ADMISSIBLE W -GRAPHS. John R. Stembridge

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1 ADMISSIBLE W -GRAPHS John R. Stemridge jrs@umich.edu Contents. General W -Graphs. Admissile W -Graphs 3. The Agenda 4. The Admissile Cells in Rank 5. Cominatorial Characterization

2 . General W -Graphs Let (W, S) e a Coxeter system, S = {s,..., s n }. Primarily, W = a finite Weyl group. Let H = H(W, S) = the associated Iwahori-Hecke algera over Z[q ±/ ]. = T,..., T n (T i q)(t i + ) = 0, raid relations. Definition. An S-laeled graph is a triple Γ = (V, m, τ), where V is a (finite) vertex set, m : V V Z[q ±/ ] (i.e., a matrix of edge-weights), τ : V S = [n]. Notation. Write m(u v) for the (u, v)-entry of m. Let M(Γ) = free Z[q ±/ ]-module with asis V. Introduce operators T i on M(Γ): { T i (v) = qv v+q / u:i/ τ(u) m(v u)u if i / τ(v), if i τ(v). Definition (K-L). Γ is a W -graph if this yields an H-module. Note: (T i q)(t i + ) = 0 (always), so W -graph raid relations.

3 Remarks. { T i (v) = Kazhdan-Lusztig use T t i, not T i. qv if i / τ(v), v+q / u:i/ τ(u) m(v u)u if i τ(v). () Restriction: for J S, Γ J := (V, m, τ J ) is a W J -graph. At q =, we get a W -representation. However, raid relations at q = W -graph: If τ(v) τ(u), then () does not depend on m(v u). Convention. m(v u) := 0 whenever τ(v) τ(u). Definition. A W -cell is a strongly connected W -graph. For every W -graph Γ, M(Γ) has a filtration whose suquotients are cells. Typically, cells are not irreducile as H-reps or W -reps. However (Gyoja, 984): if W is finite every irrep may e realized as a W -cell.

4 . Admissile W -graphs H has a distinguished asis {C w : w W } (the Kazhdan-Lusztig asis). The action of T i on C w is encoded y a W -graph Γ W = (W, m, τ), where τ(v) = {s S : l(sv) < l(v)} (left descent set), m is determined y the Kazhdan-Lusztig polynomials: m(u v) = { µ(u, v)+µ(v, u) if τ(u) τ(v), 0 if τ(u) τ(v), where µ(u, v) = coeff. of q (l(v) l(u) )/ in P u,v (q) (= 0 unless u v). Remarks. This graph is generally very sparse, and has edge weights in Z. The cells of Γ W decompose the regular representation of H. These cells are often not irreducile as H-reps or W -reps. For all W of interest (finite or crystallographic), we know that P u,v (q) has nonnegative coefficients. These W -graphs are edge-symmetric; i.e., m(u v) = m(v u) if τ(u) τ(v) and τ(v) τ(u). () If µ(u, v) 0, then l(u) l(v) mod, so these graphs are ipartite. (Vogan) Similar W -graphs, cells, and K-L polynomials exist for Harish-Chandra modules ( real Lie group reps).

5 03: 0:8 0:35 00:96 98:96 99:60 97:60 96:560 95:560 93:560 94:567 9:00 90:39 88:375 9:00 89:65 87:405 77:00 86:340 83:340 84:340 76:340 75:340 8:340 8: : :340 85:55 80:89 78:3640 7:89 79: : :5040 6:400 6:7560 7: : : : : :835 64: : :4536 5:400 57:400 59:400 55: :8800 5: : :778 54:00 50:400 4:400 46:400 44: : :6075 4: : :835 45: :400 34:3500 3: : : : :3500 8:7560 3:5040 7:3640 6:89 30:89 4:340 5:3640 3:3640 9:340 6:340 :340 0:340 9:340 :340 33:55 8:65 5:405 7:00 :39 4:375 :567 0:00 3:00 6:560 8:560 9:560 7:60 4:60 5:96 3:96 :35 :8 0:

6 Definition. An S-laeled graph Γ = (V, m, τ) is admissile if it is edge-symmetric; i.e., m(u v) = m(v u) if τ(u) τ(v) and τ(v) τ(u), all edge weights m(u v) are nonnegative integers, and it is ipartite. Main Contention. These axioms capture the W -graphs that we care aout, and are sufficiently rigid that there should e few synthetic cells. Sufficient understanding of admissile W -cells could yield constructions of K-L cells without having to compute K-L polynomials. Example. The admissile A 4 -cells: All of these are K-L cells; none are synthetic.

7 The admissile D 4 -cells (three are synthetic):

8 3. The Agenda Prolem (W finite). Are there finitely many admissile W -cells? Confirmed for A,..., A 9, B, B 3, D 4, D 5, D 6, E 6, and rank. What aout W W -cells? Prolem. Classify/generate all admissile W -cells. Are the only admissile A n -cells the K-L cells? Caution (McLarnan-Warrington): Interesting things happen in A 5. Prolem 3. Understand cominatorial rigidity for cells. Rigidity means M(Γ ) = M(Γ ) (as W -reps) Γ = Γ. Example: Are K-L cells rigid? True for A n. Admissile W -cells are not rigid in general. Prolem 4. Understand compressiility of cells. A given W -cell or W -graph should e reconstructile from a small amount of data. (One possile approach: ranching rules).

9 4. The Admissile Cells in Rank Consider W = I (m), m <. (When m =, anything goes.) Given an I (m)-graph, partition the vertices according to τ: φ Focus on non-trivial cells: τ(v) = {} or {} for all v V. Encode edge weights {} {} (resp., {} {}) y a matrix A (resp. B). The conditions on A and B are as follows: m = : A = 0, B = 0. m = 3: AB =, BA =. m = 4: ABA = A, BAB = B. m = 5: ABAB 3AB + = 0, BABA 3BA + = 0. Remarks.. If we assume only Z-weights, no classification is possile (cf. m = 3). Edge symmetry A = B t. When m = 3, edge weights Z 0 edge symmetry, ut not in general.

10 Theorem. A -colored graph is an admissile I (m)-cell iff it is a properly -colored A-D-E Dynkin diagram whose Coxeter numer divides m. Example. The Dynkin diagrams with Coxeter numer dividing 6 are A, A, D 4, and A 5. Therefore, the (nontrivial) admissile G -cells are Note: The nontrival K-L cells for I (m) are paths of length m. Proof Sketch. Let Γ e any properly -colored graph. [ ] 0 B Let M = encode the edge weights of Γ. A 0 Let φ m (t) e the Cheyshev polynomial such that φ m ( cos θ) = Then Γ is an I (m)-cell φ m (M) = 0 sin mθ sin θ. M is diagonalizale with eigenvalues { cos(πj/m) : j < m}. Now assume Γ is admissile (M = M t, Z 0 -entries). If Γ is an I (m)-cell, then M is positive definite. Hence, M is a (symmetric) Cartan matrix of finite type. Conversely, let A e any Cartan matrix of finite type (symmetric or not). Then the eigenvalues of A are cos(πe j /h), where e, e,... are the exponents and h is the Coxeter numer.

11 5. Cominatorial Characterization For simplicity, we assume W is raid-finite: s i s j has finite order for all i, j. Theorem. If (W, S) is raid-finite, then an admissile S-laeled graph is a W -graph if and only if it satisfies the Compatiility Rule, the Simplicity Rule, the Bonding Rule, and the Polygon Rule. The Compatiility Rule (applies to all W -graphs for all W ): If m(u v) 0, then every i τ(u) τ(v) is onded to every j τ(v) τ(u). Necessity follows from analyzing commuting raid relations. Reformulation: Define the compatiility graph Comp(W, S): vertex set S = [n], edges I J when I J and every i I J is onded to every j J I. Compatiility means that τ : Γ Comp(W, S) is a graph morphism.

12 Compatiility graphs for A 3, A 4, and D 4 : 3 a 3 a 3 a 3 a 3 c 4 34 a 34 c a 3 c a a c 3 4 a 3 c 4 a 4 c 34 a 3 c a 3 c 3 c 0 a a ac c a a c c 3 3 c ac a 0 a c 3 ac

13 The Simplicity Rule (applies only in the raid-finite case): All edges are either simple or are inclusion arcs. That is, m(u v) 0 implies m(u v) = m(v u) = or τ(u) τ(v). Necessity follows from Theorem. The Bonding Rule: If s i s j has order p ij 3, then the cells of Γ must e {i,j} singletons with τ = or τ = {i, j}, and A-D-E Dynkin diagrams with Coxeter numer dividing p ij. Necessity again follows from Theorem. Example. If p ij = 3, then the nontrivial cells in Γ {i,j} are {i} {j}. Equivalently (for onds with p ij = 3): if i τ(u), j / τ(u) then there is a unique vertex v adjacent to u such that i / τ(v), j τ(v). Remark. The Compatiility, Simplicity, and Bonding Rules suffice to determine all admissile A 3 -cells.

14 The Polygon Rule: [Compare with G. Lusztig, Represent. Theory (997), Prop. A.4.] Define V ij := {v V : i τ(v), j τ(v)}, V i j := {v V : i τ(v), j / τ(v)}, V ij := {v V : i / τ(v), j / τ(v)}. A path u v v r v is alternating of type (i, j) if u V ij, v V i j, v V j i, v 3 V i j, v 4 V j i,..., v V ij. Set Nij r (u, v) := m(u v )m(v v ) m(v r v) (sum over all r-step alternating paths of type (i, j)). Then: Nij(u, r v) = Nji(u, r v) for r p ij. Example. 3-step alternating paths u i,j i/j j/i j/i i/j v