalls of the (cid:96) -alcoves to produce a multiset Y . Finally, the third step extends our multiset withinthe interior of each (cid:96) -alcove to produce a multiset Z . In the third step, certain labelled pointsarising in the second step (seeds) gives rise to two spirals inside the interior of two adjacentalcoves which move like a billiard. Finally we consider (cid:101) Z := Z \ X, which is the object of our conjecture. Consider the full subgraph of X + consisting of multiples of (cid:36) :0 (cid:36) Consider a labelled point m = (cid:0) µ, m (cid:0) v k (cid:1)(cid:1) such that µ belongs to this full subgraph. A smallstep produces the labelled point (cid:0) µ (cid:48) , ( m +2) (cid:0) v k (cid:1)(cid:1) where µ → µ (cid:48) is the unique edge with source µ .Consider the set X obtained by repeatedly taking small steps beginning with the labelled point (cid:0) , (cid:0) v (cid:1)(cid:1) . In other words, X is the set X = (cid:8)(cid:0) kω , k (cid:0) v (cid:1)(cid:1) | k ∈ Z ≥ (cid:9) . (4.1)We designate the labelled points of the form ( (cid:96)k, k(cid:96) ( v )) with k > seeds . Remark 4.1.
Of course we could have defined X via (4.1) directly. We prefer the inductivedefinition, as it is closer in spirit to the more complicated definitions which will occur in thenext two steps. G. Lusztig and G. Williamson Figure 2.
The directed graph Γ wall . From now on assume that (cid:96) ≥
3. Consider the subgraph of Γ with vertices λ ∈ X + such that (cid:104) λ, α ∨ (cid:105) ∈ (cid:96) Z for some α ∈ Φ + and edges ( λ, λ (cid:48) ) such that (cid:104) λ, α ∨ (cid:105) = (cid:104) λ (cid:48) , α ∨ (cid:105) ∈ (cid:96) Z for some α ∈ Φ + . Let Γ wall denote the graph obtained by removing the gray edges and vertices as inFig. 2. Points in X + or Γ wall such that (cid:104) λ, α ∨ (cid:105) ∈ (cid:96) Z for all α ∈ Φ + are called corner points .A point µ in X + of Γ wall is an almost corner if there exists a corner point c and an arrow c → µ in X + .Fix a labelled point (cid:0) µ, n (cid:0) v k (cid:1)(cid:1) with µ ∈ Γ wall . We assume that µ is such that there is a uniqueedge with source µ . From (cid:0) µ, n (cid:0) v k (cid:1)(cid:1) one may obtain new labelled points as follows:1. A rest produces the labelled point (cid:0) µ, ( n + 3) (cid:0) v k +1 (cid:1)(cid:1) .2. A small step produces (as above) the labelled point (cid:0) µ (cid:48) , ( n + 2) (cid:0) v k (cid:1)(cid:1) , where µ → µ (cid:48) is theunique edge in Γ wall with source µ .3. A giant leap produces either one or two new labelled points, and is only possible if µ isnot a corner or almost corner point. Define d to be the direction of the unique arrowwith source µ . First we proceed j < (cid:96) − wall in direction d untilwe reach a corner point. We then proceed for another (cid:96) − − j steps in all directionsfrom the corner point which do not agree with the direction d . (There are either one ortwo such directions.) A giant leap consists of the resulting points, which are labelled by( n + 2 (cid:96) + 1) (cid:0) v k +1 (cid:1) (see Fig. 3).Now suppose that we are given a labelled point q = ( µ, n, v k ) as above (i.e., such that there isa unique edge in Γ wall with source µ ). We now describe a way of producing a multiset of labelledpoints (all with multiplicity 1) beginning with q , some of which are designated as seeds :1. If µ is a corner point. We produce (cid:96) new points as follows: we take (cid:96) − m = m , . . . , m (cid:96) − and then rest to produce a new labelledpoint m (cid:96) . The final labelled point m (cid:96) is designated as a seed. (See Fig. 4.)2. If µ is an almost corner point. We produce (cid:96) new points as follows: let m = m and restonce to produce a new labelled point m , now take (cid:96) − m toilliards and Tilting Characters for SL µn ( v k ) µ (cid:48)(cid:48) ( n + 2 (cid:96) + 1)( v k +1 ) µ (cid:48) ( n + 2 (cid:96) + 1)( v k +1 ) or µ n ( v k ) µ (cid:48) ( n + 2 (cid:96) + 1)( v k +1 ) Figure 3.
Performing a giant leap with (cid:96) = 5. (Squares denote corner points.) v )12( v )14( v )16( v )18( v ) 21( v ) 54( v )57( v )59( v )61( v )63( v ) 66( v ) Figure 4.
An illustration of cases (1) and (2) when (cid:96) = 5. Seeds are underlined. produce (cid:96) − m , . . . , m (cid:96) − and finally rest once more to produce a finallabelled point m (cid:96) . The final labelled point m (cid:96) is designated as a seed. (See Fig. 4.)3. If µ is neither a corner nor an almost corner point. We take a giant leap to produce oneor 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 , Q , Q , . . . , where Q (resp. Q i for i >
1) is obtained by applying the above procedure to q (resp. to each seed in Q i − ). We say that the union (as multisets) Q = (cid:91) i ≥ Q i is the result of applying dynamics on the walls to the seed q . Example 4.2.
Fig. 5 gives an example with (cid:96) = 5. Our initial seed is the unique point withlabel 10 (cid:0) v (cid:1) . We illustrate a few iterations of the above algorithm. Seeds are underlined. Notethat one more iteration of the algorithm will produce the labelled point (cid:0) (cid:36) + 7 (cid:36) , (cid:0) v (cid:1)(cid:1) with multiplicity 2.Now consider our set X from the previous step. We apply dynamicson walls to each seedin X (i.e., each labelled point of the form (cid:0) k(cid:96)(cid:36) , k(cid:96) (cid:0) v (cid:1)(cid:1) with k >
0) to generate a multiset Y k .We now define a new multiset Y := X ∪ (cid:91) k> Y k . We also remember (for the purposes of the next step) which elements of Y were designated seeds.0 G. Lusztig and G. Williamson v )77( v )54( v )57( v ) 59( v ) 61( v ) 63( v )66( v )10( v )12( v )14( v )16( v )18( v )21( v ) 54( v )57( v )59( v )61( v )63( v )66( v ) 43( v ) 77( v )43( v ) 77( v ) Figure 5.
Dynamics on the walls with (cid:96) = 5.
Remark 4.3.
Consider dynamics on the wall restricted to the full subgraph displayed in Fig. 6with seed q = (cid:0) µ, n (cid:0) v k (cid:1)(cid:1) . Then 4 iterations of the above algorithm yields the following pointsand labels (see Fig. 6): u, u (cid:48) with label ( n + 11) (cid:0) v k +1 (cid:1) ,u i , u (cid:48) i with label ( n + 14 + 2( i − (cid:0) v k +2 (cid:1) for i = 1 , , , ,v, v (cid:48) with label ( n + 23) (cid:0) v k +3 (cid:1) ,w, w (cid:48) with label ( n + 34) (cid:0) v k +4 (cid:1) , and then µ with label ( n +45) (cid:0) v k +5 (cid:1) and multiplicity 2. (The point is that there are two directedpaths leading from µ back to itself, which causes the multiplicity to double.) Repeating thisalgorithm 4 i times leads to q i := (cid:0) µ, n + 45 i (cid:0) v k +5 i (cid:1)(cid:1) with multiplicity 2 i . From this observationilliards and Tilting Characters for SL qw uu u u u vw (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) u (cid:48) v (cid:48) Figure 6.
Subgraph demonstrating exponential growth of multiplicities for (cid:96) = 5. one deduces easily that for (cid:96) = 5 the set Y contains labelled points whose multiplicity grows ex-ponentially in n . Similar considerations show that the same statement about Y is true for any (cid:96) . 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 X + as embedded in thevector space X R := X ⊗ Z R , which we regard as a Euclidean space for some W f -invariant bilinear form. An (cid:96) -alcove isa connected component of the complement X R \ (cid:91) α ∈ Φ ∨ + (cid:8) λ | (cid:104) α ∨ , λ (cid:105) ∈ (cid:96) Z (cid:9) . An (cid:96) -regular point is a point which belongs to some (cid:96) -alcove. An (cid:96) -alcove is dominant if it iscontained in the real cone generated by X + .Now consider a seed q = ( µ, m ) ∈ Y . We want to associate a multiset of labelled points Λ q to q . If µ is a corner or almost corner, we set Λ q := ∅ .From now on we assume that q is neither a corner or almost corner. In this case µ belongs tothe closure of two dominant (cid:96) -alcoves, A (cid:48) and A (cid:48)(cid:48) . We perform an identical procedure for bothalcoves, so fix one such alcove and call it A . Let ∆ denote the full subgraph of Γ consisting ofpoints belonging to A . Then ∆ has one of the following forms: (cid:96) + 1points A but not to A are called wall points . Note that our starting point µ isa wall point. Variables and their initialisations:
Let ( µ start , m start ) := ( µ, m ) and let d start denote theunique edge of ∆ with source µ and target an interior point of ∆. To begin with, set Λ := { ( µ start , m start ) } . In the algorithm we need the following variables, which are initialised asfollows:CurrentPoint := µ start , CurrentLabel := m start , CurrentDirection := d start , CurrentSign := 1 . The loop:
The multiset of labelled points is obtained by repeating the following ad infinitum.Let µ new denote the point obtained by moving from CurrentPoint one step in CurrentDirection.Write CurrentLabel as n (cid:0) v k (cid:1) . Two possibilities may occur:1. µ new is (cid:96) -regular:CurrentLabel := ( n + 2) (cid:0) v k (cid:1) , CurrentPoint := µ new , add (CurrentPoint , CurrentLabel) to Λ . (The variables CurrentDirection and CurrentSign remain unchanged.)2. CurrentPoint is (cid:96) -regular and µ new is a wall point:CurrentLabel := ( n + 3) (cid:0) v k +CurrentSign (cid:1) , CurrentDirection := d new , CurrentSign := − CurrentSign , add (CurrentPoint , CurrentLabel) to Λ . Here d new denotes the vector obtained by reflecting CurrentDirection in the wall onwhich µ new lies. (The variable CurrentPoint remains unchanged.) Example 4.4.
We illustrate the first few steps of the algorithm in some examples. In each casethe starting point µ start and its label are underlined. The successive values of CurrentLabel andCurrentPoint are obtained by following the arrows.1. A reasonably generic example with (cid:96) = 11: ( ∗ ) v ) 18( v ) 20( v ) 22( v ) 24( v ) 26( v ) 28( v )31( v )33( v )35( v )38( v )40( v )42( v )44( v )46( v )48( v )50( v )53( v ) 55( v ) 57( v )60( v )62( v )64( v )66( v )68( v )70( v )72( v )75( v )77( v ) . . . v ) illiards and Tilting Characters for SL (cid:8) v ) , (cid:0) v (cid:1) , v ) , (cid:0) v (cid:1) , v ) , (cid:0) v (cid:1) , v ) , (cid:0) v (cid:1) , v ) , . . . (cid:9) .
2. An interesting example with (cid:96) = 5 (we display both alcoves): ∗∗ ∗ ∗ ∗ ∗∗∗ ... ∗∗ ∗ ∗ ∗ v ) ∗∗ ∗∗ ∗ ... ∗ ∗∗∗ ∗∗ ∗ ∗ In either alcove the sequence of labels obtained by following the arrows beginning at 21( v )is as follows:23( v ) , v ) , v ) , i ( v (cid:1) , v ) , v ) , v ) , (cid:0) v (cid:1) , v ) , v ) , v ) , (cid:0) v (cid:1) , v ) , . . . .
3. Another example with (cid:96) = 5 (again we display both alcoves): ∗ ... ∗∗ ∗ ∗ ∗∗ ∗∗∗ ∗∗ ∗∗ ... v ) In either alcove the sequence beginning at 77 (cid:0) v (cid:1) is79 (cid:0) v (cid:1) , (cid:0) v (cid:1) , (cid:0) v (cid:1) , (cid:0) v (cid:1) , (cid:0) v (cid:1) , (cid:0) v (cid:1) , . . . . (cid:96) = 3 (again we display both alcoves): v ) ∗ ∗ ∗∗ . . . ∗ ∗∗ ∗ ... The sequence at either middle vertex is: 15( v ), 18( v ), 21( v ), 24( v ), 27( v ), . . . .Now, with q = ( µ, m ) as above (so that µ is neither a corner or almost corner) we apply theabove algorithm to both alcoves which contain µ in their closure to produce multisets Λ (cid:48) and Λ (cid:48)(cid:48) .We then remove q from both, and define Λ q to be the union of the resulting multisets. (Thus Λ q contains infinitely many points from both alcoves, but does not contain q itself.)The above algorithm produces, for each each seed q ∈ Y , a multiset Λ q . We define Z to bethe union Z := Y ∪ (cid:91) seeds q ∈ Y Λ q . Finally, we define (cid:101) Z := Z \ X . Example 4.5.
In Fig. 1 we display (cid:101) Z for (cid:96) = 5 and all labelled points (cid:0) µ, n (cid:0) v k (cid:1)(cid:1) with n ≤ µ ∈ X + are displayed in thesmaller of the two alcoves contained in B µ . The reader is referred to [25] for further examples. Remark 4.6.
For all (cid:0) µ, n (cid:0) v k (cid:1)(cid:1) ∈ (cid:101) Z it is easy to see that µ ∈ X ++ := Z > (cid:36) ⊕ Z > (cid:36) . In this section we outline an alternative construction of the multiset (cid:101) Z , which is more naturalin some respects.In the previous section we regarded the dominant weights X + as the vertices of a direc-ted graph. An important difference in the current construction is that now we consider all weights X . That is, we consider the elements of X as the vertices of a directed graph withedges λ → λ + γ if λ, λ + γ ∈ X + and γ ∈ { (cid:36) , (cid:36) − (cid:36) , − (cid:36) } . We write µ → µ (cid:48) to indicatethat there is a directed edge from µ to µ (cid:48) .We use similar terminology to earlier: the notions of a wall point , corner point , almost corner and (cid:96) -regular point extend in an obvious way to X . Note that almost corners are necessarilywall points. For us a rooted tree is what many call an “arborescence”: a directed graph with a distinguishedvertex (the source ) such that there is a unique directed path from the source to any other vertex. This is equivalent in an obvious way to the usual notion of rooted tree. illiards and Tilting Characters for SL A and B , denote by A ∗ B the rooted tree obtained by adjoining onecopy of B to each sink in A at the source of B . That is, if t , . . . , t k denote the sinks of A andwe denote by b , . . . , b k the sources in B (cid:116) B (cid:116) · · · (cid:116) B ( k factors), then A ∗ B := ( A (cid:116) B (cid:116) B (cid:116) · · · (cid:116) B ) / ( t i ∼ b i ) . This operation is associative. The image of A (resp. a copy of B ) under this quotient map willbe called a component of type A (resp. B ).Recall that (cid:96) ≥ I := . . .. . .. . . . . .. . . J := (cid:96) − (cid:96) (cid:96) + 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 (cid:96) = 3, the graph I ∗ J ∗ J looks as follows: . . .. . .. . .. . .. . .. . . Now define: J ∞ := (cid:0) I ∗ J ∗ (cid:96) − (cid:1) ∗∞ := (cid:0) I ∗ J ∗ (cid:96) − (cid:1) ∗ (cid:0) I ∗ J ∗ (cid:96) − (cid:1) ∗ (cid:0) I ∗ J ∗ (cid:96) − (cid:1) ∗ · · · . Recall the notion of component from above. We say that v ∈ J ∞ is of wall type if it lies ina component of type I or is the source in a component of type J . We say that v is of almostcorner type if it is the source in a component of type I . (Thus if v ∈ J ∞ is of almost corner typethen it is also of wall type.) Example 5.2.
We continue Example 5.1 with (cid:96) = 3. The points of wall type (resp. of almost6 G. Lusztig and G. Williamsoncorner type) in J ∞ are depicted as open (resp. filled) circles:. . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . .. . .. . . Fix an almost corner λ . We will construct a mapΦ = Φ λ : J ∞ → X . It will have the following two properties:1. If v → v (cid:48) in J ∞ then either Φ( v ) = Φ( v (cid:48) ) or Φ( v ) → Φ( v (cid:48) ) in X .2. For v ∈ J ∞ , Φ( 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 ∞ is the union of components of types I or J it will suffice to define the imageof Φ on the source of J ∞ , and then define it on each component of type I or J inductively. Thisis what we do now: Rule : Let v denote the source of J ∞ . Set Φ( v ) = λ (our fixed almost corner). Rule : Let I (cid:48) ⊂ J ∞ denote a component of type I and let v (cid:48) denote the source of I (cid:48) . Supposethat µ := Φ( v (cid:48) ) is defined, but that Φ is not defined on the rest of I (cid:48) . We define Φ on I (cid:48) asfollows: µ → µ → µ → . . . → µ (cid:96) − → µ (cid:96) − → µ (cid:96) − . Here µ = µ , µ , . . . , µ (cid:96) are uniquely determined by the requirement that µ = µ and µ → µ →· · · → µ (cid:96) − are all wall points. (The fact that µ , . . . , µ (cid:96) − are well defined is a consequence ofproperty (2) above: Φ( v (cid:48) ) is an almost corner.) Rule : Let J (cid:48) ⊂ J ∞ denote a component of type J and let v (cid:48) denote the source of J (cid:48) .Suppose that µ := Φ( v (cid:48) ) is defined, but that Φ is not defined on all of J (cid:48) .By property (2) above µ is a wall-point but is not an almost corner. Hence there exist twoedges d (cid:48) , d (cid:48)(cid:48) with source µ and image an (cid:96) -regular point. For d ∈ { d (cid:48) , d (cid:48)(cid:48) } , define a sequence { ( µ i , d i ) } i ≥ as follows:1) ( µ , d ) := ( µ, d );2) if µ i + d i is an interior point, define ( µ i +1 , d i +1 ) := ( µ i + d i , d i );illiards and Tilting Characters for SL µ i + d i is a wall point, define ( µ i +1 , d i +1 ) := ( µ i , r ( d i )), where r denotes the reflectionin the (unique) wall containing µ i + d i .Denote by { ( µ (cid:48) i , d (cid:48) i ) } i ≥ (resp. { ( µ (cid:48)(cid:48) i , d (cid:48)(cid:48) i )) } i ≥ ) the sequences associated to d = d (cid:48) (resp. d = d (cid:48)(cid:48) ).We define Φ on J (cid:48) as follows: µ (cid:48) = µ (cid:48)(cid:48) µ (cid:48) µ (cid:48)(cid:48) µ (cid:48) µ (cid:48)(cid:48) . . .. . . µ (cid:48) (cid:96) − µ (cid:48)(cid:48) (cid:96) − µ (cid:48) (cid:96) µ (cid:48)(cid:48) (cid:96) µ (cid:48) (cid:96) +1 µ (cid:48)(cid:48) (cid:96) +1 . . .. . . µ (cid:48) (cid:96) − + d (cid:48) (cid:96) − µ (cid:48)(cid:48) (cid:96) − + d (cid:48)(cid:48) (cid:96) − (Note that µ (cid:48) (cid:96) − + d (cid:48) (cid:96) − and µ (cid:48)(cid:48) (cid:96) − + d (cid:48)(cid:48) (cid:96) − 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 fromthe definitions. Property (2) for almost corner points follows from the following observation:suppose that J (cid:48) ⊂ J ∞ is a component of type J , let v (cid:48) denote its source, and let v (cid:48) , v (cid:48)(cid:48) denotethe two sinks (so that v (cid:48) and v (cid:48)(cid:48) are of wall type in J ∞ ). Then if Φ( v (cid:48) ) is of distance k ≤ (cid:96) − v (cid:48) ) and Φ( v (cid:48)(cid:48) ) are of distance k − Example 5.3.
We illustrate Φ on a component of type J with (cid:96) = 5 (the source is marked witha circle): ∗ ... ∗∗ ∗ ∗ ∗∗ * * ∗∗∗ ∗∗ ∗∗ ... Note the similarity to the previous section. The only difference is the arrow joining an (cid:96) -regularpoint to a wall point.
Let v denote the source of J ∞ . For any choice of label n (cid:0) v k (cid:1) , we inductively extend Φ toproduce a map (cid:101) Φ = (cid:101) Φ λ,n ( v k ) : J ∞ → X × M (cid:101) Φ( v ) = (cid:0) λ, n (cid:0) v k (cid:1)(cid:1) . We proceed as follows: Suppose that v → v (cid:48) in J ∞ and that (cid:101) Φ isdefined on v but not on v (cid:48) , and let (cid:101) Φ( v ) = (cid:0) Φ( v ) , m (cid:0) v k (cid:1)(cid:1) . Then (cid:101) Φ( v (cid:48) ) = (cid:0) Φ( v (cid:48) ) , m (cid:48) (cid:0) v k (cid:48) (cid:1)(cid:1) with m (cid:48) := (cid:40) m + 2 if Φ( v ) (cid:54) = Φ( v (cid:48) ), m + 3 if Φ( v ) = Φ( v (cid:48) )and k (cid:48) := k if Φ( v ) (cid:54) = Φ( v (cid:48) ), k + 1 if Φ( v ) = Φ( v (cid:48) ) and v → v (cid:48) belongs to a component of type I,k ± v ) = Φ( v (cid:48) ) and v → v (cid:48) belongs to a component of type J .
The sign ambiguity in the final case is resolved as follows: suppose that J (cid:48) ⊂ J ∞ is a componentof type J , v (cid:48) is its initial vertex, and (cid:101) Φ( v (cid:48) ) = (cid:0) µ, m (cid:0) v k (cid:1)(cid:1) . Then (cid:101) Φ takes values in X × Z ≥ × (cid:8) v k , v k +1 (cid:9) on J (cid:48) . Consider our mapΦ = Φ λ : J ∞ → X (which depended on the choice of a fixed almost corner λ ). We now “prune” our tree J ∞ toproduce a new tree J λ . Consider the set K λ := (cid:26) v ∈ J ∞ (cid:12)(cid:12)(cid:12)(cid:12) there exists v , . . . , v m such that Φ( v ) / ∈ X ++ and v → v → . . . → v m = v (cid:27) and define J λ to be the full subgraph with vertices J ∞ \ K λ . (That is, we remove all branchesfrom J ∞ that contain elements which are mapped to weights which are not strictly dominant.)The restriction of Φ to J λ defines a mapΦ : J λ → X ++ . Restricting the extension (cid:101)
Φ (which depended on an additional choice of label n (cid:0) v k (cid:1) ) of Φto J ∞ , ++ yields a map: (cid:101) Φ = (cid:101) Φ λ,n ( v k ) : J λ → X ++ × M . For any positive integer m let λ m := ( (cid:96) − m(cid:36) + (cid:36) and g m := (cid:0) λ m , (2 m(cid:96) − (cid:0) v − (cid:1)(cid:1) . The previous constructions provide us with a rooted tree J λ m and a map (cid:101) Φ g m : J λ m → X ++ × M . Consider the union of multisets Z (cid:48) := (cid:91) m ≥ (cid:8)(cid:101) Φ g m ( v ) | v ∈ J λ m (cid:9) . The following lemma implies that the above algorithm provides an alternative construction of (cid:101) Z .illiards and Tilting Characters for SL Lemma 5.4.
We have (cid:101) Z = Z (cid:48) \ { g m | m ∈ Z ≥ } . Proof (sketch).
Fix m ≥ (cid:101) Φ := (cid:101) Φ g m as defined above. Also, set q = (cid:0) (cid:96)m(cid:36) , (cid:0) v (cid:1)(cid:1) and consider the set Λ q defined as above. We claim that we haveΛ q \ { q } = (cid:8)(cid:101) Φ g m ( v ) | v ∈ J (cid:96)m(cid:36) (cid:9) \ { g m } , (5.1)which implies the lemma. The equality (5.1) follows from the following local considerations:1. The restriction of (cid:101) Φ 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 areremoved in (5.1).2. The restriction of (cid:101) Φ to any component of type I other than the initial segment takes thesame values as those produced by part (2) of “Dynamics on the walls”.3. If v (resp. v (cid:48) , v (cid:48)(cid:48) ) denotes the source (resp. sinks) of a component of type J then (cid:101) Φ( v (cid:48) )and (cid:101) Φ( v (cid:48)(cid:48) ) are obtained from (cid:101) Φ( v ) by a giant leap, i.e., part (3) of “Dynamics on the walls”.4. For each component of type J in J ∞ , if we consider the full subgraph. . .. . . . . .. . . J int :=then the restriction of (cid:101) Φ to each branch of J int produces the same set as the algorithm inSection 4.4 (“Billiards in an alcove”). (cid:4) Recall the notion of an (cid:96) -alcove in X R from above. An alcove is a 1-alcove. The alcove A := { λ | < (cid:104) α ∨ , λ (cid:105) < α ∨ ∈ Φ ∨ + } is the fundamental alcove . The map x (cid:55)→ x A givesa bijection between the affine Weyl group W and the set of alcoves. It restricts to a bijectionbetween f W and the dominant alcoves.Given µ ∈ X the open box B µ := (cid:8) λ ∈ X R | (cid:104) α ∨ , µ (cid:105) < (cid:104) α ∨ , λ (cid:105) < (cid:104) α ∨ , µ (cid:105) + 1 for all α ∨ ∈ Σ ∨ (cid:9) contains exactly 2 alcoves. In this way we obtain a map W → X by sending x ∈ W to theunique µ ∈ X such that B µ contains x A . It restricts to a map κ : f W → X + . Let s denote the simple affine reflection. Consider the elements x := id , x = s , x = s s , x = s s s , x := s s s s , . . . of W . (These are the alcoves along one edge of the dominant cone.) We have κ ( x i ) = κ ( x i +1 ) = i(cid:36) for all i ≥ κ . Recall that X ++ = Z > (cid:36) ⊕ Z > (cid:36) denotes thestrictly dominant weights. Consider µ ∈ X ++ and let x, x (cid:48) denote the two elements of f W indexing alcoves contained in B µ . If R ( z ) = { s ∈ S | zs < z } denotes the right descent set thenit is easy to see that we have R ( x ) (cid:116) R ( x (cid:48) ) = { s , s , s } = S (disjoint union) . µ ∈ X ++ , s ∈ S there is a unique element x sµ ∈ f W such that s ∈ R ( x sµ ) and µ = κ ( x sµ ).Recall our prime p from above, and consider the multiset (cid:101) Z constructed in the previoussection with (cid:96) = p . We now describe how to use (cid:101) Z to define new elements in the anti-sphericalmodule AS v . Consider the Z -linear map ϕ : Z [ v ] → Z (cid:2) v ± (cid:3) given by v (cid:55)→ v i (cid:55)→ v i + v − i for i >
0. Set p ζ := n x . For any i > s ∈ S denote the unique element of R ( x i ) andconsider the element: p ζ i := n x i + (cid:88) ( µ,n ( vk )) ∈ (cid:101) Z ; n ∈{ i,i − ,i − } ϕ (cid:0) v k (cid:1) n x sµ . Conjecture 6.1.
We have: p ζ i = p n x i for ≤ i < p ( p + 1) ; p ζ i = p n x i for all ≤ i . Remark 6.2.
Some remarks on the conjecture:1. For an example of (cid:101) Z the reader is referred to Example 4.5.2. It is a nice exercise to compare our conjecture to the results of J.G. Jensen [12] and Par-ker [20]. For example, in Fig. 1, the results of Jensen and Parker are explained by theunique alcoves labelled 12(1), 14(1), 16(1), 18(1) and 21( v ).3. Our conjecture (in particular the definition of the set (cid:101) Z ) does not seem to make sense for p = 2.4. We have verified part (1) of the conjecture for p = 3 , p = 7 bycomputer. (In fact these calculations led to the conjecture.)5. To determine the p -canonical basis in AS v it is enough to know the elements p n x i for all i >
0. (After exploiting the automorphism s (cid:55)→ s , s (cid:55)→ s , s (cid:55)→ s this can be deducedfrom a v -analogue of the fact that one can apply the tilting tensor product theorem todetermine all tilting characters, provided one knows the tilting characters along the wallsand in the ( p − ρ -shift of the fundamental box, see [24, Section 1.6]. One can check byhand that one has p n x = n x for all x ∈ { id , s , s s s s , s s s s s } and so the only remaining cases are p n x i for i > p n x for x ∈ f W it should be enough to know p n x i for all i .Thus our conjecture gives a formula for the p n x (which currently have no other rigorousdefinition ).7. Recall that the exists a bijection between two-sided cells in the affine Weyl group andnilpotent orbits in the dual group [16]. Moreover, every two sided cell intersects f W ina left cell (the canonical left cell) [19]. The “difficult” elements x i for i > . This suggests that theproblem of determining tilting characters should be related to the geometry of nilpotent or-bits. Related results (connecting the p -canonical basis to coherent sheaves on the Springerresolution) may be found in [3, 4]. However see the last sentence of Remark 2.1. illiards and Tilting Characters for SL T pk(cid:36) grow (at least) exponentially in k . Thus our con-jecture implies that decomposition numbers for symmetric groups S n grow exponentiallyin n (see [7, 10] for the connection between tilting module characters and decompositionnumbers), and that the dimension of the tilting module T k(cid:36) grows exponentially in k .Neither of these statements is true for SL or for the quantum group of SL at a p th rootof unity.9. The powers of v which occur after n iterations of the algorithm of Section 4.3 are all atleast n . (By contrast, the algorithm Section 4.4 only ever changes the power of v by ± p n x i involves arbitrarily high powers of v as i grows. This in turn implies that certain structure constants for the action of 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 basisfor the Hecke algebra H of W is unbounded.10. Fix m = (cid:0) µ, n (cid:0) v k (cid:1)(cid:1) ∈ (cid:101) Z and let x , x (cid:48) index the two alcoves contained in B µ , chosen suchthat R ( x n ) ⊂ R ( x ). The formula for p ζ i above implies that m contributes ϕ (cid:0) v k (cid:1) to thecoefficient of n x (resp. n x (cid:48) , n x ) in p ζ n (resp. p ζ n +1 , p ζ n +2 ). A key step in arriving at themultiset (cid:101) Z is to observe that the elements of the p -canonical basis { p n x i } may be decom-posed 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 generalresults 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 thebeginning of this paper (which is conjecturally described by (cid:101) Z ). It was this procedure (combinedwith heuristics as to what constitutes generation 2) that led us to our conjecture.Fix an alcove y A (for y ∈ f W ) which is not on a wall. For all i ≥ p n y,x i (cid:54) = 0, weconsider the polynomial obtained from p n y,x i by discarding negative powers of v , and write i ( f )in the alcove y A . This produces a diagram, in which the strictly dominant alcoves are decoratedby symbols of the form i ( f ). Now, for all µ ∈ X ++ we replace each “triple” of the form i ( f ) , ( i + 2)( f )( i + 1)( f ) or i ( f ) , ( i + 2)( f )( i + 1)( f ) 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. For an example of such a diagram for p = 5 see . References [1] Achar P.N., Makisumi S., Riche S., Williamson G., Free-monodromic mixed tilting sheaves on flag varieties,arXiv:1703.05843.[2] Achar P.N., Makisumi S., Riche S., Williamson G., Koszul duality for Kac–Moody groups and characters oftilting modules, arXiv:1706.00183.[3] Achar P.N., Rider L., The affine Grassmannian and the Springer resolution in positive characteristic,
Com-pos. Math. (2016), 2627–2677, arXiv:1408.7050.[4] Achar P.N., Rider L., Reductive groups, the loop Grassmannian, and the Springer resolution,arXiv:1602.04412.[5] Andersen H.H., Filtrations and tilting modules,
Ann. Sci. ´Ecole Norm. Sup. (4) (1997), 353–366.[6] Angiono I.E., A quantum version of the algebra of distributions of SL , Publ. Res. Inst. Math. Sci. (2018),141–161, arXiv:1607.04869.[7] Donkin S., On tilting modules for algebraic groups, Math. Z. (1993), 39–60.[8] Elias B., Quantum Satake in type A . Part I, J. Comb. Algebra (2017), 63–125, arXiv:1403.5570.[9] Elias B., Losev I., Modular representation theory in type A via Soergel bimodules, arXiv:1701.00560.[10] Erdmann K., Symmetric groups and quasi-hereditary algebras, in Finite-Dimensional Algebras and RelatedTopics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , Vol. 424, Kluwer Acad. Publ.,Dordrecht, 1994, 123–161.[11] James G., The decomposition matrices of GL n ( q ) for n ≤ Proc. London Math. Soc. (1990), 225–265.[12] Jensen J.G., On the character of some modular indecomposable tilting modules for SL , J. Algebra (2000), 397–419.[13] Jensen L.T., p -Kazhdan–Lusztig theory, Ph.D. Thesis, Max Planck Institute for Mathematics, Bonn, 2017.[14] Jensen L.T., Williamson G., The p -canonical basis for Hecke algebras, in Categorification and HigherRepresentation Theory, Contemp. Math. , Vol. 683, Amer. Math. Soc., Providence, RI, 2017, 333–361,arXiv:1510.01556.[15] Libedinsky N., Williamson G., The anti-spherical category, arXiv:1702.00459.[16] Lusztig G., Cells in affine Weyl groups. IV,
J. Fac. Sci. Univ. Tokyo Sect. IA Math. (1989), 297–328.[17] Lusztig G., On the character of certain irreducible modular representations, Represent. Theory (2015),3–8, arXiv:1407.5346.[18] Lusztig G., Williamson G., On the character of certain tilting modules, Sci. China Math. (2018), 295–298,arXiv:1502.04904.[19] Lusztig G., Xi N.H., Canonical left cells in affine Weyl groups, Adv. Math. (1988), 284–288.[20] Parker A., Some remarks on a result of Jensen and tilting modules for SL ( k ) and q − GL ( k ),arXiv:0809.2249.[21] Riche S., Williamson G., Tilting modules and the p -canonical basis, arXiv:1512.08296.[22] Soergel W., Charakterformeln f¨ur Kipp–Moduln ¨uber Kac–Moody-Algebren, Represent. Theory (1997),115–132.[23] Soergel W., Kazhdan–Lusztig polynomials and a combinatoric for tilting modules, Represent. Theory (1997), 83–114.[24] Williamson G., Algebraic representations and constructible sheaves, Jpn. J. Math. (2017), 211–259,arXiv:1610.06261.[25] Williamson G., Examples of p -canonical bases for the anti-spherical module for SL , available at