W-graph representations for Coxeter groups and … · W-graph representations for Coxeter groups and Hecke algebras ... st whenever m st > 3. A Coxeter system (W,S) ... then dimH

W-graph representations for

Coxeter groups and Hecke algebras

Yunchuan Yin

A thesis submitted in fulfilmentof the requirements for the degree of

Doctor of Philosophy

School of Mathematics and StatisticsThe University of Sydney

July 2004

Acknowledgements

The work in this thesis was supported by IPRS and University ofSydney IPA scholarships. I thank the University of Sydney for theseoffers, and in particular the School of Mathematics and Statistics.

During the period of research and preparation of this thesis, A/ProfR.B.Howlett held the position of supervisor. I appreciate his abundantexpertise and the way of guidance. Throughout, he has always beenwilling to discuss any matter, small or large, at any time and shownan abundance of patience and good humour. For his many invaluablecomments, and also for carefully reading the thesis and correcting myEnglish, I would like to thank him very much.

I wish to thank my family for all their love and support.

i

Contents

Acknowledgements i

Introduction v

Chapter 1. Coxeter groups, Hecke algebras and theirrepresentations 1

1. Coxeter groups and Weyl groups 12. Representations of Coxeter groups and Hecke algebras 4

Chapter 2. Inducing W-graphs 91. Induced modules 92. The elements Rx,δ,y,γ 103. The construction of the W-graph basis 124. Inducing ordered W-graphs 225. Inducing bipartite W-graphs 246. Inducing cells 267. WK-cells in induced W-graphs 278. Connection with Kazhdan-Lusztig polynomials 34

Chapter 3. An inversion formula and duality 371. Duality of W-graphs 372. Further properties of the polynomials Rx,δ,y,γ 403. An inversion formula for the generalized Kazhdan-Lusztig

polynomials 46

Chapter 4. Irreducible W-graphs for types D4 and D5 511. W -graph conditions 512. The Weyl group of type Dn 583. Type D4 594. Type D5 645. Tables for types D4 and D5 66

Chapter 5. W-graphs for types E6 and E7 711. Using Magma to induce W-graphs 712. Decomposing the induced graphs 743. Tables for type E6 77

iii

iv CONTENTS

4. The construction of irreducible W-graphs for type E6 835. The construction of irreducible W-graphs for type E7 856. Tables for type E7 88

Bibliography 91

Introduction

The “W-graph” concept was introduced by Kazhdan and Lusztigin their influential paper [16]. If W is a Coxeter group then a W-graph provides a method for constructing a matrix representation ofthe Hecke algebra � associated with W , the degree of the represen-tation being the number of vertices of the W-graph. In [16] Kazhdanand Lusztig showed, among other things, that any irreducible repre-sentation of Hecke algebra of type An can be afforded by a W-graph.In references [1, 6, 17] W-graphs are found for all the irreducible rep-resentations of Hecke algebras of type Bn = Cn(n � 5 ), F4, H3, H4

and I2(m). In 1984, A. Gyoja [14] showed that any irreducible repre-

sentation of Hecke algebra of any finite Weyl group can be afforded bya W-graph.

The principal aim of this thesis is to explicitly construct W-graphsfor the irreducible representations of � when W is of type E6 and E7.

In order to investigate W-graphs for arbitrary finite Coxeter groups,we make extensive use of induced representations. In general, if A is analgebra and B a subalgebra of A, then whenever V is a left B-modulewe define the A-module induced from V to be the space A⊗

B V withthe obvious left A-action.

In the case that G is a group and H a subgroup of G of index n,the above construction can be made explicit as follows. Let t1, · · · , tnbe a system of coset representatives of H in G, so that

G = Ht1 ∪Ht2 ∪ · · · ∪Htn.Suppose that B is a matrix representation of H of degree d. For eachx ∈ G we define A(x) = (B(tixtj

−1))n×n, an n×n array of d×d blocks,where

B(tixtj−1) =

{B(tixtj

−1) tixtj−1 ∈ H

0 tixtj−1 /∈ H.

Then A is a matrix representation of G of degree dn. Note that A isusually reducible.

It is natural to ask whether there is aW-graph analogue of the aboveconstruction. We shall show that in fact there is. To be precise, let �

v

vi INTRODUCTION

be the Hecke algebra associated with a finite Coxeter group W , let WJ

be a parabolic subgroup of W and V a module for the correspondingHecke algebra�J . Then a WJ -graph structure for V gives rise to a W-graph structure for the induced module � ⊗�J

V . The vertices of theinduced W-graph can be identified with the ordered pairs (w, γ), whereγ runs through vertices of the WJ -graph and w through representativesof the cosets of WJ in W .

In the case that WJ is the identity subgroup and V has dimen-sion 1, the construction coincides with that given by Kazhdan andLusztig [16] for the regular representation, while for arbitrary J andV of dimension 1 it concides with construction given by Couillens [5]and Deodhar [10].

Of course the representation obtained in this way is usually re-ducible; so in order to use this construction to find W-graphs for irre-ducible representations of W , we need a method of finding irreducibleconstituents of W-graph representations. Our chief tool in this direc-tion is the concept of cells in W-graphs, introduced by Kazhdan andLusztig [16]. The computational algebra package Magma1 has beenused to assist in finding Kazhdan-Lusztig cells in induced W-graphs.

Here is the review of the contents.Chapters 1,2 and 3 form the theoretical part. Chapter 1 contains

the necessary background material on Coxeter groups and Hecke alge-bras. In Chapter 2, we shall generalize the notion of Kazhdan-Lusztigpolynomials to suit our induced � -modules, and prove an existenceand uniqueness result that yields a W-graph basis for the induced mod-ule. We shall pay attention to the cells embedded in the inducedW-graph, and investigate relations between our polynomials and theregular Kazhdan-Lusztig ones.

In Chapter 3, we shall investigate duality of W-graphs. In partic-ular, if Γ and Γ′ are dual WJ -graphs we investigate the relationshipbetween the generalized Kazhdan-Lusztig polynomials associated withΓ and those associated with Γ′. An inversion formula is derived andduality of the induced W-graphs is proved.

The second part contains Chapter 4, 5 and 6, in which the mainaim is the construction of W-graphs for the irreducible representationsfor E6 and E7.

Chapter 4 commences with an investigation of general properties ofW-graphs for groups with simply laced Coxeter diagrams. In this caseit is not hard to find necessary and sufficient conditions for a graphto be a W-graph. For groups of low rank it is possible to use these

1Developed by J. J. Cannon et al.; see http://www.magma.maths.usyd.edu.au

INTRODUCTION vii

conditions to find W -graphs for the irreducible representations. We dothis for types D4 and D5, since the W -graphs for D5 form the startingpoint for our E6 calculations.

For E6 and E7 it was necessary to use machine calculations, sincesome of the irreducible representations have large degrees. For example,one of the cases in type E7 has degree 512, and it is not practical toconstruct its W-graph by hand. A Magma program was written tocalculate induced W -graphs, using the theory described in Chapter 2.The cells in these induced W -graphs are easily found using one ofMagma’s inbuilt functions.

Our method is as follows. Consider W (D5) as a parabolic subgroup

of W (E6), so the index is #W (E6)#W (D5)

= 27. If Γ is a W-graph correspond-

ing to an irreducible d-dimensional representation of W (D5), so that

Γ has d vertices, then the induced W-graph IndW (E6) Γ has 27d ver-tices. By finding cells in such induced graphs, we are able to findW-graphs for some of the irreducible constituents of the induced repre-sentation. (Calculations with inner products of characters tell us whichirreducibles appear in which induced representations.) Although someof the irreducible representations are not directly obtainable in thisway, in practice it is usually not difficult to produce W-graphs also forthe irreducible constituents that are not cells. More precisely, we areable to make any of the irreducible constituents appear as cells by sim-ple basis changes that modify the induced W-graph in question. Thesame method can be applied to type E7.

The W-graphs for the irreducible representations of E6 and E7, aswell as the Magma programs used in the calculations, are all on theattached CD.

REMARK: Sections 1, 2 and 3 in Chapter2 constitute a modi-fied version of the paper “Inducing W-graphs” by R. B. Howlett andYunchuan Yin, which was published in Mathematische Zeitschrift 244,415-431 (2003).

CHAPTER 1

Coxeter groups, Hecke algebras and their

representations

This chapter contains some necessary background material for thisthesis. We start with some general discussion of Coxeter groups andHecke algebras, and then we introduce the explicit representations ofthem due to Kazhdan and Lusztig [16].

1. Coxeter groups and Weyl groups

The pair (W,S ) is called a Coxeter system if W has a presentation

W = 〈s ∈ S | s2 = 1, (st)mst = 1 for all s, t ∈ S with s �= t〉with mst = mts positive integers greater than 1, or mst = ∞. Thisthesis concentrates on finite Coxeter groups; for these mst = ∞ doesnot occur.

The Coxeter graph associated to the given Coxeter system consistsof a vertex set which is in bijective correspondence with S, the verticescorresponding to s and t in S are joined by an edge if and only ifmst � 3. Each edge is labelled by the number mst whenever mst > 3.

A Coxeter system (W,S) is defined to be indecomposable if itsCoxeter graph is connected, i.e., for any s, t ∈ S, there exists a sequences0 = s, s1, · · · , sr = t in S for some r � 0, so that {si−1, si} is an edgefor every i, 1 � i � r.

The indecomposable Coxeter groups of finite order are classified asfollows

An ( n � 1 vertices )

Bn ( n � 2 vertices )4

Dn ( n � 4 vertices )

E6

1

2 1. COXETER GROUPS AND HECKE ALGEBRAS

E7

E8

F44

G26

H35

H45

I2(p) (p = 5 or p � 7)p

Assume that

W = 〈s ∈ S | s2 = 1, (st)mst = 1, for s �= t〉is a finite Coxeter group. Then W can be described as a group gener-ated by reflections in a finite dimensional Euclidean space.

Let V be a vector space over the real field R with basis {αs | s ∈ S}.We define a bilinear form on V , (v, v′) �→ 〈v, v′〉 by

〈αs, αt〉 = − cosπ

mst

and extended by linearity.This form is symmetric since mst = mts. Note also that 〈αs, αs〉 = 1

for each s. If Hs is the subspace of V given by

Hs = { v ∈ V ; 〈αs, v〉 = 0 }then dimHs = l − 1 and we have V = Rαs ⊕Hs.

We now define a linear map τs : V → V by

τs(v) = v − 2〈αs, v〉αs.

Then τs(αs) = −αs and τs(v) = v whenever v ∈ Hs. Thus τs is thereflection in the hyperplane Hs. Clearly τs

2 = 1. It is also true that(τsτt)

mst = 1 if s �= t, so there exists a homomorphism

θ : W → 〈τs | s ∈ S〉

1. COXETER GROUPS AND WEYL GROUPS 3

from W to the group of transformations of V generated by the τs givenby θ(s) = τs, for s ∈ S. Since 〈τsv, τsv′〉 = 〈v, v′〉 for all v, v′ ∈ V ,this homomorphism θ gives rise to a representation of W as a group ofisometries of V . Moreover, the representation θ is faithful (see Bour-baki [2] or Humphreys [15]).

Thus we claim that any finite Coxeter group can be described asa reflection group in finite dimensional Euclidean space. The set S iscalled the set of simple generating reflections.

It is a straightforward consequence of the faithfulness of the reflec-tion representation of W that if J is any subset of the set S of simplereflections the the subgroup WJ of W generated by J is isomorphic tothe Coxeter group

W = 〈s ∈ J | s2 = 1, (st)mst = 1 for s, t ∈ J , s �= t〉.We call such subgroups of W parabolic subgroups.

For any element of a Coxeter group (W,S), we define �(w) to bethe smallest number r such that w = s1s2 · · · sr with all si ∈ S. Thenumber �(w) is called the length of w. The expression w = s1s2 · · · sr

is called a reduced form of w if r = �(w) and si ∈ S, 1 � i � r.The following two results are well known. Proofs can be found, for

example, in the books of Bourbaki and Humphreys cited above.

Lemma 1.1. Let (W,S) be a Coxeter system and J an arbitrarysubset of S. Then each coset wWJ in W contains a unique element ofthe set DJ = { d ∈ W | �(ds) � �(d) for all s ∈ J }. Furthermore, forall d ∈ DJ and w ∈WJ we have �(dw) = �(d) + �(w).

The set DJ defined in Lemma 1.1 is called the set of minimal cosetrepresentatives for W/WJ .

Lemma 1.2. Let (W,S) be a Coxeter system and let J ⊆ S be suchthat WJ is finite. The subgroup WJ has a unique element wJ such that�(swJ) � �(wJ) for all s ∈ J . Furthermore, for all w ∈ WJ we have�(wwJ) = �(wJ) − �(w).

Combining Lemmas 1.1 and 1.2 we deduce readily that if WJ is afinite parabolic subgroup of W then every coset wWJ contains a uniqueelement of maximal length as well as a unique element of minimallength: if the minimal length element is d, the maximal length elementis dwJ .

A Coxeter group is called crystallographic if mst ∈ {2, 3, 4, 6 } forall s �= t. Thus the Coxeter groups of type Al, Bl, Dl, E6, E7, E8, F4, G2

are crystallographic. The finite crystallographic Coxeter groups willfrom now on be called Weyl groups.


2. Representations of Coxeter groups and Hecke algebras

We start by recalling some of the results from the paper of Kazhdanand Lusztig [16]. Let (W,S) be a Coxeter system (as defined above)and let F be the ring Z[q−1/2, q1/2] (where q1/2 is an indeterminate).The Hecke algebra � of (W,S) is the algebra generated by (Ts)s∈S

subject to

Ts2

= q + (q − 1)Ts;

TrTsTr · · ·︸︷︷︸mrs factors

= TsTrTs · · ·︸︷︷︸mrs factors

for all r, s ∈ S. It is well known that if w = s1s2s3 · · · sl ∈ W with�(w) = �, then the element Tw = Ts1Ts2 · · · Tsl

∈ � is independentof the choice of reduced expression s1s2s3 · · · sl for w. Moreover theelements Tw, w ∈W , form a basis of � .

For any w ∈ W , we define qw = q�(w) and εw = (−1)�(w). Let � bethe Bruhat order relation on W . That is, for y, w ∈ W , we say y � wif there exist reduced forms w = s1s2 · · · sr and y = si1si2 · · · sit withall si lying in S such that i1, i2, · · · , it is a subsequence of 1, 2, · · · , r.Let a �→ a be the involution of the ring F defined by q1/2 = q−1/2, thisextends to an involution h �→ h of � , defined by∑

awTw =∑

awTw.

where Tw = T−1w−1 . We can now state

Theorem 1.3 (Kazhdan-Lusztig [16, Theorem 1.1]). For every el-ement w ∈W , there is a unique element Cw ∈� such that

Cw = Cw,

Cw =∑y�w

εyεwqw1/2qy

−1Py,wTy

where Py,w ∈ � is a polynomial in q of degree less than or equal to12(�(w) − �(y) − 1) for y < w, and Pw,w = 1.

The polynomials Py,w in the theorem are called Kazhdan-Lusztigpolynomials.

Observe that if we put Ts = q−1/2Ts, then

Ts2 = q−1Ts

2= q−1(q + (q − 1)Ts) = 1 + (q1/2 − q−1/2)Ts.

We find it convenient to change our notation and replace q1/2 byq, taking � to be an algebra over A = Z[q−1, q], the ring of Laurentpolynomials with integer coefficients in the indeterminate q.

2. REPRESENTATIONS OF COXETER GROUPS 5

The defining relations for � now are as follows

Ts2 = 1 + (q − q−1)Ts

TrTsTr · · ·︸︷︷︸mrs factors

= TsTrTs · · ·︸︷︷︸mrs factors

(for all r, s ∈ S). Moreover, � has A-basis { Tw | w ∈ W } whereTw = Ts1Ts2 · · ·Tsl

whenever s1s2 · · · sl is a reduced expression for w,and

TsTw =

{Tsw if �(sw) > �(w)

Tsw + (q − q−1)Tw if �(sw) < �(w),

for all w ∈W and s ∈ S.We also define A+ = Z[q], the ring of polynomials in q with integer

coefficients, and let a �→ a be the involutory automorphism of A suchthat q = q−1. This involution on A extends to an involution on �satisfying Ts = T−1

s = Ts +(q−1−q) for all s ∈ S. This gives Tw = T−1w−1

for all w ∈W .For each J ⊆ S define WJ = 〈J〉, the corresponding parabolic

subgroup of W , and let DJ = {w ∈ W | �(ws) > �(w) for all s ∈ J },the set of minimal coset representatives ofW/WJ . Let�J be the Heckealgebra associated with WJ . As is well known, �J can be identifiedwith a subalgebra of � .

We can now restate Theorem 1.3

Theorem 1.4. The algebra � has a unique basis {Cw | w ∈ W }such that Cw = Cw for all w and Cw =

∑y∈W py,wTy for some elements

py,w ∈ A+ with the following properties:

(i) py,w = 0 if y � w;(ii) pw,w = 1;(iii) py,w has zero constant term if y �= w.

The polynomials py,w are related to the polynomials Py,w of [16] (the

Kazhdan-Lusztig polynomials) by py,w(q) = (−q)�(w)−�(y)Py,w(q2). Thatis, to get py,w from Py,w replace q by q2, apply the bar involution, andthen multiply by (−q)�(w)−�(y). The quantity µ(y, w), which is the coef-

ficient of q12(�(w)−�(y)−1) in Py,w, is the coefficient of q in (−1)�(w)−�(y)py,w.

However, since Kazhdan and Lusztig show that µ(y, w) is nonzero onlywhen �(w)−�(y) is odd, µ(y, w) can also be described as the coefficientof q in −py,w.

2.1. Definition of W -graph. Since we have slightly modified thedefinition of Hecke algebra used in [16], we are forced to also slightlyalter the definition of W-graph. We define a W-graph datum to be


a triple (Γ, I, µ) consisting of a set Γ (the vertices of the graph), afunction

I : γ �→ Iγ

from Γ to the set of all subsets of S, and a function

µ : Γ × Γ → Z

such that µ(δ, γ) �= 0 if and only if {δ, γ} is an edge of the graph. Thesedata are subject to the requirement that AΓ, the free A-module on Γ,has an � -module structure satisfying

(1) Tsγ =

{−q−1γ if s ∈ Iγ

qγ +∑

{δ∈Γ|s∈Iδ} µ(δ, γ)δ if s /∈ Iγ ,

for all s ∈ S and γ ∈ Γ. If τs is the A-endomorphism of AΓ suchthat τs(γ) is the right-hand side of Eq. (1) then this requirement isequivalent to the condition that for all s, t ∈ S such that st has finiteorder,

τsτtτs . . .︸︷︷︸m factors

= τtτsτt . . .︸︷︷︸m factors

where m is the order of st. (Note that the definition of τs guaranteesthat (τs + q−1)(τs − q) = 0 for all s ∈ S.)

For simplicity, if (Γ, I, µ) is a W-graph datum, we say that Γ isW-graph. We call Iγ the descent set of the vertex γ ∈ Γ, and we callµ(δ, γ) and µ(γ, δ) the edge weights associated with the edge {δ, γ}. Inalmost all the cases we consider it turns out that µ(γ, δ) = µ(δ, γ).

Given a W-graph Γ we define

Γ−s = { γ ∈ Γ | s ∈ Iγ },

Γ+s = { γ ∈ Γ | s /∈ Iγ }.

Observe that the involution a �→ a on A determines a semilinear in-volution v �→ v on AΓ with the property that γ = γ for all γ ∈ Γ. Ifs ∈ S and γ ∈ Γ then

Tsγ = Tsγ = Tsγ + (q−1 − q)γ;

thus if γ ∈ Γ−s it follows that

Tsγ = −q−1γ + (q−1 − q)γ = −qγ = Tsγ,

2. REPRESENTATIONS OF COXETER GROUPS 7

while if γ ∈ Γ+s we find that

Tsγ =(qγ +

∑δ∈Γ−

s

µ(δ, γ)δ)

+ (q−1 − q)γ

= q−1γ +∑δ∈Γ−

s

µ(δ, γ)δ

= Tsγ.

Since � is generated by { Ts | s ∈ S }, the following proposition is animmediate consequence of these calculations.

Proposition 1.5. If Γ is a W-graph then the associated� -moduleAΓ admits an involution v �→ v that fixes all elements of Γ and iscompatible with the involution h �→ h of � , in the sense that hv = h vfor all h ∈� and v ∈ AΓ.

For use in Chaper 2, we make the following definition.

Definition 1.6. An ordered W -graph is a set Γ with a partial order� and a W -graph datum (Γ, I, µ) satisfying the following conditions:

(i) for all θ, γ ∈ Γ such that µ(θ, γ) �= 0, either θ < γ or γ < θ;(ii) for all γ ∈ Γ+

s , s ∈ S, the set { θ ∈ Γ−s | γ < θ and µ(θ, γ) �= 0 }

is either empty or consists of a single element sγ;(iii) for all s ∈ S and γ ∈ Γ+

s , if sγ exists then µ(sγ, γ) = 1.

2.2. Cells in W-graphs. Following [16], given any W-graph Γ wedefine a preorder relation ≤ on Γ as follows: for γ, γ′ ∈ Γ we say thatγ ≤Γ γ

′ if there exists a sequence of vertices γ = γ0, γ1, · · · γn = γ′ suchthat for each i (1 � i � n), we have both (γi−1, γi) ∈ Θ and Iγi−1

� Iγi.

We shall refer to ≤Γ as the Kazhdan-Lusztig preorder on Γ.Let ∼ be the equivalence relation on Γ associated to the Kazhdan-

Lusztig preorder; thus γ ∼ γ′ means that γ ≤Γ γ′ and γ′ ≤Γ γ. The

corresponding equivalence classes are called the cells of Γ.In the terminology of graph theory, the cells of Γ are the strong

components of the directed graph with vertex set Γ and edge set con-sisting of those ordered pairs (δ, γ) such that µ(δ, γ) �= 0 and Iδ � Iγ.There is an efficient algorithm, due to R. E. Tarjan [18], for findingthe strong components of a directed graph, and Magma has an imple-mentation of this algorithm. We made use of this feature of Magma inour computations for types E6 and E7.

Let ∆ be a subset of Γ with the property that for all γ, δ ∈ Γ, ifδ ∈ ∆ and γ ≤Γ δ then γ ∈ ∆. It follows immediately from Eq. (1)that A∆ is an� -submodule of AΓ; hence (∆, I∆, µ∆×∆) is a W-graphdatum (where I∆ and µ∆×∆ denote the restrictions of I and µ).


If Ξ is a cell in Γ and we define

∆ = {δ ∈ Γ | δ /∈ Ξ and δ ≤Γ γ for some γ ∈ Ξ}then A(Ξ∪∆) and A∆ are both submodules of AΓ. It follows readilythat (IΞ, µΞ×Ξ) is a W-graph datum, the associated � -module beingisomorphic to A(Ξ ∪ ∆)/A∆.

It is immediately clear that if Γ is an ordered W-graph in the senseof Definition 1.6, then the cells in Γ are also ordered W-graphs.

CHAPTER 2

Inducing W-graphs

Let � be the Hecke algebra associated with a Coxeter group W .This thesis is concerned with the problem of finding W-graphs that re-alize given representations of� . In [16], Kazhdan and Lusztig showedthat the regular representation of � has an associated W-graph. Weshall show that if WJ is a parabolic subgroup of W and V is a modulefor the corresponding Hecke algebra�J , then a WJ -graph structure forV gives rise to a W-graph structure for the induced module � ⊗�J

V .Our construction closely parallels that of Kazhdan and Lusztig, whichcan be regarded as the special case J = ∅ and dimV = 1. It alsogeneralizes the work of Couillens [5] and Deodhar [10], correspondingto dimV = 1 and arbitrary J .

1. Induced modules

Suppose now that Γ is a WJ -graph (so that AΓ is an �J-module)and let M = � ⊗�J

AΓ be the � -module induced from AΓ. Then,identifying AΓ with an A-submodule of M in the obvious way, M hasan A-basis { Tdγ | d ∈ DJ , γ ∈ Γ }, where DJ is the set of minimalcoset representatives for W/WJ . We can define an involution on M by

setting∑

d,γ ad,γTdγ =∑

d,γ ad,γTdγ for all d ∈ DJ and γ ∈ Γ. SinceT1 is the identity element of � this extends the involution on AΓdescribed in Proposition 1.5, and clearly Tdv = Tdv for all d ∈ DJ andv ∈ AΓ. Thus for all d ∈ DJ and u ∈ WJ we have

Tduγ = TdTuγ = Td(Tuγ) = Td(Tuγ) = Tdu γ (for all γ ∈ Γ),

and hence Tduv = Tduv for all v ∈ AΓ. Thus hv = h v for all h ∈ �and v ∈ AΓ, and so we obtain the following result.

Proposition 2.1. The involution on M defined above is compatiblewith the involution on � .

Proof. Let h ∈ � and m ∈ M be arbitrary. Then m = kv forsome k ∈� and v ∈ AΓ, and so

hm = h(kv) = h(k v) = (hk)v = hkv = hm,

as required. �9

10 2. INDUCING W-GRAPHS

Our aim is to associateM with a W-graph by finding an appropriatebasis of M . In particular, elements of this W-graph basis will be fixedby the involution.

The following result is well known.

Lemma 2.2 (Deodhar [9, Lemma 3.2]). Let J ⊆ S and s ∈ S, anddefine

D−J,s = { d ∈ DJ | �(sd) < �(d) },

D+J,s = { d ∈ DJ | �(sd) > �(d) and sd ∈ DJ },

D0J,s = { d ∈ DJ | �(sd) > �(d) and sd /∈ DJ },

so that DJ is the disjoint union D−J,s ∪D+

J,s ∪D0J,s. Then sD+

J,s = D−J,s,

and if d ∈ D0J,s then sd = dt for some t ∈ J .

2. The elements Rx,δ,y,γ

For all x, y ∈ DJ and γ, δ ∈ Γ we define elements Rx,δ,y,γ ∈ A bythe formula

(2) Tyγ =∑

x∈DJ , δ∈Γ

Rx,δ,y,γTxδ.

We begin by deriving formulas which provide an inductive procedurefor calculating these elements.

If y = 1 then Tyγ = γ, and hence

Rx,δ,1,γ =

{1 if x = 1 and δ = γ

0 otherwise.

Suppose now that 1 �= y ∈ DJ , so that we may choose s ∈ S with�(sy) = �(y) − 1. Then by Lemma 2.2 we have y = sv with v ∈ D+

J,s

and �(y) = �(v) + 1, and

Tyγ = Ts(Tvγ) =∑

x∈DJ , δ∈Γ

Rx,δ,v,γT−1s Txδ.

Each x in this expression lies in exactly one of the sets D+J,s, D

−J,s or

D0J,s. When x ∈ D0

J,s we write t = x−1sx (an element of J); in this case

T−1s Tx = TxT

−1t . When x ∈ D−

J,s we have T−1s Tx = Tsx, while x ∈ D+

J,s

2. THE ELEMENTS Rx,δ,y,γ 11

gives T−1s Tx = Tsx + (q−1 − q)Tx. Thus we obtain

Tyγ =∑δ∈Γ

∑x∈D+

J,s

Rx,δ,v,γ(Tsx + (q−1 − q)Tx)δ

+∑δ∈Γ

∑x∈D−

J,s

Rx,δ,v,γTsxδ +∑δ∈Γ

∑x∈D0

J,s

Rx,δ,v,γTxT−1t δ

=∑δ∈Γ

∑x∈D−

J,s

Rsx,δ,v,γTxδ +∑δ∈Γ

∑x∈D+

J,s

(q−1 − q)Rx,δ,v,γTxδ

+∑δ∈Γ

∑x∈D+

J,s

Rsx,δ,v,γTxδ +∑

x∈D0J,s

∑δ∈Γ−

t


+∑

x∈D0J,s

∑δ∈Γ+

t


=∑δ∈Γ

∑x∈D−

J,s


∑x∈D+

J,s


+∑δ∈Γ

∑x∈D+

J,s

Rsx,δ,v,γTxδ −∑

x∈D0J,s

∑δ∈Γ−

t

qRx,δ,v,γTxδ

+∑

x∈D0J,s

∑δ∈Γ+

t

Rx,δ,v,γTx

(q−1δ +

∑θ∈Γ−

t

µ(θ, δ)θ)

=∑δ∈Γ

∑x∈D−

J,s


∑x∈D+

J,s


+∑δ∈Γ

∑x∈D+

J,s

Rsx,δ,v,γTxδ −∑

x∈D0J,s

∑δ∈Γ−

t

qRx,δ,v,γTxδ

+∑

x∈D0J,s

∑δ∈Γ+

t

q−1Rx,δ,v,γTxδ +∑

x∈D0J,s

∑θ∈Γ+

t

∑δ∈Γ−

t

µ(δ, θ)Rx,θ,v,γTxδ.

Comparing this with Eq. (2) gives us the following result.

Proposition 2.3. Let γ, δ ∈ Γ and x, y ∈ DJ . If s ∈ S is suchthat �(sy) < �(y) then

Rx,δ,y,γ =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Rsx,δ,sy,γ if x ∈ D−J,s

Rsx,δ,sy,γ + (q−1 − q)Rx,δ,sy,γ if x ∈ D+J,s

q−1Rx,δ,sy,γ if x ∈ D0J,s and δ ∈ Γ+

t

−qRx,δ,sy,γ +∑

θ∈Γ+t

µ(δ, θ)Rx,θ,sy,γ if x ∈ D0J,s and δ ∈ Γ−

t ,

where t = x−1sx.


We can use induction on �(y) to establish that Rx,δ,y,γ = 0 unlessx � y in the Bruhat partial order on W ; this follows from the fact thatif sy � y and x � sy then both x � y and sx � y (see Deodhar [9,Theorem 1.1]). It is also easily seen that

Rx,δ,x,γ =

{1 if δ = γ

0 if δ �= γ,

and if �(y) − �(x) = k then the coefficient of qj in Rx,δ,y,γ is zero for|j| > k, and also zero for |j| = k if δ �= γ.

3. The construction of the W-graph basis

As is the preceding section, we assume that Γ is a WJ -graph andM the induced � -module.

Theorem 2.4. The � -module M induced from AΓ has a uniquebasis {Cw,γ | w ∈ DJ , γ ∈ Γ } such that Cw,γ = Cw,γ for all w ∈ DJ

and γ ∈ Γ, and

Cw,γ =∑

y∈DJ ,δ∈Γ

Py,δ,w,γTyδ

for some elements Py,δ,w,γ ∈ A+ with the following properties:

(i) Py,δ,w,γ = 0 if y � w;

(ii) Pw,δ,w,γ =

{1 if δ = γ,

0 if δ �= γ;

(iii) Py,δ,w,γ has zero constant term if (y, δ) �= (w, γ).

We shall show that the basis elements Cw,γ give M the structure of aW-graph module. That is, there is a W-graph Λ with vertex set DJ ×Γsuch that the A-isomorphism A(DJ ×Γ) → M taking (w, γ) ∈ DJ ×Γto Cwγ is an � -isomorphism. Before giving the proof of Theorem 2.4,we describe the edge weights and descent sets for Λ.

Given y, w ∈ DJ and δ, γ ∈ Γ with (y, δ) �= (w, γ), we definean integer µ(y, δ, w, γ) as follows. If y < w then µ(y, δ, w, γ) is thecoefficient of q in −Py,δ,w,γ, and if w < y then it is the coefficient of qin −Pw,γ,y,δ. If neither y < w nor w < y then

µ(y, δ, w, γ) =

{µ(δ, γ) if y = w,

0 if y �= w.

We write (y, δ) ≺ (w, γ) if y < w and µ(y, δ, w, γ) �= 0.The descent set associated with the vertex (w, γ) of Λ is

I(w, γ) = { s ∈ S | �(sw) < �(w) or sw = wt for some t ∈ Iγ }

3. THE CONSTRUCTION OF THE W-GRAPH BASIS 13

and the edge weight for ((y, δ), (w, γ)) is µ(y, δ, w, γ) (as defined above).Thus {(y, δ), (w, γ)} is an edge of Λ if and only if µ(y, δ, w, γ) �= 0, andthis occurs if and only if either (y, δ) ≺ (w, γ) or (w, γ) ≺ (y, δ), ory = w and {δ, γ} is an edge of Γ.

In accordance with the notation introduced in Section 2, we define

Λ−s = { (w, γ) ∈ DJ × Γ | s ∈ I(w, γ) }

= { (w, γ) | w ∈ D−J,s or w ∈ D0

J,s with t ∈ Iγ },and similarly Λ+

s = { (w, γ) ∈ DJ × Γ | s /∈ I(w, γ) }.Our proof of Theorem 2.4 will also incorporate a proof of the fol-

lowing result, which will be an important component of the subsequentproof that Λ is a W-graph.

Theorem 2.5. Let v ∈ DJ and γ ∈ Γ. Then for all s ∈ S suchthat �(sv) > �(v) and sv ∈ DJ we have

TsCv,γ = qCv,γ + Csv,γ +∑

µ(z, δ, v, γ)Cz,δ,

where the sum is over all (z, δ) ∈ Λ−s such that (z, δ) ≺ (v, γ).

Proof. We address the uniqueness part of Theorem 2.4 first. Fixw ∈ DJ and γ ∈ Γ, and observe that the equation Cw,γ = Cw,γ can bewritten in the form∑

x∈DJδ∈Γ

Px,δ,w,γTxδ =∑y∈DJθ∈Γ

Py,θ,w,γ

∑x∈DJδ∈Γ

Rx,δ,y,θTxδ,

or, equivalently, as

Px,δ,w,γ =∑y∈DJ

∑θ∈Γ

Py,θ,w,γRx,δ,y,θ

for all x ∈ DJ and δ ∈ Γ. Recall that Rx,δ,x,δ = 1, and if (y, θ) �= (x, δ)then Rx,δ,y,θ = 0 unless x < y. Since also Py,θ,w,γ is required to be zerounless y � w, we obtain

(3) Px,δ,w,γ − Px,δ,w,γ =∑

{y,θ |x<y�w}Py,θ,w,γRx,δ,y,θ.

Conditions (ii) and (iii) in Theorem 2.4 specify the elements Px,δ,w,γ

when x = w, and in view of Condition (iii) they are then recursivelydetermined for x < w by Eq. (3): the point is that the right hand sideis known by the inductive hypothesis, and since Px,δ,w,γ is required tobe in A+ and have zero constant term it must equal the sum of theterms on the right hand side of Eq. (3) with positive exponent of q. Sothere is at most one family of elements Px,δ,w,γ satisfying the requiredconditions.


Turning now to the existence part of the proof, we give a recursiveprocedure for constructing elements Px,δ,w,γ satisfying the requirementsof Theorem 2.4. We start with the definition

Py,δ,1,γ =

{0 if (y, δ) �= (1, γ),

1 if (y, δ) = (1, γ).

for all y ∈ DJ and γ, δ ∈ Γ. This gives C1,γ = γ, so that Cw,γ = Cw,γ

holds for w = 1, as do Conditions (i), (ii) and (iii).Now assume that w �= 1 and that for all v ∈ DJ with �(v) < �(w)

the elements Px,δ,v,γ have been defined (for all x ∈ DJ and γ, δ ∈ Γ) sothat the requirements of Theorem 2.4 are satisfied. Thus the elementsCv,γ are known when �(v) < �(w). We may choose s ∈ S such thatw = sv with �(w) = �(v) + 1; note that v ∈ DJ by Lemma 2.2. Inaccordance with the formula in Theorem 2.5 we define

(4) Cw,γ = (Ts − q)Cv,γ −∑

(z,θ)≺(v,γ)

(z,θ)∈Λ−s

µ(z, θ, v, γ)Cz,θ.

Since Ts − q = Ts − q, induction immediately gives Cw,γ = Cw,γ. Wedefine P ′

y,δ,w,γ and P ′′y,δ,w,γ by

(Ts − q)Cv,γ =∑y∈DJδ∈Γ

P ′y,δ,w,γTyδ(5)

∑(z,θ)≺(v,γ)

(z,θ)∈Λ−s

µ(z, θ, v, γ)Cz,θ =∑y∈DJδ∈Γ

P ′′y,δ,w,γTyδ(6)

and define Py,δ,w,γ = P ′y,δ,w,γ − P ′′

y,δ,w,γ.If y ∈ DJ then

(Ts − q)Ty =

⎧⎪⎨⎪⎩Tsy − qTy if y ∈ D+

J,s

Tsy − q−1Ty if y ∈ D−J,s

Ty(Tt − q) if y ∈ D0J,s

where we have written t = y−1sy ∈ J in the case y ∈ D0J,s. Thus we

see that

(Ts − q)Cv,γ =∑

y∈D+J,s

δ∈Γ

Py,δ,v,γ(Tsy − qTy)δ +∑

y∈D−J,s

δ∈Γ

Py,δ,v,γ(Tsy − q−1Ty)δ

+∑

y∈D0J,s

δ∈Γ

Py,δ,v,γTy(Tt − q)δ


=∑

y∈D−J,s

δ∈Γ

(Psy,δ,v,γ − q−1Py,δ,v,γ)Tyδ +∑

y∈D+J,s

δ∈Γ

(Psy,δ,v,γ − qPy,δ,v,γ)Tyδ

+∑

y∈D0J,s

θ∈Γ

Py,θ,v,γTy(Tt − q)θ.

Now for all t ∈ J and θ ∈ Γ,

(Tt − q)θ =

⎧⎪⎨⎪⎩

(−q − q−1)θ if θ ∈ Γ−t∑

δ∈Γ−t

µ(δ, θ)δ if θ ∈ Γ+t ,

and therefore∑θ∈Γ

Py,θ,v,γTy(Tt − q)θ=∑θ∈Γ−

t

(−q − q−1)Py,θ,v,γTyθ +∑θ∈Γ+

t

δ∈Γ−t

µ(δ, θ)Py,θ,v,γTyδ

=∑δ∈Γ−

t

((−q − q−1)Py,δ,v,γ +

∑θ∈Γ+

t

µ(δ, θ)Py,θ,v,γ

)Tyδ.

Now comparing Eq. (5) with the expression for (Ts − q)Cv,γ obtainedabove we obtain the following formulas for the cases y ∈ D+

J,s (case (a)),

y ∈ D−J,s (case (b)), y ∈ D0

J,s and δ ∈ Γ−t (case (c)) and y ∈ D0

J,s and

δ ∈ Γ+t (case (d)):

(7) P ′y,δ,w,γ =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Psy,δ,v,γ − qPy,δ,v,γ (case (a)),

Psy,δ,v,γ − q−1Py,δ,v,γ (case (b)),

(−q − q−1)Py,δ,v,γ +∑

θ∈Γ+t

µ(δ, θ)Py,θ,v,γ (case (c)),

0 (case (d)).

Since Cz,θ =∑

y,δ Py,δ,z,θTyδ, we have∑(z,θ)≺(v,γ)

(z,θ)∈Λ−s

µ(z, θ, v, γ)Cz,θ =∑y∈DJδ∈Γ

∑(z,θ)≺(v,γ)

(z,θ)∈Λ−s

µ(z, θ, v, γ)Py,δ,z,θTyδ

and by comparison with Eq. (6)

(8) P ′′y,δ,w,γ =

∑(z,θ)≺(v,γ)

(z,θ)∈Λ−s

µ(z, θ, v, γ)Py,δ,z,θ.


We must check that with P ′y,δ,w,γ and P ′′

y,δ,w,γ given by Eq’s (7) and (8),the elements Py,δ,w,γ = P ′

y,δ,w,γ−P ′′y,δ,w,γ lie in A+ and satisfy Conditions

(i), (ii) and (iii) of Theorem 2.4.In cases (b) and (c) of Eq. (7), observe that y /∈ Γ+

s whereas v ∈ Γ+s .

Hence y �= v, and the inductive hypothesis guarantees that Py,δ,v,γ is anelement of A+ with zero constant term; so q−1Py,δ,v,γ ∈ A+. It followsthat P ′

y,δ,w,γ ∈ A+ in all cases, and since also P ′′y,δ,w,γ ∈ A+ we deduce

that Py,δ,w,γ ∈ A+.With regard to Condition (i), the inductive hypothesis tells us that

the right hand side of Eq. (7) is nonzero only if y � v or sy � v.Since w = sv with �(w) = �(v)+1, both of these conditions imply thaty � w. Hence P ′

y,δ,w,γ = 0 unless y � w. Similarly, the right hand sideof Eq. (8) is nonzero only if y � z for some z < w; so P ′′

y,δ,w,γ = 0 unlessy < w. Hence Condition (i) is satisfied.

The above remarks show, in particular, that P ′′w,δ,w,γ = 0 in all cases.

Since w � v we see that Pw,δ,v,γ = 0, and since w ∈ D−J,s (by the choice

of s) the second case in Eq. (7) gives

Pw,δ,w,γ = P ′w,δ,w,γ = Pv,δ,v,γ =

{1 if δ = γ

0 if δ �= γ.

Hence Condition (ii) is satisfied.It remains to check that Condition (iii) is satisfied. We may as-

sume that y < w, since otherwise the required conclusion follows fromConditions (i) and (ii).

So suppose that y < w, and consider first the case that y ∈ D+J,s.

Then (z, θ) = (y, δ) cannot occur in the sum in Eq. (8), since (z, θ) ∈ Λ−s

implies that z /∈ D+J,s. Hence all the summands have zero constant term

(by the inductive hypothesis), and so P ′′y,δ,w,γ has zero constant term.

Furthermore, y �= w gives sy �= v; so Psy,δ,v,γ has zero constant term,and hence so does P ′

y,δ,w,γ. So Condition (iii) holds in this case.

Next, suppose that y ∈ D−J,s and (y, δ) ⊀ (v, γ). In this case it is

again true that (z, θ) = (y, δ) cannot occur in Eq. (8), and so P ′′y,δ,w,γ

has zero constant term. Furthermore, (y, δ) ⊀ (v, γ) also implies thatthe coefficient of q in Py,δ,v,γ is zero, whence q−1Py,δ,v,γ has zero constantterm. Again Psy,δ,v,γ has zero constant term since sy �= v; so P ′

y,δ,w,γ

has zero constant term, and the desired conclusion follows.If y ∈ D−

J,s and (y, δ) ≺ (v, γ) then (z, θ) = (y, δ) does arise inEq. (8). Since Py,δ,y,δ = 1, the corresponding summand is exactlyµ(y, δ, v, γ). Since all the other summands have zero constant term itfollows that the constant term of P ′′

y,δ,w,γ is µ(y, δ, v, γ). This is also


the constant term of P ′y,δ,w,γ, since µ(y, δ, v, γ) is the coefficient of q

in −Py,δ,v,γ while Psy,δ,v,γ has zero constant term. So Py,δ,w,γ has zeroconstant term.

Finally, suppose that y ∈ D0J,s. If δ ∈ Γ+

t —that is, t /∈ Iδ—then(y, δ) /∈ Λ−

s , and so (z, θ) = (y, δ) is not allowed in Eq. (8). HenceP ′′

y,δ,w,γ has zero constant term. Since in this case we also have thatP ′

y,δ,w,γ = 0, the desired conclusion follows. So it remains to consider

δ ∈ Γ−t . In this case (z, θ) = (y, δ) occurs in Eq. (8) if and only if

(y, δ) ≺ (v, γ). So, as above, we see that P ′′y,δ,w,γ has constant term

µ(y, δ, v, γ) if (y, δ) ≺ (v, γ), and zero in the other case. Turning toP ′

y,δ,w,γ, we see that the summands µ(δ, θ)Py,θ,v,γ all have zero constant

term, while the constant term of (−q − q−1)Py,δ,v,γ is the coefficient ofq in Py,δ,v,γ, which is µ(y, δ, v, γ) if (y, δ) ≺ (v, γ) and zero otherwise.So Py,δ,w,γ = P ′

y,δ,w,γ − P ′′y,δ,w,γ has zero constant term in either case, as

required. �Observe that the formula for Cw,γ in Theorem 2.4 may be written

asCw,γ = Twγ +

∑{y,δ|y<w}

Py,δ,w,γTyδ,

and inverting this gives

(9) Twγ = Cw,γ +∑

{y,δ|y<w}Qy,δ,w,γCy,δ

where the elements Qy,δ,w,γ (defined whenever y < w) are given recur-sively by

Qy,δ,w,γ = −Py,δ,w,γ −∑

{z,θ|y<z<w}Qy,δ,z,θPz,θ,w,γ.

In particular, Qy,δ,w,γ is in A+, has zero constant term, and has coeffi-cient of q equal to µ(y, δ, w, γ).

We now state our main result.

Theorem 2.6. The basis elements Cw,γ give M the structure of aW-graph module, as described above.

Proof. For all (z, θ), (w, γ) ∈ DJ × Γ we define ξ(z, θ, w, γ) ∈ Zas follows: if z � w we put

ξ(z, θ, w, γ) = µ(z, θ, w, γ)

and if z � w we put

ξ(z, θ, w, γ) =

{1 if (z, θ) = (rw, γ) and �(z) > �(w) for some r ∈ S,

0 otherwise.


We start by using induction on �(w) to prove that for all s ∈ S

(10) TsCw,γ =

⎧⎨⎩−q−1Cw,γ if (w, γ) ∈ Λ−

s ,

qCw,γ +∑

(z,θ)∈Λ−s

ξ(z, θ, w, γ)Cz,θ if (w, γ) /∈ Λ−s .

If w ∈ D+J,s then (w, γ) /∈ Λ−

s , and Eq. (10) follows immediately from

Theorem 2.5 (applied with v replaced by w), since the only (z, θ) ∈ Λ−s

with ξ(z, θ, w, γ) �= 0 and �(z) � �(w) is (z, θ) = (sw, γ).

If w ∈ D−J,s, which implies that (w, γ) ∈ Λ−

s , then writing v = swand applying Theorem 2.5 gives

Cw,γ = (Ts − q)Cv,γ −∑

µ(z, δ, v, γ)Cz,δ,

where (z, δ) ≺ (v, γ) and (z, δ) ∈ Λ−s for all terms in the sum. The

inductive hypothesis thus gives TsCz,δ = −Cz,δ, and since we also haveTs(Ts − q) = −q−1(Ts − q) it follows that TsCw,γ = −q−1Cw,γ, as re-quired.

Now suppose that w ∈ D0J,s, and as usual let us write sw = wt,

where t ∈ J . Suppose first that t ∈ Iγ, so that (w, γ) ∈ Λ−s . By Eq. (9)

above,

Cw,γ = Twγ −∑

{y,δ|y<w}Qy,δ,w,γCy,δ,

and since TsTwγ + q−1Twγ = Tw(Ttγ + q−1γ) = 0 we find that

(11) TsCw,γ + q−1Cw,γ = −∑

{y,δ|y<w}Qy,δ,w,γ(TsCy,δ + q−1Cy,δ).

By the inductive hypothesis,

TsCy,δ+q−1Cy,δ =

⎧⎨⎩

0 if (y, δ) ∈ Λ−s

(q + q−1)Cy,δ +∑

(z,θ)∈Λ−s

ξ(z, θ, y, δ)Cz,θ if (y, δ) /∈ Λ−s ,

and so Eq. (11) gives

(12) TsCw,γ + q−1Cw,γ = −∑

(y,δ)/∈Λ−s

y<w

Qy,δ,w,γ(q + q−1)Cy,δ + X


for some X in the A-submodule spanned by the elements Cz,θ for(z, θ) ∈ Λ−

s . Now since Ts = T−1s + (q − q−1) it follows that

(Ts + q−1)Cw,γ = (Ts + q−1)Cw,γ

= −∑

(y,δ)/∈Λ−s

y<w

Qy,δ,w,γ(q−1 + q)Cy,δ + X,

and comparing with Eq. (12) shows that for all (y, δ) with y < w and(y, δ) /∈ Λ−

s ,

(13) Qy,δ,w,γ = Qy,δ,w,γ.

Since Qy,δ,w,γ is in A+ and has zero constant term, Eq. (13) forcesQy,δ,w,γ to be zero whenever y < w and (y, δ) /∈ Λ−

s . Therefore theright hand side of Eq. (11) is zero, since TsCy,δ + Cy,δ = 0 whenever(y, δ) ∈ Λ−

s . So

TsCw,γ = −q−1Cw,γ,

as required.

Now suppose that t /∈ Iγ , so that (w, γ) /∈ Λ−s . Replacing γ by θ in

Eq. (9) we obtain

Cw,θ = Twθ −∑

{y,δ|y<w}Qy,δ,w,θCy,δ,

for all θ ∈ Γ. It follows that

(14) (Ts − q)Cw,γ −∑θ∈Γ−

t

µ(θ, γ)Cw,θ

is the sum of

(15) (Ts − q)Twγ −∑θ∈Γ−

t

µ(θ, γ)Twθ

and

−∑

{y,δ|y<w}

(Qy,δ,w,γ(Ts − q)Cy,δ −

∑θ∈Γ−

t

µ(θ, γ)Qy,δ,w,θCy,δ

).


Using the inductive hypothesis to evaluate (Ts − q)Cy,δ, this last ex-pression can be written as the sum of the following three terms:

−∑

(y,δ)∈Λ+s

y<w

∑(z,θ)∈Λ−

s

Qy,δ,w,γξ(z, θ, y, δ)Cz,θ,(16)

∑(y,δ)∈Λ−

sy<w

Qy,δ,w,γ(q−1 + q)Cy,δ,(17)

∑{y,δ)|y<w}

θ∈Γ−t

µ(θ, γ)Qy,δ,w,θCy,δ.(18)

Now the expression (15) is zero, since

(Ts − q)Twγ −∑θ∈Γ−

t

µ(θ, γ)Twθ = Tw

(Ttγ − qγ −

∑θ∈Γ−

t

µ(θ, γ)θ),

and t /∈ Iγ . Observe that the coefficient of each Cy,δ in the sum of theexpressions (16), (17) and (18) is in A+, and the only contributionsto the constant terms of these coefficients come from (17) in the case(y, δ) ≺ (w, γ). However, the expression (14) is invariant under theinvolution m �→ m; hence the total coefficient of each Cy,δ in the sumof (16), (17) and (18) must be a constant (since no other elements ofA+ are invariant under the involution). So we conclude that

(Ts − q)Cw,γ −∑θ∈Γ−

t

µ(θ, γ)Cw,θ =∑

(y,δ)∈Λ−s

(y,δ)≺(w,γ)

µ(y, δ, w, γ)Cy,δ.

Since µ(θ, γ) = µ(w, θ, w, γ), and the condition θ ∈ Γ−t is equivalent to

(w, θ) ∈ Λ−s , this may be rewritten as

TsCw,γ = qCw,γ +∑

µ(y, δ, w, γ)Cy,δ.

where the sum is over all (y, δ) ∈ Λ−s such that (y, δ) ≺ (w, γ) or y = w.

To deduce that Eq. (10) holds, it remains to check that that there isno z ∈ DJ such that (z, γ) ∈ Λ−

s and �(z) = �(w) + 1, with z = rw forsome r ∈ S.

Clearly these conditions cannot hold with r = s, as sw /∈ DJ ; sowe may suppose that r �= s. Now (z, γ) ∈ Λ−

s implies that either�(sz) < �(z) or sz = zu for some u ∈ Iγ. In the former case z must bethe longest element ofW{r,s}z, since both �(sz) < �(z) and �(rz) < �(z),and this gives �(srz) = �(z) − 2, a contradiction since srz = rzt andrz ∈ DJ . Similarly, the other case gives �(s(zu)) = �(z) < �(zu) and�(r(zu)) = �(wu) < �(zu), so that �(srzu) = �(zu) − 2, contradicting


the fact that srzu = swu = wtu has length at least �(w) = �(z) − 1(since w ∈ DJ and tu ∈WJ).

We have now completed the proof of Eq. (10), and to complete theproof of Theorem 2.6 it remains to show that for all s ∈ S we haveξ(z, θ, w, γ) = µ(z, θ, w, γ) whenever (z, θ) ∈ Λ−

s and (w, γ) /∈ Λ−s . This

is true by definition whenever �(z) � �(w). If �(z) > �(w) then bothsides are zero unless (w, γ) ≺ (z, θ).

So our task is to show that (w, γ) ≺ (z, θ) with (z, θ) ∈ Λ−s and

(w, γ) /∈ Λ−s implies that (z, θ) = (rw, γ) with �(z) = �(w)+1, for some

r ∈ S, and µ(z, θ, w, γ) = 1. In fact we shall show that this holds withr = s (which is the only possibility, as could be shown directly by anargument similar to the one used above).

Since (z, θ) ∈ Λ−s we have that TsCz,θ = −Cz,θ, whence

(19)∑

y∈DJ ,δ∈Γ

Py,δ,z,θTsTyδ = −∑

y∈DJ ,δ∈Γ

Py,δ,z,θTyδ.

If w ∈ D0J,s, so that (w, γ) /∈ Λ−

s gives γ /∈ Γ−t (where t = w−1sw), then

a comparison of the coefficients of Twγ shows that Pw,γ,z,θ = 0 (sinceTsTwγ = TwTtγ = qTwγ + X, where X is a combination of terms ofthe form Twδ with δ ∈ Γ−

t ). This contradicts (w, γ) ≺ (z, θ). The onlyalternative is w ∈ D+

J,s, and in this case comparison of the coefficientsof Tswγ on the two sides of Eq. (19) gives

(q − q−1)Psw,γ,z,θ + Pw,γ,z,θ = −q−1Psw,γ,z,θ,

which reduces toqPsw,γ,z,θ = −Pw,γ,z,θ.

Now since (w, γ) ≺ (z, θ) the coefficient of q in Pw,γ,z,θ is nonzero;so the constant term of Psw,γ,z,θ is nonzero. So (sw, γ) = (z, θ) and−Pw,γ,z,θ = q, whence µ(w, γ, z, θ) = 1, as required. �

It is convenient to distinguish three kinds of edges of the W -graphΛ. Firstly, there is an edge from the vertex (w, γ) to the vertex (w, δ)whenever there is an edge from γ to δ in Γ. We call these horizontaledges. Next, if s ∈ S and w is in either D+

J,s or D−J,s then there is an

edge joining (w, γ) and (sw, γ). We call these vertical edges. All otheredges are called transverse.

Proposition 2.7. Suppose that vertices (w, γ) and (z, θ) of Λ arejoined by a transverse edge, and suppose that �(w) � �(z). ThenI(z, θ) ⊆ I(w, γ).

Proof. Let s ∈ I(z, θ), and suppose, for a contradiction, thats is not in I(w, γ). Since the edge is not horizontal we have either


(w, γ) ≺ (z, θ) or (z, θ) ≺ (w, γ), and the assumption �(w) � �(z)means that the former alternative holds. So we have (w, γ) ≺ (z, θ),with (z, θ) ∈ Λ−

s and (w, γ) ∈ Λ+s . We showed in the last part of the

previous proof that these conditions imply that (z, θ) = (sw, γ). Thismeans that the edge {(w, γ), (z, θ)} is vertical rather than transverse,and so we have the desired contradiction. �

4. Inducing ordered W-graphs

Proposition 2.8. Suppose that the WJ -graph Γ admits a partialorder � satisfying the conditions of Definition 1.6. Then the inducedW -graph Λ admits a partial order � satisfying Definition 1.6 and hav-ing the following properties:

(i) if δ, γ ∈ Γ and y, w ∈ DJ are such that y � w and δ � γ,then (y, δ) � (w, γ);

(ii) if δ, γ ∈ Γ and y, w ∈ D+J,s for some s ∈ S, then (y, δ) � (w, γ)

implies that (sy, δ) � (sw, γ);(iii) if y ∈ D0

J,s and w ∈ D+J,s for some s ∈ S, then (y, δ) � (w, γ)

implies that (y, tδ) � (sw, γ), for all γ ∈ Γ and δ ∈ Γ+t such

that tδ exists, where t = y−1sy;(iv) if (y, δ), (w, γ) ∈ Λ satisfy Py,δ,w,γ �= 0 then (y, δ) � (w, γ).

Proof. We define � on Λ to be the minimal transitive relationsatisfying the requirements (i), (ii) and (iii). Then it is clear that(y, δ) � (w, γ) implies that y � w, with equality only if δ � γ. Hencethe fact that the relation � on Γ is antisymmetric implies the same forthe relation � on Λ.

We prove first that Condition (iv) is satisfied, using induction on�(w). In the case �(w) = 0 the assumption that Py,δ,w,γ �= 0 forces(y, δ) = (w, γ). So suppose that �(w) > 0, and choose s ∈ S with�(sw) < �(w). Recall that Py,δ,w,γ = P ′

y,δ,w,γ − P ′′y,δ,w,γ; hence either

P ′′y,δ,w,γ �= 0 or P ′

y,δ,w,γ �= 0.

If P ′′y,δ,w,γ �= 0 then it follows from Eq. (8) there exists (z, θ) with

(z, θ) ≺ (sw, γ) and Py,δ,z,θ �= 0. The inductive hypothesis then yieldsboth (y, δ) � (z, θ) and (z, θ) � (sw, γ), and since (sw, γ) � (w, γ)it follows that (y, δ) � (w, γ), as required. So we may assume thatP ′

y,δ,w,γ �= 0.

Suppose first that y ∈ D+J,s. By Eq. (7) either Py,δ,sw,γ �= 0 or

Psy,δ,sw,γ �= 0, and so the inductive hypothesis tells us that either(y, δ) � (sw, γ) or (sy, δ) � (sw, γ). Since (y, δ) � (sy, δ) we obtain(y, δ) � (sw, γ) in either case, and hence (y, δ) � (w, γ).

4. INDUCING ORDERED W-GRAPHS 23

Now suppose that y ∈ D−J,s. Once more Eq. (7) and the inductive

hypothesis yield that either (y, δ) � (sw, γ) or (sy, δ) � (sw, γ). Theformer alternative gives (y, δ) � (w, γ) as in the previous cases, whilethe latter alternative gives the same result since Property (ii) aboveholds.

Finally, suppose that y ∈ D0J,s, and let t = y−1sy ∈ J . By Eq. (7)

we see that either Py,δ,sw,γ �= 0, which yields (y, δ) � (w, γ) as in theprevious cases, or else δ ∈ Γ−

t and µ(δ, θ)Py,θ,sw,γ �= 0 for some θ ∈ Γ+t .

Thus {θ, δ} is an edge of Γ with t ∈ Iδ and t /∈ Iθ, and by Conditions (i),(ii) of Definition 1.6 it follows that either δ = tθ or δ � θ. Moreover,since Py,θ,sw,γ �= 0 the inductive hypothesis yields that (y, θ) � (sw, γ).If δ � θ then (y, δ) � (y, θ), and so (y, δ) � (sw, γ) � (w, γ). Ifδ = tθ then (y, δ) � (w, γ) follows from (y, θ) � (sw, γ), in view ofProperty (iii) above.

It remains to show that Λ is an ordered W -graph in the sense ofDefinition 1.6.

Let (y, δ), (w, γ) ∈ Λ with µ(y, δ, w, γ) �= 0. In the case y = wwe have µ(y, δ, w, γ) = µ(δ, γ), and since Γ is an ordered WJ -graph itfollows that δ and γ are comparable, whence so are (y, δ) = (w, δ) and(w, γ). If y �= w then µ(y, δ, w, γ) is a coefficient of one or other of thepolynomials Py,δ,w,γ and Pw,γ,y,δ, and so (iv) above implies that (w, γ)and (y, δ) are comparable. So Condition (i) of Definition 1.6 holds.

Let s ∈ S and (w, γ) ∈ Λ+s , and suppose that (y, δ) ∈ Λ−

s with(w, γ) < (y, δ) and µ(y, δ, w, γ) �= 0. We must show that (y, δ) is theunique such element of Λ−

s .

Suppose first of all that the edge {(y, δ), (w, γ)} is transverse. Sinces ∈ I(y, δ) and s /∈ I(w, γ), it follows that I(y, δ) � I(w, γ); so byProposition 2.7 we must have (y, δ) ≺ (w, γ). But this implies thatPy,δ,w,γ �= 0, and in view of (iv) this contradicts the assumption that(w, γ) < (y, δ). So {(y, δ), (w, γ)} is either vertical or horizontal.

If the edge {(y, δ), (w, γ)} is vertical then δ = γ and y = rw forsome r ∈ S. Since (w, γ) < (y, γ) we have w � y; so �(w) � �(rw).Now since s ∈ I(rw, γ) and s /∈ I(w, γ) it follows readily that r = s.So (y, δ) = (sw, γ); moreover, this case can only arise if w ∈ D+

J,s.

Now suppose that {(y, δ), (w, γ)} is horizontal, so that y = w and{δ, γ} is an edge of Γ. Since Γ is an ordered WJ -graph, Condition (i)of Definition 1.6 yields that either γ < δ or δ < γ; however, the latteralternative would give (w, δ) < (w, γ), contradicting our assumptionthat (w, γ) < (y, δ) = (w, δ). Now since s ∈ I(w, δ) and s /∈ I(w, γ)we see that w ∈ D0

J,s, and t = w−1sw is in Iδ and not in Iγ. Since Γsatisfies Condition (ii) of Definition 1.6 it follows that δ = tγ.


We have shown that

(y, δ) =

{(sw, γ) if w ∈ D+

J,s

(w, tγ) if w ∈ D0J,s

where t = w−1sw. So (y, δ) is uniquely determined. In accordance withDefinition 1.6, we write (y, δ) = s(w, γ).

It remains to check that Λ satisfies Condition (iii) of Definition 1.6;that is, we must show that if (w, γ) ∈ Λ+

s and (y, δ) = s(w, γ) thenµ(y, δ, s, γ) = 1. If w ∈ D0

J,s with w−1sw = t then s(w, γ) is definedif and only if tγ is defined, in which case s(w, γ) = (w, tγ). Moreover,in this case we have that µ(w, tγ, w, γ) = µ(tγ, γ) = 1, since Γ satisfiesCondition (iii) of Definition 1.6. On the other hand, if w ∈ D+

J,s thens(w, γ) = (sw, γ), and the desired conclusion that µ(sw, γ, w, γ) = 1follows from Theorem 2.5. �

5. Inducing bipartite W-graphs

Definition 2.9. A W-graph is called bipartite if its vertex set Γis the disjoint union of nonempty sets Γ1,Γ2 such that µ(δ, γ) = 0whenever δ, γ ∈ Γ1 or δ, γ ∈ Γ2.

We assume that a WJ -graph Γ is bipartite and let Γ1,Γ2 be the twoparts of the vertex set. Then the vertex set of the induced W-graph Λ,namely {(w, γ) | γ ∈ Γ, w ∈ DJ}, is the disjoint union of the followingtwo sets:

Λ1 = {(w, γ) | �(w) is even and γ ∈ Γ1 or �(w) is odd and γ ∈ Γ2};Λ2 = {(w, γ) | �(w) is even and γ ∈ Γ2 or �(w) is odd and γ ∈ Γ1}.Proposition 2.10. Assume that Γ = Γ1 ∪Γ2 is bipartite as above.

Then

(i) if δ, γ are in the same part Γi of Γ and �(w) − �(y) is even,or δ, γ are in different Γi and �(w) − �(y) is odd, then thepolynomial Py,δ,w,γ involves only even powers of q.

(ii) if δ, γ are in different parts of Γ and �(w)−�(y) is even, or δ, γare in the same part and �(w)−�(y) is odd, then the polynomialPy,δ,w,γ involves only odd powers of q.

Proof. Use induction on �(w). If �(w) = 0, it follows from (i) and(ii) of Theorem 2.4. So assume that �(w) > 0 and let w = sv wheres ∈ S and �(v) = �(w) − 1.

Suppose first that δ, γ are in the same part of Γ and �(w)− �(y) iseven, which is one of the cases in Part (i). The inductive hypothesisimmediately implies that the terms on the right hand side of Eq. (7)

5. INDUCING BIPARTITE W-GRAPHS 25

involve only even powers of q, with the possible exception of the termsµ(δ, θ)Py,θ,v,γ in the sum that appears in case (c) (when y ∈ D0

J,s and

δ ∈ Γt−). But if µ(δ, θ) �= 0 then θ and δ must be in different parts

of Γ, which also implies that θ, γ are in different parts of Γ; so Py,θ,v,γ

(where �(v)−�(y) is odd) involves only even powers of q by the inductivehypothesis. Hence P ′

y,δ,w,γ involves only even powers of q.Let us consider the powers of q in P ′′

y,δ,w,γ. The nonzero terms inEq. (8) correspond to quadruples (z, θ, v, γ) such that Pz,θ,v,γ has anonzero coefficient of q (since this coefficient is −µ(z, θ, v, γ)). Hence,by the inductive hypothesis, Pz,θ,v,γ involves only odd powers of q.There are now two possible cases.

(1) If �(v)− �(z) is even, then θ, γ must in different parts of Γ; soθ, δ are in different parts of Γ and

�(z) − �(y) = (�(w) − �(y)) − (�(v) − �(z)) − 1

is odd. So Py,δ,z,θ involves only even powers of q, by the induc-tive hypothesis.

(2) If �(v)− �(z) is odd, then θ, γ must be in the same part of Γ;so θ, δ are in the same part of Γ and �(z) − �(y) is even. Soagain Py,δ,z,θ involves only even powers of q, by the inductivehypothesis.

Hence P ′′y,δ,w,γ, like P ′

y,δ,w,γ, involves only even powers of q.The other cases are all similar. If δ, γ are in the same part of Γ and

�(w) − �(y) is odd the inductive hypothesis immediately implies thatthe terms on the right hand side of Eq. (7) involve only odd powersof q, with the possible exception of the terms µ(δ, θ)Py,θ,v,γ in the sumthat appears in case (c). But when µ(δ, θ) �= 0 then θ and δ are indifferent parts of Γ, which also implies that θ, γ are in different partsof Γ; so by the inductive hypothesis Py,θ,v,γ involves only odd powersof q, since this time �(v)− �(y) is even. For P ′′

y,δ,w,γ the calculations arethe same as above, except that the parity of �(z)−�(y) is reversed sincethe parity of �(v) − �(z) has been reversed. The result is that all theterms µ(z, θ, v, γ)Py,δ,z,θ involve only odd powers of q. Finally, whenδ, γ are in different parts of Γ the inductive hypothesis yields that theterms on the right hand side of Eq. (7) involve only even powers of qif �(w)− �(y) is odd and odd powers of q if �(w)− �(y) is even, and allnonzero terms µ(z, θ, v, γ)Py,δ,z,θ in P ′′

y,δ,w,γ involve only even powers ofq if �(w)− �(y) is odd and odd powers of q if �(w)− �(y) is even, since�(w) − �(z) is necessarily even. �

As an immediate consequence of Proposition 2.10 we have the fol-lowing result.


Theorem 2.11. Assume that WJ -graph Γ is bipartite. Then theinduced W-graph Λ is bipartite.

6. Inducing cells

Let (w, γ) ∈ DJ × Γ, and let s ∈ S. If (w, γ) ∈ Λ−s then we have

TsCw,γ = −q−1Cw,γ, and so

(20) −q−1∑y∈DJδ∈Γ

Py,δ,w,γTyδ =∑y∈DJδ∈Γ

Py,δ,w,γTsTyδ.

We also have

TsTyδ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩Tsyδ if y ∈ D+

J,s

Tsyδ + (q − q−1)Tyδ if y ∈ D−J,s

−q−1Tyδ if y ∈ D0J,s and δ ∈ Γ−

t

qTyδ +∑

θ∈Γ−tµ(θ, δ)Tyθ if y ∈ D0

J,s and δ ∈ Γ+t

where t = y−1sy. Substituting this into Eq. (20) and equating coeffi-cients yields a proof of the following result.

Proposition 2.12. Let s ∈ S and (w, γ) ∈ Λ−s . If y ∈ D+

J,s then

Py,δ,w,γ = −qPsy,δ,w,γ for all δ ∈ Γ. If y ∈ D0J,s and δ ∈ Γ+

t , where

t = y−1sy, then Py,δ,w,γ = 0.

Note that this simplifies our original inductive formulas for the poly-nomials Py,δ,w,γ. In particular, in the situation of Eq. (8) we have thatP ′′(y, δ, w, γ) = 0 when y ∈ D0

J,s and δ ∈ Γ+t .

Let ≤Γ be the Kazhdan-Lusztig preorder on Γ, so that δ ≤Γ γ if andonly if there exists a finite sequence δ = γ0, γ1, . . . , γk = γ of elementsof Γ with µ(γi−1, γi) �= 0 and I(γi−1) � I(γi) for all i ∈ {1, 2, . . . , k}.

Proposition 2.13. Let y, w ∈ DJ and δ, γ ∈ Γ with δ �≤Γ γ. ThenPy,δ,w,γ = 0.

Proof. Use induction on �(w). Since δ �= γ the case �(w) = 0follows from (i) and (ii) of Theorem 2.4. So assume that �(w) > 0, andlet w = sv where s ∈ S and �(v) = �(w) − 1.

The inductive hypothesis immediately implies that the terms on theright hand side of Eq. (7) are zero, with the possible exception of theterms µ(δ, θ)Py,θ,v,γ in the sum that appears case (c) (when y ∈ D0

J,s

and δ ∈ Γ−t ). In all of these terms we have that Iδ � Iθ, since t ∈ Iδ and

t /∈ Iθ. So either δ ≤Γ θ or else µ(δ, θ) = 0. By the inductive hypothesis,either θ ≤Γ γ or else Py,θ,v,γ = 0. But since δ �≤Γ γ we cannot haveboth δ ≤Γ θ and θ ≤Γ γ; so either µ(δ, θ) = 0 or Py,θ,v,γ = 0. So all theterms µ(δ, θ)Py,θ,v,γ are zero, and so P ′

y,δ,w,γ = 0.

7. WK -CELLS IN INDUCED W-GRAPHS 27

All the elements z appearing on the right hand side of Eq. (8)satisfy �(z) � �(v), and so the inductive hypothesis tells us that ifδ �≤Γ θ then Py,δ,z,θ = 0. Furthermore, if θ �≤Γ γ then Pz,θ,v,γ = 0, andso µ(z, θ, v, γ) = 0. Since δ �≤Γ γ we must have either θ �≤Γ γ or δ �≤Γ θ,and so all the terms µ(z, θ, v, γ)Py,δ,z,θ are zero. So P ′′

y,δ,w,γ = 0, andhence Py,δ,w,γ = 0, as required. �

Suppose now that (z, θ) and (w, γ) are vertices of Λ that are ad-jacent and satisfy I(z, θ) � I(w, γ). If w = z then s ∈ I(w, θ) ands /∈ I(w, γ) forces sw = wt for some t ∈ Iθ with t /∈ Iγ . So in thiscase θ and γ are adjacent vertices of Γ with Iθ � Iγ . In particular,θ ≤Γ γ. The same conclusion holds trivially if the edge {(z, θ), (w, γ)}is vertical, since in this case θ = γ. If the edge is transverse thenby Proposition 2.7 we deduce that �(z) < �(w), and so we must have(z, θ) ≺ (w, γ). Thus Pz,θ,w,γ �= 0, and so θ ≤Γ γ by Proposition 2.13.

Let ≤Λ be the Kazhdan-Lusztig preorder on the W -graph Λ. Thus≤Λ is generated by the requirement that (z, θ) ≤Λ (w, γ) whenever(z, θ) and (w, γ) are adjacent and I(z, θ) � I(w, γ). The above calcu-lations have proved the following theorem.

Theorem 2.14. If (z, θ) and (w, γ) are vertices of Λ such that(z, θ) ≤Λ (w, γ) then θ ≤Γ γ.

Recall from Section 2.2 of Chapter 1 that vertices θ, γ ∈ Γ lie inthe same cell of Γ if and only if θ ≤Γ γ and γ ≤Γ θ. Similarly, (z, θ)and (w, γ) are in the same cell of Λ if and only if (z, θ) ≤Λ (w, γ) and(w, γ) ≤Λ (z, θ). Theorem 2.14 shows that if ∆ is a cell in Γ then theset { (w, γ) | w ∈ DJ and γ ∈ ∆ } is a union of cells in Λ. In the casethat Γ is the Kazhdan-Lusztig WJ -graph for the regular representation,this result (and Theorem 2.14) have been proved by Meinolf Geck [13].

7. WK-cells in induced W-graphs

Let J, K ⊆ S, and let ρ be a representation of WJ . Inducing to Wand then restricting to WK yields a representation ResW

WK(IndW

WJ(ρ)),

and by Mackey’s formula we have

(21) ResWWK

(IndWWJ

(ρ)) ∼=∑

d

IndWK

WK∩dWJd−1(ResdWJd−1

WK∩dWJd−1(dρ))

where d runs through a set of representatives of the WK\W/WJ doublecosets, and dρ is the representation of dWJd

−1 defined by

(dρ)x = ρ(d−1xd)


for all x ∈ dWJd−1. Our aim in this section is to describe a W -graph

version of Eq. (21). We need first to review some well known factsabout double cosets of parabolic subgroups.

Recall that if J ⊆ S then each W/WJ coset contains a uniqueelement of DJ , and �(xt) = �(x) + �(t) whenever x ∈ DJ and t ∈ WJ .It follows that each WJ\W coset contains a unique element of the setD−1

J = { x−1 | x ∈ DJ }, and �(ty) = �(t) + �(y) whenever t ∈ WJ andy ∈ D−1

J . Now if J, K ⊆ S and x ∈ DJ we can write x = ud withu ∈WK and d ∈ D−1

K and �(x) = �(u)+�(d), and since x ∈ DJ it followsfrom Lemma 2.2 that also d ∈ DJ . Thus d is in the setDKJ = D−1

K ∩DJ .So for every w ∈ W there exists an element d ∈ DKJ and elementsu ∈WK and t ∈ WJ with w = udt and �(w) = �(u) + �(d) + �(t).

The following result is proved in [4, Theorem 2.7.4].

Proposition 2.15 (Kilmoyer). Let K and J be subsets of S. Theneach WK\W/WJ double coset contains a unique element of the set DKJ .Moreover, whenever d ∈ DKJ we have WK ∩ dWJd

−1 = WL, whereL = K ∩ dJd−1.

Note that, as a consequence of Proposition 2.15, if d ∈ DKJ thenthe isomorphism z �→ d−1zd from WK ∩ dWJd

−1 to d−1WKd ∩ WJ

preserves lengths of elements.

Definition 2.16. Whenever L ⊆ K ⊆ S we define DKL = WK∩DL,

the set of minimal length coset representatives for WK/WL.

Let J, K ⊆ S and w ∈ W , and let d ∈ WKwWJ ∩ DKJ . Supposethat u ∈ WK is such that ud ∈ DJ . Writing L = K ∩ dJd−1, we canexpress u in the form u′v with u′ ∈ DK

L and v ∈WL and then we have

ud = u′vd = u′dv′

where v′ = d−1vd ∈ d−1WKd ∩WJ and

�(ud) = �(u) + �(d) = �(u′) + �(v) + �(d) = �(u′) + �(d) + �(v′).

Since ud ∈ DJ and v′ ∈WJ this forces �(v′) = 0. We conclude that

(22) DJ = { ud | d ∈ DKJ and u ∈ DKK∩dJd−1 }.

Returning now to W-graphs, we start with a trivial observation.

Proposition 2.17. Any W-graph becomes a WL-graph if the ele-ments of S\L are ignored.

In other words, if (Γ, I, µ) is a W-graph datum and for each γ ∈ Γwe define I ′γ = Iγ ∩ L then (Γ, I ′, µ) is a WL-graph datum. We write

ResSL(Γ) for the WL-graph obtained in this way. It is trivial to check


that the WL-module obtained from ResSL(Γ) is simply the restriction of

the W-module obtained from Γ.Now let Γ be a WJ -graph and Λ = IndS

J (Γ) the induced W -graph,constructed as in Section 3. The vertex set of Λ can be identified withthe set DJ × Γ = { (x, γ) | x ∈ DJ , γ ∈ Γ }, which is in one to onecorrespondence with { (u, d, γ) | u ∈ DK

K∩dJd−1, d ∈ DKJ , γ ∈ Γ }.Consider a fixed d, and put L = K ∩ dJd−1. The vertices (ud, γ)

of ResSK(Λ), as u ∈ DK

L and γ ∈ Γ vary, span a subgraph of ResSK(Λ),

which we refer to as the d-subgraph of ResSK(Λ). We shall show that

the d-subgraph of ResSK(Λ) is a WK-graph.

Because d−1Ld ⊆ J and the the isomorphism z �→ dzd−1 fromWd−1Ld to WL is length preserving, the Wd−1Ld-graph ResJ

d−1Ld imme-diately gives rise to a WL-graph, which, for brevity, we refer to as dΓ.We write the vertices of dΓ as pairs dγ, where γ varies over verticesof Γ. The descent set of dγ ∈ dΓ is

Idγ = { dsd−1 | s ∈ Iγ ⊆ J and dsd−1 ∈ K } ⊆ K ∩ dJd−1,

and the edges and edge weights of dΓ correspond exactly to those of Γ:

µ(dγ, dγ′) = µ(γ, γ′)

for all γ, γ′ ∈ Γ. The vertex set of the induced WK-graph IndKL (dΓ)

is { (u, dγ) | u ∈ DKL and γ ∈ Γ }, which is in obvious one to one

correspondence with the vertex set of the d-subgraph of ResSK(Λ). We

shall show that these graphs are actually isomorphic.

Lemma 2.18. The descent set of the vertex (u, dγ) of IndKL (dΓ)

equals the descent set of the vertex (ud, γ) of ResSK(Λ).

Proof. The descent set of (u, dγ) consists of the s ∈ K such thateither �(su) < �(u) or u−1su ∈ Idγ , and the descent set of (ud, γ) con-sists of the s ∈ K such that either �(sud) < �(ud) or (ud)−1s(ud) ∈ Iγ.It is clear from the fact that ud ∈ DJ that �(su) < �(u) if and only if�(sud) < �(ud). Moreover, the definition of dΓ gives Idγ = d(J∩Iγ)d−1;so u−1su ∈ Idγ immediately implies that (ud)−1s(ud) ∈ Iγ . On theother hand, since u ∈ WK and Iγ ⊆ J , if (ud)−1s(ud) = s′ ∈ Iγ thends′d−1 = u−1su ∈ WK ∩ dWJd

−1 = WL, whence �(ds′d−1) = �(s′) = 1,giving u−1su ∈ L ∩ dIγd−1 = d(J ∩ Iγ)d−1. �

The following result shows that the edges and edge weights ofIndK

L (dΓ) and the d-subgraph of ResSK(Λ) also agree.

Lemma 2.19. Let J, K, d, L, Γ, dΓ be as in the discussion above.Let Py,δ,w,γ (for y, w ∈ DJ and δ, γ ∈ Γ) be the polynomials appearingin the construction of IndS

J (Γ), and let PKy,dδ,w,dγ (for y, w ∈ DK

L and


dδ, dγ ∈ dΓ) be the corresponding polynomials in the construction ofIndK

L (dΓ). Then PKy,dδ,w,dγ = Pyd,δ,wd,γ, for all y, w ∈ DK

L and γ, δ ∈ Γ.

Note that Eq. (22) above shows that yd, wd ∈ DJ , as necessary forthe statement to make sense.

The proof of Lemma 2.18 is a straightforward induction on �(w).Since yd � d, Theorem 2.4 gives

PKy,dδ,1,dγ = Pyd,δ,d,γ =

{1 if (y, dδ) = (1, dγ)

0 otherwise,

which starts the induction. Turning to the inductive step, let w ∈ DKL

with �(w) � 1, and write w = sv with �(v) < �(w). Note that s ∈ K,since w ∈ WK . Now for all y ∈ DK

L we see that y−1sy ∈ L if and onlyif (yd)−1s(yd) ∈ J , and it follows readily that the three cases y ∈ D+

L,s,

y ∈ D−L,s, y ∈ D0

L,s correspond to the three cases yd ∈ D+J,s, yd ∈ D−

J,s,

yd ∈ D0J,s. When y ∈ D0

L,s we write t = y−1sy; note that (for any

δ ∈ Γ) we have dδ ∈ (dΓ)+t if and only if δ ∈ Γ+

d−1td. Following theterminology of Eq. (7), we call the cases y ∈ D+

L,s, y ∈ D−L,s, y ∈ D0

J,s

with dδ ∈ (dΓ)−t and y ∈ D0J,s with dδ ∈ (dΓ)+

t respectively cases (a),(b), (c) and (d). Then, as in Eq. (7),

PK′y,dδ,w,dγ =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

PK′sy,dδ,v,dγ − qPK′

y,dδ,v,dγ (case (a)),

PK′sy,dδ,v,dγ − q−1PK′

y,dδ,v,dγ (case (b)),

(−q − q−1)PK′y,dδ,v,dγ +

∑dθ∈dΓ+

t

µ(dδ, dθ)PK′y,dθ,v,dγ (case (c)),

0 (case (d)).

Since for all the terms in the sum in case (c) we have µ(dδ, dθ) = µ(δ, θ),with θ running through all elements of Γ+

d−1td as dθ runs through allelements of (dΓ)+

t , it follows from the inductive hypothesis that theright hand side above equals the corresponding formula for P ′

yd,δ,wd,γ

obtained from Eq. (7). Thus PK′y,dδ,w,dγ = P ′

yd,δ,wd,γ.

In a similar fashion, it follows from Eq. (8) that PK′′y,dδ,w,dγ = P ′′

yd,δ,wd,γ.

The point is that (zd, θ) ∈ Λ−s if and only if zd ∈ D−

J,s or zd ∈ D0J,s with

(zd)−1s(zd) ∈ Iθ, and this corresponds to (z, dθ) ∈ (IndKL )−s ; more-

over, the inductive hypothesis gives ν(zd, θ, vd, γ) = νK(z, dθ, v, dγ)and (zd, θ) ≺ (vd, γ) if and only if (z, dθ) ≺ (v, dγ). So it follows thatPK

y,dδ,w,dγ = Pyd,δ,wd,γ, as required.

Since the coefficients of the polynomials PKy,dδ,w,dγ and Pyd,δ,wd,γ de-

termine the edges and edge weights of IndKL (dΓ) and the d-subgraph of


ResSK(Λ), Lemmas 2.18 and 2.19 combine to show that these graphs are

isomorphic. In particular, the d-subgraph of ResSK(Λ) is a WK-graph.

In view of Eq. (22) we see that the vertex set of ResK(Λ) is thedisjoint union of the vertex sets of its d-subgraphs, as d runs throughall elements of DKJ . It seems reasonable to expect, therefore, that eachd-subgraph is a union of WK-cells of ResK(Λ). To prove this, we makeuse of the following result.

Lemma 2.20 (Deodhar[9, Lemma 3.5]). Let d ∈ D−1K and w ∈ W,

and write w = ue with e ∈ D−1K and u ∈ WK . Then d � w if and only

if d � e.

As above, let Γ be a WJ -graph and Λ = IndSJ (Γ), and consider

the WK-graph ResSK(Λ). Let d, e be distinct elements of DKJ with

�(d) � �(e). We show that if there is an edge of ResSK(Λ) joining

a vertex α of the d-subgraph and a vertex β of the e-subgraph thene � d, and the descent set of α is a subset of the descent set of β.

We may write α = (ud, γ) and β = (ve, δ), where u ∈ DKK∩dJd−1 and

v ∈ DKK∩eJe−1, and γ, δ ∈ Γ. Note that since d �= e the edge joining α

and β is not horizontal. Suppose first that it is transverse. Then eitherud � ve or ve � ud. But the former alternative would give d � ve andhence d � e by Lemma 2.20, contradicting to our assumptions thatd �= e and �(e) � �(d). So we must have ve � ud, and, by the sameargument, e � d. Moreover, I(ud, γ) ⊆ I(ve, δ), by Proposition 2.7,and so I(ud, γ) ∩ K, which is the descent set of α in ResS

K(Λ), is asubset of I(ve, δ) ∩K, the descent set of β.

We now consider the case that {α, β} is vertical, which means thatδ = γ and ud = sve for some s ∈ S. We either have ud � ve or ve � ud,depending on whether �(sve) = �(ve) − 1 or �(sve) = �(ve) + 1. As inthe last paragraph, the former alternative gives d � e, contradictingour hypotheses. So ve � ud, and e � d.

Suppose, for a contradiction, that I(ud, γ) ∩K � I(ve, δ) ∩K, sothat there exists an r ∈ K with r ∈ I(ud, γ) and r /∈ I(ve, γ). Observefirst that r �= s, since otherwise we would have

WKdWJ = WKudWJ = WKrudWJ = WKveWJ = WKeWJ ,

contradicting the assumption that d and e are distinct elements ofDKJ .Now �(rve) > �(ve), since r /∈ I(ve, δ). Since also �(sve) > �(ve), itfollows that �(rsve) = �(ve) + 2; that is, �(rud) = �(ud) + 1. Sincer ∈ I(ud, γ) this forces rud = udt for some t ∈ Iγ ⊆ J . Now udt mustbe the longest element in W{r,s}udt, since �(rudt) = �(du) < �(dut) and

�(sdut) = �(vet) = �(ve) + 1 = �(du) < �(dut).


Moreover, ve = (sr)(udt) is the minimal length element in W{r,s}udtsince, as noted above, �(rve) > �(ve) and �(sve) > �(ve). Thus sris the longest element of W{r,s}, and it follows that rs = sr. Thusrve = rsud = srud = sudt = vet, and since t ∈ Iγ this shows thatr ∈ I(ve, γ), contradicting our assumptions.

Proposition 2.21. Let J, K ⊆ S and let Γ be a WJ -graph. Foreach d ∈ DKJ , the d-subgraph of ResS

K(IndSJ (Γ)) is a union of cells.

Proof. Let α be a vertex in the d-subgraph. We must prove thatany vertex β that is in the same cell of ResS

K(IndSJ (Γ)) as α is also in the

d-subgraph. Recall that the vertex set of ResSK(IndS

J (Γ)) is the disjointunion of the vertex sets of its e-subgraphs, as e runs through DKJ ; soβ must lie in the e-subgraph for some e ∈ DKJ .

Since α and β are in the same cell we have that α ≤ β and β ≤ α,where ≤ is the Kazhdan-Lusztig preorder on ResS

K(IndSJ (Γ)). So there

exists a sequence of vertices α = α0, α1, α2, . . . , αn = β with αi−1

and αi adjacent and I(αi−1) � I(αi) for 1 � i � n, and another suchsequence β = β0, β1, β2, . . . , βm = α with βj−1 and βj adjacent andI(βj−1) � I(βj) for 1 � j � m.

Let αi lie in the di-subgraph and βi in the ej-subgraph, wheredi, ej ∈ DKJ (for all i ∈ {0, 1, . . . , n} and j ∈ {0, 1, . . . , m}). Sinceαi−1 and αi are adjacent and I(αi−1) � I(αi) the argument precedingthis proposition shows that either di−1 = di or �(di−1) < �(di). So�(di) � �(di−1), and di−1 � di in the Bruhat order. Thus it follows thatd = d0 � e = dn. But the same reasoning applied to the sequence ofβj’s gives e � d. Hence e = d, as required. �

We give an example to illustrate the distribution of WK-cells in[IndW

WJ(Γ)]WK

. Let W be the Weyl group of type D4, with generatorsr, s, t and u, where r, s, u correspond to the end nodes of the Coxetergraph. Let J = {r, s, t} (of type A3) and Γ the WJ -graph consistingof two vertices γ, δ such that Iγ = {r, s}, Iδ = {t} and µ(δ, γ) =µ(γ, δ) = 1. Then DJ = {1, u, tu, rtu, stu, rstu, trstu, utrstu}. LetK = {r, t, u}. Then there are two WK\W/WJ double cosets, withshortest elements d1 = 1 and d2 = stu. We find that K ∩ d1Jd1

−1 ={r, t} and K ∩ d2Jd2

−1 = {u, t}; so we have DKK∩d1Jd1

= {1, u, tu, rtu}and DK

K∩d2Jd2−1 = {1, r, tr, utr}. The vertex set of the d1-subgraph of

ResSK(IndS

J (Γ)) is

{(1, γ), (u, γ), (tu, γ), (rtu, γ), (1, δ), (u, δ), (tu, δ), (rtu, δ)}


and the vertex set of the d2-subgraph is

{(stu, γ), (rstu, γ), (trstu, γ), (utrstu, γ),(stu, δ), (rstu, δ), (trstu, δ), (utrstu, δ)}.

The diagram below shows IndSJ (Γ) (on the left) and ResS

K(IndSJ (Γ))

(obtained by removing s from all the descent sets of IndSJ (Γ)). The

circles denote vertices of the graphs, and the generators written insidea circle comprise the descent set of the vertex. All edge weights are 1.

s,r t

r,s,u u

t t,u

r,t r,u

s,t s,u

r,s r,s,u

t,s,r t

r,s,u t,u

1

u

tu

rtu

stu

rstu

trstu

utrstu

γ δ

r t

r,u u

t t,u

r,t r,u

t u

r r,u

t,r t

r,u t,u

γ δ

The W -graph IndSJ (Γ) has two cells of size 3, namely

{(1, γ), (1, δ), (u, δ)}and

{(trstu, γ), (utrstu, γ), (utrstu, δ)},with the remaining 10 vertices constituting a third cell. There are sixcells in ResS

K(IndSJ (Γ)), as follows:

{(1, γ), (1, δ), (u, δ)},{(u, γ), (tu, γ)},

{(tu, δ), (rtu, γ), (rtu, δ)},{(stu, γ), (stu, δ), (rstu, γ)},

{(rstu, δ), (trstu, δ)},{(trstu, γ), (utrstu, γ), (utrstu, δ)}.


The first three of these are in the d1-subgraph, the other three in thed2-subgraph. Observe that for every edge joining a vertex α of thed1-subgraph and a vertex β of the d2-subgraph we have I(β) ⊆ I(α),in accordance with the results proved above (since �(d2) � �(d1)).

8. Connection with Kazhdan-Lusztig polynomials

The results of the preceding sections can be applied with J = φ (sothat WJ = {1}, the trivial subgroup of W ) and Γ the trivial WJ -graphconsisting of a single vertex (and no edges). In this case �J � A andthe �J-module AΓ is simply a 1-dimensional A-module. Note alsothat DJ = W .

By Theorem 2.4 the � -module M induced from AΓ has a uniqueA-basis {Cw | w ∈ W} such that Cw = Cw for all w ∈ W andCw =

∑y∈W py,wTy for some elements py,w ∈ A+ with the following

properties:

(i) py,w = 0 if y � w;(ii) pw,w = 1;(iii) py,w has zero constant term if y �= w.

That is, we recover Theorem 1.4 (our reformulation of [16, Theorem1.1]).

The elements Cw form a W -graph basis for � , and Eq. (2.3a) of[16] (or Theorem 2.5 above) shows the W -graph is ordered, in the senseof Definition 1.6, relative to the Bruhat order on W .

Applying Theorem 1.4 with W replaced by WJ yields a WJ -graphbasis for the regular representation of �J . The representation of �obtained by inducing the regular representation of �J is, of course,the regular representation of � . Applying our procedure for inducingW -graphs yields a W -graph basis for � consisting of elements Cw,γ

(for w ∈ DJ and γ ∈WJ) such that Cw,γ = Cw,γ and

(23) Cw,γ =∑y∈DJ

∑δ∈WJ

Py,δ,w,γTyCδ,

where the polynomials Py,δ,w,γ satisfy the conditions given in Theo-rem 2.4. By Proposition 2.8 the set Λ = {Cw,γ | w ∈ DJ , γ ∈ WJ }has a partial order such that for all y, w ∈ DJ and δ, γ ∈WJ ,

(i) if y � w and δ � γ then Cy,δ � Cw,γ,(ii) if Cy,δ �Cw,γ and y, w∈D+

J,s for some s ∈ S then Csy,δ � Csw,γ,

(iii) if Cy,δ � Cw,γ with w ∈ D+J,s and y ∈ D0

J,s for some s ∈ S, and

if also tδ > δ where t = y−1sy, then Cy,tδ � Csw,γ.

8. CONNECTION WITH KAZHDAN-LUSZTIG POLYNOMIALS 35

Furthermore, the partial order on Λ is defined to be the minimal partialorder satisfying these three properties.

Note that Λ is in bijective correspondence with W via Cw,γ ↔ wγ.

Proposition 2.22. The above partial order on Λ corresponds ex-actly to the Bruhat order on W, in the sense that Cy,δ � Cw,γ if andonly if yδ � wγ in W .

Proof. Let us check first that the Bruhat order on W does satisfythe properties (i), (ii) and (iii) above. With regard to (i), it is certainlytrue that y � w and δ � γ implies that yδ � wγ. Turning to (ii),suppose that y, w ∈ D+

J,s and δ, γ ∈ WJ satisfy yδ � wγ. Sincew < sw ∈ DJ we see that

�(swγ) = �(sw) + �(γ) = 1 + �(w) + �(γ) = 1 + �(wγ),

and �(syδ) = 1 + �(yδ) similarly. So syδ � swγ, by Deodhar [7,Theorem 1.1]. For (iii), suppose that w ∈ D+

J,s and y ∈ D0J,s, and let

δ, γ ∈ WJ be such that yδ � wγ. Suppose also that tδ > δ, wheret = y−1sy ∈ J . Then

�(syδ) = �(ytδ) = �(y) + �(tδ) = 1 + �(y) + �(δ) = 1 + �(yδ),

and since also �(swγ) = 1 + �(wγ) as above, Deodhar [7, Theorem 1.1]again gives the desired conclusion that ytδ = syδ � swγ.

Since the partial order on Λ is generated by the properties (i), (ii)and (iii), and since also the Bruhat order on W satisfies the sameproperties, it follows that Cy,δ � Cw,γ implies that yδ � wγ for ally, w ∈ DJ and δ, γ ∈WJ .

We must show, conversely, that yδ � wγ implies that Cy,δ � Cw,γ.In view of statement IV in [7, Theorem 1.1] it is sufficient to do thiswhen �(wγ) = �(yδ) + 1. Making this assumption, we argue by in-duction on �(w). Observe that if �(w) = 0 then wγ = γ ∈ WJ , andsince yδ � wγ it follows that yδ ∈ WJ . Hence y = 1, and Cy,δ � Cw,γ

by Property (i). So suppose that �(w) > 0, and choose s ∈ S withsw < w.

Consider first the possibility that syδ > yδ. Then we must infact have syδ = wγ, since, using the terminology of [7, Theorem 1.1],Property Z(s, syδ, wγ) implies that syδ � wγ. So either sy = w andδ = γ, in which case Cy,δ � Cw,γ by Property (i), or else y = w andγ = tδ, where t = y−1sy ∈ J , and again Property (i) gives Cy,δ � Cw,γ.

The only alternative is that syδ < yδ, and in this case we have thatsyδ � swγ (by Z(s, yδ, wγ), in Deodhar’s terminology). If y ∈ D−

J,s

then the inductive hypothesis yields that Csy,δ � Csw,γ, and Property(ii) gives Cy,δ � Cw,γ. Since y ∈ D+

J,s is not possible given syδ < yδ,


it remains to deal with the case y ∈ D0J,s. Writing t = y−1sy we have

syδ = ytδ � swγ, and the inductive hypothesis gives Cy,tδ � Csw,γ.Note that here tδ < δ and sw ∈ D+

J,s; so applying Property (iii) weobtain the desired conclusion that Cy,δ � Cw,γ. �

Equation (23) and Theorem 1.4 give Cδ =∑

θ∈WJpθ,δTθ, and we

deduce thatCw,γ =

∑y∈DJ

∑δ,θ∈WJ

Py,δ,w,γpθ,δTyθ,

since TyTθ = Tyθ for all y ∈ DJ and θ ∈ WJ . The coefficient ofTyθ in this expression is

∑δ∈WJ

Py,δ,w,γpθ,δ, and for this to be nonzerothere must exist a δ ∈WJ such that Py,δ,w,γ and pθ,δ are both nonzero.Now pθ,δ �= 0 implies that θ � δ by Theorem 1.4, and Py,δ,w,γ �= 0gives yδ � wγ, by Propositions 2.8 and 2.22. These combine to giveyθ � yδ � wγ. So if the coefficient of Tyθ in Cw,γ is nonzero thenyθ � wγ. Furthermore, the coefficient is a polynomial in q whoseconstant term is nonzero only if there exists a δ ∈WJ such that Py,δ,w,γ

and pθ,δ both have nonzero constant terms. This only occurs when(y, δ) = (w, γ) and θ = δ; that is, the constant term is nonzero only ifyθ = wγ. Hence by the uniqueness assertion in Theorem 1.4 we deducethat Cw,γ = Cwγ, and

(24) pyθ,wγ =∑

δ∈WJ

Py,δ,w,γpθ,δ

for all y, w ∈ DJ and θ, γ ∈WJ .Since the elements Cw,γ produced by our construction coincide with

the elements Cwγ of the Kazhdan-Lusztig construction, the W -graphdata of our construction must also agree with Kazhdan-Lusztig. So ifyθ � wγ then µ(yθ, wγ), the coefficient of q in −pyθ,wγ, must equalthe element µ(y, θ, w, γ) of our construction. That is, if y < w thenµ(yθ, wγ) equals the coefficient of q in −Py,θ,w,γ, while if y = w thenit equals µ(θ, γ), which is the coefficient of q in −pθ,γ. Eq. (24) aboveconfirms this.

CHAPTER 3

An inversion formula and duality

1. Duality of W-graphs

Throughout this chapter we assume that W is finite. We shallprove an inversion formula, which can be considered as a generaliza-tion of [16, Theorem 3.1]. It also generalizes the inversion formulaof Douglass [11] that relating the two set of relative Kazhdan-Lusztigpolynomials defined by Deodhar [10].

Recall that for all J ⊆ S we have defined wJ to be the longestelement in WJ . In particular, wS denotes the longest element in W .Since the longest element is unique, but �(w−1) = �(w) for all w ∈ W ,each wJ is an involution. Recall also that �(wSw) = �(wS) − �(w) forall w ∈ W , and that elements of the set DJ are characterized by theproperty that �(dx) = �(d) + �(x) for all x ∈ WJ . It follows that ifd ∈ DJ and x ∈WJ then

�((wSdwJ)x) = �(wS) − �(dwJx)

= �(wS) − �(d) − �(wJx)

= (�(wS) − �(d) − �(wJ)) + �(x)

= �(wSdwJ) + �(x),

and hence wSdwJ ∈ DJ . Furthermore, if d1, d2 ∈ DJ with d1 � d2

then d1wJ � d2wJ , and so wS(d2wJ) � wS(d1wJ). Hence d �→ wSdwJ

defines a poset anti-automorphism of DJ .

Lemma 3.1. Let d ∈ DJ and s ∈ S, and put e = wSdwJ andu = wSswS.

(1) If d ∈ D+J,s, then e ∈ D−

J,u.

(2) If d ∈ D−J,s, then e ∈ D+

J,u.

(3) If d ∈ D0J,s, and sd = dt (where t ∈ J), then e ∈ D0

J,u andue = er, where r = wJtwJ ∈ J .

Proof. It follows from Lemma 1.2 that �(wSwwS) = �(w) for allw ∈ W . So wSSwS = S, and, in particular, u ∈ S. Suppose that

37

38 3. AN INVERSION FORMULA AND DUALITY

d ∈ D+J,s. Then

�(ue) = �(wSsdwJ)

= �(wS) − �(sdwJ)

= �(wS) − �(sd) − �(wJ)

= (�(wS) − �(d) − �(wj)) − 1

= �(e) − 1.

Hence e ∈ D−J,u. The case d ∈ D−

J,s is similar. Finally, suppose d ∈ D0J,s

and sd = dt (where t ∈ J). Then

ue = (wSswS)(wSdwJ) = wS(sd)wJ

= wS(dt)wJ = (wSdwJ)(wJtwJ) = er

where r = wJtwJ . �

Define the function πS : W → W by πS(w) = wSwwS. Note thatsince πS is a length preserving automorphism of W , it follows thatTw �→ TπS(w) yields an automorphism of � . Note also that if d ∈ DJ

and e = wSdwJ then, by Lemma 3.1,

πS({ s ∈ S | d ∈ D+J,s }) ⊆ { u ∈ S | e ∈ D−

J,u },(25)

πS({ s ∈ S | d ∈ D−J,s }) ⊆ { u ∈ S | e ∈ D+

J,u },(26)

and for all I ⊆ J

πS({ s ∈ S | d ∈ D0J,s, d

−1sd ∈ I })⊆ { u ∈ S | e ∈ D0

J,u, e−1ue ∈ πJ(I) }.(27)

Moreover, since e = wSdwJ gives also d = wSewJ , the reverse inclusionsalso hold, so that in fact we have equality in (25), (26) and (27).

Now let (Γ, I, µ) be a W-graph datum. Let Γ′ = {γ′ | γ ∈ Γ} be aset in one to one correspondence with Γ, and define

µ′(γ′, δ′) = −µ(δ, γ),

and

I ′γ′ = {πS(s) | s ∈ S \ Iγ}for all γ, δ ∈ Γ.

Proposition 3.2. The triple (Γ′, I ′, µ′) defined above is a W-graphdatum.

Proof. Since

Ts(Ts − (q − q−1)) = 1

1. DUALITY OF W-GRAPHS 39

we see that the generators of � are all invertible. Furthermore

(−Ts−1)2 = 1 + (q − q−1)(−Ts

−1)

for all s ∈ S, and

((−Tr−1)(−Ts

−1)(−Tr−1) . . .)m = ((−Ts

−1)(−Tr−1)(−Ts

−1) . . .)m.

where r, s ∈ S and m is the order of rs.Thus Ts �→ −Ts

−1 yields an automorphism of � , and therefore sodoes Ts �→ −T−1

πS(s) Let τ be the matrix representation of� associated

with Γ, so that for each s ∈ S the matrix τ(Ts) has rows and columnsindexed by elements of Γ, the entry corresponding to the pair (γ, δ)being given by

(28) τγδ(Ts) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩−q−1 if γ = δ and s ∈ Iδ,

q if γ = δ and s /∈ Iδ,

µ(γ, δ) if s ∈ Iγ and s /∈ Iδ,

0 otherwise.

For each s ∈ S, let σ(Ts) be the transpose of τ(−T−1πS(s)). In view of

the automorphism above the matrices τ(−T−1πS(s)) satisfy the defining

relations of � , and transposing, we deduce that the matrices σ(Ts)do also. Hence s �→ σ(Ts) determines a matrix reprresentation of � .Furthermore, since −Ts

−1 = Ts + (q − q−1), we find that

σ(Ts) = −τ(TπS(s))T + (q − q−1)I

(where T denotes transpose and I is the identity matrix). Thus

σγδ(Ts) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩−q + (q − q−1) if γ = δ and πS(s) /∈ Iδ,

q−1 + (q − q−1) if γ = δ and πS(s) ∈ Iδ,

−µ(δ, γ) if πS(s) ∈ Iδ and πS(s) /∈ Iγ,

0 otherwise.

That is, since I ′δ′ is the complement of πS(Iδ),

σγδ(Ts) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩−q−1 if γ = δ and s ∈ I ′δ′ ,q if γ = δ and s /∈ I ′δ′ ,µ′(γ′, δ′) if s ∈ I ′γ′ and s /∈ I ′δ′ ,0 otherwise.

It follows that Γ′ is a W-graph with σ the associated representationof � . �

Definition 3.3. We call the W -graph Γ′ constructed from Γ asabove the dual of Γ.


Proposition 3.4. Suppose that Γ and Γ′ are dual WJ -graphs. TheW -graph IndS

J (Γ) has vertex set { (w, γ) | w ∈ DJ , γ ∈ Γ } and theW -graph IndS

J (Γ′) has vertex set { (w, γ′) | w ∈ DJ , γ ∈ Γ }. For eachw ∈ DJ let w′ = wSwwJ . Then for each γ ∈ Γ, the descent set ofthe vertex (w′, γ′) of IndS

J (Γ′) is equal to the descent set of the vertex(w, γ)′ of the dual of IndS

J (Γ).

Proof. According to the definition of the descent set of (w, γ) inIndS

J (Γ) we have

I(w, γ) = { s ∈ S | w ∈ D−J,s } ∪ { s ∈ S | w ∈ D0

J,s, w−1sw ∈ Iγ }.

So

S\I(w, γ) = { s ∈ S | w ∈ D+J,s }∪{ s ∈ S | w ∈ D0

J,s, w−1sw ∈ J \Iγ }.

Thus it follows that the descent set of (w, γ)′ in the dual of IndSJ (Γ) is

the union of

πS({ s ∈ S | w ∈ D+J,s })(29)

and

πS({ s ∈ S | w ∈ D0J,s, w

−1sw ∈ J \ Iγ }).(30)

Similarly, the descent set of (wSwwJ , γ′) in IndS

J (Γ′) is the union of

{ u ∈ S | wSwwJ ∈ D−J,u }(31)

and

{ u ∈ S | wSwwJ ∈ D0J,u, (wSwwJ)−1u(wSwwJ) ∈ πJ(J \ Iγ) }.(32)

But as was shown above (see Eq. (25) and Eq. (27)) the sets (29) and(31) are equal, as are the sets (30) and (32); hence the result follows. �

In view of Proposition 3.4 it seems reasonable to expect that thedual of an induced W -graph is isomorphic to the W -graph inducedfrom the dual. We shall investigate this question later in this chapter.

2. Further properties of the polynomials Rx,δ,y,γ

Let (Γ, I, µ) be a WJ -graph datum, and consider again the poly-nomials Rx,δ,y,γ defined in Section 2 of Chapter 2. We derive someadditional properties of these polynomials similar to those given inProposition 2.3.

Starting with the equation

Tyγ =∑

x∈DJ ,δ∈Γ

Rx,δ,y,γTxδ.

2. FURTHER PROPERTIES OF THE POLYNOMIALS Rx,δ,y,γ 41

we multiply both sides by Ts. By calculations similar to those in theproof of Proposition 2.3, the right hand side,

∑Rx,δ,y,γTs(Txδ), is found

to be equal to∑x∈D−

J,s

δ∈Γ

(Rsx,δ,y,γ + (q − q−1)Rx,δ,y,γ)Txδ +∑

x∈D+J,s

δ∈Γ

Rsx,δ,y,γTxδ

+∑

x∈D0J,s

δ∈Γ+t

(qRx,δ,y,γ)Txδ

+∑

x∈D0J,s

δ∈Γ−t

(−q−1Rx,δ,y,γ +

∑θ∈Γ+

t

µ(δ, θ)Rx,θ,y,γ

)Txδ.

(33)

where we have written t = x−1sx ∈ S when x ∈ D0J,s. The left hand

side is

Ts(Tyγ) = (Ts + (q − q−1))Tyγ

= TsTyγ + (q − q−1)Tyγ.

We consider the different cases separately. Firstly, if y ∈ D0J,s then

TsTy = TyTu, where u = y−1sy ∈ S, and so

Ts(Tyγ) = Ty(Tuγ) + (q − q−1)Tyγ

= Ty(Tu + (q − q−1))γ

= TyTuγ

=

{−q−1Tyγ if γ ∈ Γ−

u ,

qTyγ +∑

θ∈Γ−uµ(θ, γ)Tyθ if γ ∈ Γ+

u .

We refer to these two cases as Case (a) and Case (b) respectively. Next,if y ∈ D−

J,s, which we call Case (c), then

Ts(Tyγ) = (TsTy + (q−1 − q)Ty)γ

= ((Tsy + (q − q−1)Ty) + (q−1 − q)Ty)γ

= Tsyγ.

Lastly, if y ∈ D+J,s (Case (d)) then

Ts(Tyγ) = Tsyγ + (q−1 − q)Tyγ.


Thus by Eq. (2) we find that

(34) Ts(Tyγ) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−q−1∑

x,δ Rx,δ,y,γTxδ,∑x,δ(qRx,δ,y,γ +

∑θ∈Γ−

uµ(θ, γ)Rx,δ,y,θ)Txδ,∑

x,δ Rx,δ,sy,γTxδ,∑x,δ(Rx,δ,sy,γ + (q − q−1)Rx,δ,y,γ)Txδ.

in Cases (a), (b), (c) and (d) respectively. This must equal Eq. (33)above; so we may equate coefficients of Txδ. The formulas obtainedin this way for x ∈ D−

J,s can be derived from those for x ∈ D+J,s by

replacing x by sx; similarly, the cases y ∈ D+J,s and y ∈ D−

J,s are

equivalent. Furthermore, the case y ∈ D−J,s has been dealt with in

Proposition 2.3; so it is really only the case y ∈ D0J,s that is new.

When x ∈ D+J,s the coefficient of Txδ in Eq. (33) is Rsx,δ,y,γ. Hence

(35) Rsx,δ,y,γ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩−q−1Rx,δ,y,γ (in Case (a)),

qRx,δ,y,γ +∑

θ∈Γ−uµ(θ, γ)Rx,δ,y,θ (in Case (b)),

Rx,δ,sy,γ (in Case (c)),

Rx,δ,sy,γ + (q − q−1)Rx,δ,y,γ (in Case (d)).

The following proposition re-expresses these equations in the form thatwe need later.

Proposition 3.5. Let γ, δ ∈ Γ and x, y ∈ DJ . If s ∈ S is suchthat sx ∈ DJ and �(sx) > �(x) then

Rx,δ,y,γ =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−qRsx,δ,y,γ if y ∈ D0J,s and γ ∈ Γ−

u ,

q−1Rsx,δ,y,γ +∑

θ∈Γ−u

µ(θ, γ)Rsx,δ,y,θ if y ∈ D0J,s and γ ∈ Γ+

u ,

Rsx,δ,sy,γ if y ∈ D+J,s,

Rsx,δ,sy,γ − (q − q−1)Rsx,δ,y,γ if y ∈ D−J,s,

where u = y−1sy.

Proof. The case y ∈ D0J,s and γ ∈ Γ−

u is Case (a) of Eq. (35), and

we have simply multiplied both sides by q. Similarly, the case y ∈ D0J,s

and γ ∈ Γ+u is Case (b) of Eq. (35): using Case (a), the terms Rx,δ,y,θ

in the sum have been replaced by −qRsx,δ,y,θ and moved to the otherside of the equation, which has then been multiplied by q−1. The othertwo parts are obtained by applying Eq. (35) with sy in place of y. Ify ∈ D+

J,s then sy ∈ D−J,s, and Case (c) of Eq. (35) immediately gives

Rsx,δ,sy,γ = Rx,δ,y,γ, as required. Similarly, if y ∈ D−J,s then Case (d) of

Eq. (35) gives Rsx,δ,sy,γ − (q − q−1)Rx,δ,sy,γ = Rx,δ,y,γ, and by Case (c)we can replace Rx,δ,sy,γ by Rsx,δ,y,γ. �


Proposition 3.5 can also be proved by comparing the coefficients ofTxδ in Eq.’s (33) and (34) when x ∈ D−

J,s and then replacing x by sx.

The similarities and differences between Propositions 3.5 and 2.3should be noted.

We have yet to consider the cases that x and y are both in D0J,s.

Writing t = x−1sx and u = y−1sy as above, we see that in Eq. (34)

(coefficient of Txδ) =

{−q−1Rx,δ,y,γ if γ ∈ Γ−

u ,

qRx,δ,y,γ +∑

θ∈Γ−uµ(θ, γ)Rx,δ,y,θ if γ ∈ Γ+

u ,

while in Eq. (33)

(coefficient of Txδ) =

{qRx,δ,y,γ if δ ∈ Γ+

t ,

−q−1Rx,δ,y,γ +∑

θ∈Γ+tµ(δ, θ)Rx,θ,y,γ if δ ∈ Γ−

t .

There are four cases to consider. Firstly, if γ ∈ Γ−u and δ ∈ Γ+

t thenwe must have Rx,δ,y,γ = 0. Next, if γ ∈ Γ+

u and δ ∈ Γ+t then we obtain

qRx,δ,y,γ +∑θ∈Γ−

u

µ(θ, γ)Rx,δ,y,θ = qRx,δ,y,γ,

which is no new information, since all the terms Rx,δ,y,θ are 0 by theprevious case. Similarly, if γ ∈ Γ−

u and δ ∈ Γ−t then we obtain

−q−1Rx,δ,y,γ = −q−1Rx,δ,y,γ +∑θ∈Γ+

t

µ(δ, θ)Rx,θ,y,γ,

which again is no new information, since all the terms Rx,θ,y,γ are 0.Finally, when γ ∈ Γ+

u and δ ∈ Γ−t we obtain the equation

(q + q−1)Rx,δ,y,γ =∑θ∈Γ+

t

µ(δ, θ)Rx,θ,y,γ −∑θ∈Γ−

u

µ(θ, γ)Rx,δ,y,θ.

The following proposition lists all the formulas we have obtained forthe polynomials Rx,δ,y,γ in the case that y ∈ D0

J,s.

Proposition 3.6. Let y ∈ D0J,s and u = y−1sy ∈ S.

Case (A): If γ ∈ Γ−u ,

Rx,δ,y,γ =

⎧⎪⎨⎪⎩−q−1Rsx,δ,y,γ if x ∈ D−

J,s,

−qRsx,δ,y,γ if x ∈ D+J,s,

0 if x ∈ D0J,s and δ ∈ Γ+

t where t = x−1sx.


Case (B): If γ ∈ Γ+u ,

Rx,δ,y,γ =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

qRsx,δ,y,γ − q∑

θ∈Γ−uµ(θ, γ)Rx,δ,y,θ if x ∈ D−

J,s,

q−1Rsx,δ,y,γ − q−1∑

θ∈Γ−uµ(θ, γ)Rx,δ,y,θ if x ∈ D+

J,s,

1

q + q−1

(∑θ∈Γ+

tµ(δ, θ)Rx,θ,y,γ −

∑θ∈Γ−

uµ(θ, γ)Rx,δ,y,θ

)if x ∈ D0

J,s and δ ∈ Γ−t .

Proof. The second part of Case (A) is part of Proposition 3.5,and the first part is obtained from the second by putting sx in placeof x. The second part of Case (B) is part of Proposition 3.5, modifiedusing the second part of Case (a); the first part is obtained from thesecond by putting sx in place of x. The remaining parts were provedin the discussion above. �

Lemma 3.7. Let (Γ, I, µ) and (Γ′, I ′, µ′) be dual WJ -graph data, thenotation being as in Section 1. Let the polynomials Rx,δ,y,γ be as definedin Section 2 of Chapter 2, and let R′

x,δ′,y,γ′ be defined analogously for Γ′.Then

εxεyRx,δ,y,γ = R′wSywJ ,γ′,wSxwJ ,θ′.

Proof. In order to simplify the notation, for each z ∈ DJ we writez′ = wSzwJ . As we have seen, z �→ z′ is a poset antiautomorphismof DJ . Our task is to prove that Rx,δ,y,γ = εxεyR

′y′,γ′,x′,δ′ . Recall that

Rx,δ,y,γ is zero if x � y or if x = y and δ �= γ, while Ry,γ,y,γ = 1 for ally ∈ DJ and γ ∈ Γ.

The proof uses induction on the length of y. If �(y) = 0, then y = 1and y′ = wSwJ , and

Rx,δ,1,γ =

{1 if x = 1 and δ = γ,

0 otherwise.

Moreover, if x �= 1 then x � 1, and wSwJ � x′. Thus we deduce that

εxε1R′wSwJ ,γ′,x′,δ′ =

{1 if x = 1 and δ = γ,

0 otherwise.

as required.Now let �(y) � 1, and choose an s ∈ S such that sy < y. Let

s0 = πS(s), and observe that s0y′ = (sy)′ > y′. In the same way, if

x ∈ D−J,s then s0x

′ = (sx)′ > x′, giving x′ ∈ D+J,s, and if x ∈ D+

J,s

then s0x′ = (sx)′ > x′, giving x′ ∈ D−

J,s. In the case that x ∈ D0J,s, so

that sx = xt for some t ∈ J , we write t0 = πJ(t) ∈ J , observing thats0x

′ = wSsxwJ = wSxtwJ = x′t0, and x′ ∈ D0J,s0

.


By Proposition 2.3 we find that

Rx,δ,y,γ =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Rsx,δ,sy,γ if x ∈ D−J,s

Rsx,δ,sy,γ + (q − q−1)Rx,δ,sy,γ if x ∈ D+J,s

qRx,δ,sy,γ if x ∈ D0J,s and δ ∈ Γ+

t

−q−1Rx,δ,sy,γ +∑

θ∈Γ+t

µ(δ, θ)Rx,θ,sy,γ if x ∈ D0J,s and δ ∈ Γ−

t ,

and by the inductive hypothesis this equals⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

εsxεsyR′s0y′,γ′,s0x′,δ′

εsxεsyR′s0y′,γ′,s0x′,δ′ + (q − q−1)εxεsyR

′s0y′,γ′,x′,δ′

qεxεsyR′s0y′,γ′,x′,δ′

−q−1εxεsyR′s0y′,γ′,x′,δ′ +

∑θ∈Γ+

t

µ(δ, θ)εxεsyR′s0y′,γ′,x′,θ′

in the four cases. Since εsx = −εx and εsy = −εy it follows thatεxεyRx,δ,y,γ equals⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

R′s0y′,γ′,s0x′,δ′ if x ∈ D−

J,s

R′s0y′,γ′,s0x′,δ′ − (q − q−1)R′

s0y′,γ′,x′,δ′ if x ∈ D+J,s

−qR′s0y′,γ′,x′,δ′ if x ∈ D0

J,s and δ ∈ Γ+t

q−1R′s0y′,γ′,x′,δ′ −

∑θ∈Γ+

t

µ(δ, θ)R′s0y′,γ′,x′,θ′ if x ∈ D0

J,s and δ ∈ Γ−t .

On the other hand, by Proposition 3.5 we find that R′y′,γ′,x′,δ′ equals⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

R′s0y′,γ′,s0x′,δ′ if x′ ∈ D+

J,s0,

R′s0y′,γ′,s0x′,δ′ − (q − q−1)R′

s0y′,γ′,x′,δ′ if x′ ∈ D−J,s0

−qR′s0y′,γ′,x′,δ′ if x′ ∈ D0

J,s0and δ′ ∈ Γ′−

t0,

q−1R′s0y′,γ′,x′,δ′ +

∑θ′∈Γ′−

t0

µ′(θ′, δ′)R′s0y′,γ′,x′,θ′ if x′ ∈ D0

J,s0and δ′ ∈ Γ′+

t0.

As we have observed, if x ∈ D−J,s then x′ ∈ D+

J,s0, and if x ∈ D+

J,s then

x′ ∈ D−J,s0

. The above equations show that εxεyRx,δ,y,γ = R′y′,γ′,x′,δ′ in

these two cases. Now suppose that x ∈ D0J,s with sx = xt, so that

x′ ∈ DJ,s0 and s0x′ = x′t0, where t0 = πJ(t). If δ ∈ Γ+

t then t /∈ Iδ,and by definition of Γ′ it follows that πJ (t) ∈ I ′δ′ , or, equivalently,δ′ ∈ Γ′−

t0 . Again the equations above show that εxεyRx,δ,y,γ = R′y′,γ′,x′,δ′ .

If δ ∈ Γ+t then, by the same reasoning, δ′ ∈ Γ′−

t0 ; furthermore, for allθ ∈ Γ we have µ(δ, θ) = −µ′(θ′, δ′) and θ ∈ Γ+

t if and only if θ′ ∈ Γ−t0 .

So the formulas for εxεyRx,δ,y,γ and R′y′,γ′,x′,δ′ agree in this case also,

completing the proof. �


According to the definition of polynomials Rx,δ,y,γ, we have

Tyγ =∑x,δ

Rx,δ,y,γTxδ.

Applying the bar involution to both sides gives

Tyγ =∑x,δ

Rx,δ,y,γ Txδ

=∑x,δ

Rx,δ,y,γ

(∑z,θ

Rz,θ,x,δTzθ)

=∑z,θ

(∑x,δ

Rz,θ,x,δRx,δ,y,γ

)Tzθ.

Therefore we have

Lemma 3.8.∑x,δ

Rz,θ,x,δRx,δ,y,γ =

{0 (z, θ) �= (y, γ),

1 (z, θ) = (y, γ).

3. An inversion formula for the generalized Kazhdan-Lusztigpolynomials

Let (Γ, I, µ) and (Γ′, I ′, µ′) be dual WJ -graph data. In this sectionwe prove that the induced W-graphs IndS

J (Γ) and IndSJ (Γ′) are isomor-

phic. We do this by proving an “inversion formula” relating the gener-alized Kazhdan-Lusztig polynomials Py,δ,w,γ arising in the definition of

IndSJ (Γ) to the corresponding polynomials Py′,δ′,w′,γ′ for IndS

J (Γ′).We use the same notation as in Sections 1 and 2. Thus γ �→ γ′ is a

bijective mapping from Γ to Γ′, and we have µ′(γ′, δ′) = −µ(δ, γ) andI ′γ′ = πJ (J \ Iγ) for all γ, δ ∈ Γ; moreover, for each w ∈ DJ we definew′ = wSwwJ , so that w �→ w′ is a poset antiautomorphism of DJ .

Theorem 3.9. In the above situation,

(36)∑

{z|x�z�w}θ∈Γ

εxεzPx,δ,z,θP′w′,γ′

,z′,θ′ =

{1 if (x, δ) = (w, γ),

0 if (x, δ) �= (w, γ).

for all (x, δ), (w, γ) ∈ DJ × Γ.

Proof. The proof we give is based on [16, Theorem 3.1], modi-fied appropriately. Let M(x,δ),(w,γ) be the left hand side of Eq. (36).Obviously M(x,δ),(w,γ) = 0 unless x � w; moreover, if x = w thenPx,δ,z,θP

′w′,γ′,z′,θ′ = Px,δ,z,θP

′x′,γ′,z′,θ′ can only be nonzero if x = z = w

and δ = θ = γ. So M(w,γ),(w,γ) = 1, and M(w,δ),(w,γ) = 0 if δ �= γ.

3. AN INVERSION FORMULA 47

Proceeding by induction on �(w)−�(x), we may assume that x < wand that M(t,β),(u,µ) = 0 for all t, u ∈ DJ such that x � t < u � w and�(u)− �(t) < �(w)− �(x). We start with the identities in Theorem 2.4:

Cz,θ = Cz,θ

andCz,θ =

∑y∈DJ ,ζ∈Γ

y�z

Py,ζ,z,θTyζ

for all w ∈ DJ and ζ ∈ Γ. These give∑x�z,δ

Px,δ,z,θTxδ =∑y�z,ζ

Py,ζ,z,θ Tyζ

=∑y�z,ζ

Py,ζ,z,θ

∑x�y,δ

Rx,δ,y,ζTxδ,

by Eq. (2). So we have

Px,δ,z,θ =∑y,ζ

x�y�z

Rx,δ,y,ζ Py,ζ,z,θ.

Similarly, in IndSJ (Γ′) we have

P ′w′,γ′,z′,θ′ =

∑u,κ

z�u�w

R′w′,γ′,u′,κ′P ′

u′,κ′,z′,θ′.

Therefore

M(x,δ),(w,γ) =∑z,θ

x�z�w

εxεz

(∑y,ζ

x�y�z

Rx,δ,y,ζ Py,ζ,z,θ

)(∑u,κ

z�u�w

R′w′,γ′,u′,κ′P ′

u′,κ′,z′,θ′

)

=∑

y,u,ζ,κx�y�u�w

εxεyRx,δ,y,ζR′w′,γ′,u′,κ′

(∑z,θ

y�z�u

εyεzPy,ζ,z,θ P ′u′,κ′,z′,θ′

)

=∑

y,u,ζ,κ

εxεyRx,δ,y,ζ R′w′,γ′,u′,κ′ M(y,ζ),(u,κ)

When y = x and u = w we have Rx,δ,y,ζ = R′w′,γ′,u′,κ′ = 0 unless ζ = δ

and κ = γ, in which case Rx,δ,y,ζ = R′w′,γ′,u′,κ′ = 1. For all other terms in

the sum we have �(u)−�(y) < �(w)−�(x), and the inductive hypothesisgives M(y,ζ),(u,κ) = 0 unless (y, ζ) = (u, κ), in which case M(y,ζ),(u,κ) = 1.Thus

M(x,δ),(w,γ) = M(x,δ),(w,γ) +∑y,ζ

εxεyRx,δ,y,ζ R′w′,γ′,y′,ζ′.


According to Lemma 3.7 and Lemma 3.8, we have∑y,ζ

εxεyRx,δ,y,ζ R′w′,γ′,y′,ζ′ =

∑y,ζ

εxεyRx,δ,y,ζ

(εwεyRy,ζ,w,γ

)= εxεw

∑y,ζ

Rx,δ,y,ζRy,ζ,w,γ

= 0,

since x �= w. So

M(x,δ),(w,γ) = M(x,δ),(w,γ).

But by definition M(x,δ),(w,γ) =∑

x�z�w,θ εxεzPx,δ,z,θP′w′,γ′ ,z′,θ′, which is

a polynomial in q since both Px,δ,z,θ and P ′w′,γ′ ,z′,θ′ are polynomials in q.

Moreover, since x �= w each term in the sum must either have x �= zor z �= w, which means that one or other of Px,δ,z,θ and P ′

w′,γ′ ,z′,θ′ must

have zero constant term. So M(x,δ),(w,γ) is a polynomial in q with zeroconstant term, and hence cannot be fixed by the involution a �→ aunless it is zero. Thus M(x,δ),(w,γ) = 0, as claimed. �

Let (Γ, I, µ) be a W-graph datum, and M = AΓ the corresponding� -module. For each J ⊆ S define

ΓJ = { γ ∈ Γ | Iγ = J },and suppose that (λJ

ζ,θ) is an integral unimodular matrix with rows

and columns indexed by ΓJ . Let (νJζ,θ) be the inverse of (λJ

ζ,θ). Givenγ, δ ∈ Γ, put J = Iγ and K = Iδ, and define

ξ(γ, δ) =∑θ∈ΓJ

∑ζ∈ΓK

λJγ,ζµ(ζ, θ)νJ

θ,δ.

It is clear that (Γ, I, ξ) is also a W-graph datum. Indeed, if we define

γ =∑ζ∈ΓJ

λJζ,γζ

whenever γ ∈ ΓJ ⊆ Γ, then (γ)γ∈Γ is an A-basis of M such that

Tsγ =

{−q−1γ if s ∈ Iγ

qγ +∑

{δ∈Γ|s∈Iδ}ξ(δ, γ)δ if s /∈ Iγ,

for all s ∈ S and γ ∈ Γ, showing that M can be identified with theA-module derived from (Γ, I, ξ).

The following result is a simple special case of the above construc-tion.

3. AN INVERSION FORMULA 49

Lemma 3.10. Let (Γ, I, µ) be a W -graph datum, and M the corre-sponding� -module. Let ε : Γ → {1,−1} be an arbitrary function, andfor all γ, δ ∈ Γ define ν(γ, δ) = ε(γ)ε(δ)µ(γ, δ). Then (Γ, I, ν) is alsoa W -graph datum, yielding an � -module isomorphic to M .

We are now able to describe the relationship between the dual ofan induced W-graph and the W-graph induced from the dual.

Theorem 3.11. Let (Γ, I, µ), (Γ′, I ′, µ′) be dual WJ -graph data, and(DJ ×Γ, I, µ), (DJ ×Γ′, I ′, µ′) the corresponding induced W-graph data.Let (Λ′, H, ξ) be the dual of (DJ × Γ, I, µ), so that(a) Λ′ = { (w, γ)′ | (w, γ) ∈ DJ × Γ } is in one to one correspondencewith DJ × Γ,(b) H(w,γ)′ = S \ πS(I(w,γ)) for all (w, γ) ∈ DJ × Γ, and(c) ξ((w, γ)′, (x, δ)′) = −µ((x, δ), (w, γ)) for all (w, γ), (x, δ) ∈ DJ×Γ.

Then

(1) H(w,γ)′ = I ′(w′,γ′), and

(2) ξ((w, γ)′, (x, δ)′) = εxεwµ′((w′, γ′), (x′, δ′))for all w, x ∈ DJ and γ, δ ∈ Γ (where w′ = wSwwJ and x′ = wSxwJ).

Proof. The identity H(w,γ)′ = I ′(w′,γ′) was proved in Proposi-tion 3.4; so our task is just to prove that for all x, w ∈ DJ and δ, γ ∈ Γ,

(37) −εxεwµ((x, δ), (w, γ)) = µ′((w′, γ′), (x′, δ′)).

Re-adopting the notation used in Chapter 2, let us write µ(x, δ, w, γ)and µ′(w′, γ′, x′, δ′) rather than µ((x, δ), (w, γ)) and µ′((w′, γ′), (x′, δ′)).

It will be sufficient to prove Eq. (37) in the case that x � w,since if w < x then, by definition, µ(x, δ, w, γ) = µ(w, γ, x, δ) andµ′(x′, δ′, w′, γ′) = µ′(w′, γ′, x′, δ′), while if x and w are not comparablein the Bruhat order then µ′(w′, γ′, x′, δ′) and µ(x, δ, w, γ) are both zero.

If x = w then

−µ(x, δ, w, γ) = −µ(δ, γ) = µ′(γ′, δ′) = µ(w′, γ′, x′, δ′),

as required. If x < w then µ(x, δ, w, γ) is the coefficient of q in −Px,δ,w,γ,while µ(w′, γ′, x′, δ′) is the coefficient of q in −P ′

w′,γ′,x′,δ′ . So our task isto prove that the coefficient of q in Px,δ,w,γ + εxεwP

′w′,γ′,x′,δ′ is zero. But

from Theorem 3.9, we have

(38)∑

{z|x�z�w}θ∈Γ

εxεzPx,δ,z,θP′w′,γ′ ,z′,θ′ = 0

since x �= w, and when x < z < w both Px,δ,z,θ and P ′w′,γ′ ,z′,θ′ have zero

constant term, so that the coefficient of q in the product Px,δ,z,θP′w′,γ′ ,z′,θ′


is zero. So the terms in Eq. (38) corresponding to (z, θ) = (x, δ) and(z, θ) = (w, γ) are the only two for which q can have nonzero coefficient.But

εxεxPx,δ,x,δP′w′,γ′,x′,δ′ + εxεwPx,δ,w,γP

′w′,γ′,w′,γ′ = Px,δ,w,γ + εxεwP

′w′,γ′,x′,δ′ ,

and so the coefficient of q in Px,δ,w,γ + εxεwP′w′,γ′,x′,δ′ is the same as

the coefficient of q in the left hand side of Eq. (38), which is zero, asrequired. �

CHAPTER 4

Irreducible W-graphs for types D4 and D5

Before we can find W-graphs for the irreducible representations ofthe Weyl groups of types E6 and E7, we need to know W-graphs forthe irreducible representations of D4 and D5. In this chapter we findthese.

In these low rank cases it is not difficult to determine, without theaid of a computer, whether or not W-graph conditions are satisfied.Accordingly we start by writing conditions which are necessary andsufficient to ensure that a graph is a W-graph, under the assumptionthat for all r, s ∈ S the order of rs is 2 or 3.

1. W -graph conditions

Assume that the Weyl group W has a simply laced diagram, thatis, mrs = 2 or 3 for all r, s ∈ S. Let Γ be a graph, and suppose that foreach vertex γ of Γ we are given a set Iγ ⊆ S, and for each edge {δ, γ} ofΓ we are given integers µ(δ, γ) and µ(γ, δ). We wish to find necessaryand sufficient conditions for (Γ, I, µ) to constitute a W-graph datum.

1.1. The rules for edges. For any pair {r, s} ⊆ S the vertices ofΓ can be divided into the following four subsets:

A = {x | r ∈ Ix, s /∈ Ix},B = {x | r /∈ Ix, s ∈ Ix},C = {x | r ∈ Ix, s ∈ Ix},D = {x | r /∈ Ix, s /∈ Ix}.

Then we have the following conditions for W -graph edges:

Theorem 4.1. The triple (Γ, I, µ) constitutes a W-graph datum ifand only if the following conditions hold for all pairs {r, s} ⊆ S.Case (A). Suppose that (rs)3 = 1. Then

(1) if x ∈ A, then∑y∈B

µ(x, y)µ(y, x) = 1;

(2) if x, t ∈ A and t �= x, then∑y∈B

µ(x, y)µ(y, t) = 0;

51

52 4. IRREDUCIBLE W-GRAPHS FOR TYPES D4 AND D5

(3) if x ∈ D and t ∈ C, then

∑y∈B

µ(t, y)µ(y, x) =∑y∈A

µ(t, y)µ(y, x),

∑z∈B

∑y∈A

µ(t, z)µ(z, y)µ(y, x) =∑z∈A

∑y∈B

µ(t, z)µ(z, y)µ(y, x).

Case (B). Suppose that (rs)2 = 1. Then

(4) if x ∈ D and t ∈ C, then∑y∈B


µ(t, y)µ(y, x);

(5) if x ∈ A and y ∈ B, then µ(x, y) = 0.

Proof. Let M = AΓ be the free A-module on Γ, and for eachs ∈ S define an A-endomorphism τs of M by

τs(γ) =

{−q−1γ if s ∈ Iγqγ +

∑{δ∈Γ|s∈Iδ} µ(δ, γ)δ if s /∈ Iγ

for all γ ∈ Γ. It is clear that τ 2s = 1 + (q − q−1)τs for all s ∈ S;

so Γ is a W-graph if and only if τrτs = τsτr whenever mrs = 2, andτrτsτr = τsτrτs whenever mrs = 3.

Consider first the case mrs = 3, and suppose that x ∈ A. Then

τsτrτs(x) = τsτr

(qx+

∑y∈B

µ(y, x)y +∑z∈C

µ(z, x)z)

= τs

(−x− q−1

∑z∈C

µ(z, x)z + q∑y∈B

µ(y, x)y

+∑y∈B

∑z∈A

µ(y, x)µ(z, y)z +∑y∈B

∑z∈C

µ(y, x)µ(z, y)z)

= q−2∑z∈C

µ(z, x)z −∑y∈B

µ(y, x)y − q−1∑y∈B

∑z∈C

µ(y, x)µ(z, y)z

− qx−∑y∈B

µ(y, x)y −∑z∈C

µ(z, x)z + q∑y∈B

∑z∈A

µ(y, x)µ(z, y)z

+∑y∈B

∑z∈A

∑t∈B

µ(y, x)µ(z, y)µ(t, z)t

+∑y∈B

∑z∈A

∑t∈C

µ(y, x)µ(z, y)µ(t, z)t.

1. W -GRAPH CONDITIONS 53

By similar calculations,

τrτsτr(x) = τrτs(−q−1x)

= −q−1τr

(qx+

∑y∈B

µ(y, x)y +∑z∈C

µ(z, x)z)

= q−1x+ q−2∑z∈C

µ(z, x)z −∑y∈B

µ(y, x)y

− q−1∑y∈B

∑z∈A

µ(y, x)µ(z, y)z − q−1∑y∈B

∑z∈C

µ(y, x)µ(z, y)z.

There are three terms common to τrτsτr(x) and τsτrτs(x), namely

q−2∑z∈C

µ(z, x)z, −∑y∈B

µ(y, x)y, −q−1∑y∈B

∑z∈C

µ(y, x)µ(z, y)z.

We find that τrτsτr(x) − τsτrτs(x) equals

(q + q−1)(x−

∑y∈B

∑z∈A

µ(y, x)µ(z, y)z)

+∑

t∈B∪C

µ(t, x)t−∑

t∈B∪C

∑y∈B

∑z∈A

µ(y, x)µ(z, y)µ(t, z)t.

If τrτsτr(x) = τsτrτs(x) then the coefficient of q in this expression mustbe 0, and so for all z ∈ A,

(39)∑y∈B

µ(z, y)µ(y, x) =

{1 if z = x,

0 if z �= x.

Conversely, if Eq. (39) holds then∑z∈A,y∈B

µ(y, x)µ(z, y)µ(t, z) = µ(t, x)

for all t ∈ B ∪ C, and it follows that τrτsτr(x) = τsτrτs(x).

By symmetry it follows that if x ∈ B then τrτsτr(x) = τsτrτs(x) ifand only if for all z ∈ B,

(40)∑y∈B

µ(z, y)µ(y, x) =

{1 if z = x,

0 if z �= x.

Observe also that if x ∈ C then τrτsτr(x) and τsτrτs(x) are alwaysequal, both being equal to −q−3x.


Now let us consider the case when x ∈ D.

τrτsτr(x) = τrτs

(qx+

∑y∈A

µ(y, x)y +∑z∈C

µ(z, x)z)

= qτr

(qx+

∑w∈B

µ(w, x)w +∑z∈C

µ(z, x)z)

+ q−2∑z∈C

µ(z, x)z

+ τr

(q∑y∈A

µ(y, x)y +∑y∈A

∑w∈B

µ(y, x)µ(w, y)w

+∑y∈A

∑z∈C

µ(y, x)µ(z, y)z)

= q3x+ q2∑y∈A

µ(y, x)y + q2∑z∈C

µ(z, x)z + q2∑w∈B

µ(w, x)w

+ q∑y∈A

∑w∈B

µ(y, w)µ(w, x)y+ q∑z∈C

∑w∈B

µ(w, x)µ(z, w)z

−∑z∈C

µ(z, x)z −∑y∈A

µ(y, x)y + q∑y∈A

∑w∈B

µ(y, x)µ(w, y)w

+∑t∈C

∑y∈A

∑w∈B

µ(y, x)µ(w, y)µ(t, w)t

+∑u∈A

∑y∈A

∑w∈B

µ(y, x)µ(w, y)µ(u, w)u

+ q−1∑y∈A

∑z∈C

µ(y, x)µ(z, y)z − q−2∑z∈C

µ(z, x)z.

Similarly, we have

τsτrτs(x) = q3x+ q2∑w∈B

µ(w, x)w + q2∑z∈C

µ(z, x)z + q2∑y∈A

µ(y, x)y

+ q∑y∈A

∑w∈B

µ(y, x)µ(w, y)w+ q∑z∈C

∑y∈A

µ(y, x)µ(z, y)z

−∑z∈C

µ(z, x)z −∑w∈B

µ(w, x)w + q∑y∈A

∑w∈B

µ(w, x)µ(y, w)y

+∑t∈C

∑y∈A

∑w∈B

µ(w, x)µ(y, w)µ(t, y)t

+∑u∈B

∑y∈A

∑w∈B

µ(w, x)µ(y, w)µ(u, y)u

− q−1∑w∈B

∑z∈C

µ(w, x)µ(z, w)z + q−2∑z∈C

µ(z, x)z.


Comparing these expressions, we see that they are equal if and only if

q∑z∈C

∑y∈A

µ(y, x)µ(z, y)z = q∑z∈C

∑w∈B

µ(w, x)µ(z, w)z

and∑t∈C

∑y∈A

∑w∈B

µ(w, x)µ(y, w)µ(t, y)t =∑t∈C

∑y∈A

∑w∈B

µ(y, x)µ(w, y)µ(t, w)t.

Equivalently, for all t ∈ C,∑y∈B


µ(t, y)µ(y, x),(41)

∑z∈B

∑y∈A

µ(t, z)µ(z, y)µ(y, x) =∑z∈A

∑y∈B

µ(t, z)µ(z, y)µ(y, x).(42)

So τsτrτs = τrτsτr if and only if Eq. (39) holds for all x, z ∈ A, Eq. (40)holds for all z, x ∈ B and Eq’s (41) and (42) hold for all t ∈ C andx ∈ D.

Consider now the case mrs = 2. For x ∈ A we have

τsτr(x) = −τs(x) = −qx−∑y∈C

µ(y, x)y −∑y∈B

µ(y, x)y,

while

τrτs(x) = τr

(qx+

∑y∈C

µ(y, x)y +∑y∈B

µ(y, x)y)

= −qx−∑y∈C

µ(y, x)y +∑y∈B

µ(y, x)τr(y).

It is clear that these are equal if µ(y, x) = 0 for all y ∈ B. Moreover,this condition is also necessary since in τsτr(x) the coefficient of y ∈ Bis −µ(y, x), while in τrτs(x) it is +µ(y, x). Similarly, for x ∈ B we findthat τrτs(x) = τsτr(x) if and only if µ(y, x) = 0 for all y ∈ A.

For x ∈ D we have

τsτr(x) = τs

(qx+

∑y∈A

µ(y, x)y +∑t∈C

µ(t, x)t)

= q2x+ q∑z∈B

µ(z, x)z + q∑t∈C

µ(t, x)t+ q∑y∈A

µ(y, x)y

+∑y∈A

∑z∈B

µ(y, x)µ(z, y)z +∑y∈A

∑t∈C

µ(y, x)µ(t, y)t,

− q−1∑t∈C

µ(t, x)t


and

τrτs(x) = τr

(qx+

∑z∈B

µ(z, x)z +∑t∈C

µ(t, x)t)

= q2x+ q∑y∈A

µ(y, x)y + q∑t∈C

µ(t, x)t+ q∑z∈B

µ(z, x)z

+∑z∈B

∑y∈A

µ(z, x)µ(y, z)y +∑z∈B

∑t∈C

µ(z, x)µ(t, z)t

− q−1∑t∈C

µ(t, x)t.

If µ(x, y) = µ(y, x) = 0 whenever x ∈ A and y ∈ B then the terms∑y∈A

∑z∈B µ(y, x)µ(z, y)z in τsτr(x) and

∑z∈B

∑y∈A µ(z, x)µ(y, z)y

in τrτs(x) vanish, and we find that

τsτr(x) − τrτs(x) =∑y∈A

∑t∈C

µ(y, x)µ(t, y)t−∑z∈B

∑t∈C

µ(z, x)µ(t, z)t.

So τrτs(x) = τsτr(x) if and only if

(43)∑z∈B

µ(t, z)µ(z, x) =∑z∈A

µ(t, z)µ(z, x)

for all t ∈ C.Finally, if x ∈ C then clearly τrτs(x) = τsτr(x) = q−2x. So it follows

that τrτs = τsτr if and only if µ(x, y) = µ(y, x) = 0 for all x ∈ A andy ∈ B, and Eq. (43) holds for all x ∈ D and t ∈ C. �

1.2. Induced characters and descent sets. For any W-graphdatum (Γ, I, µ), the formula Eq. (28) determines a family of matrices(τ(Ts))s∈S satisfying the defining relations of � . If we replace q by 1we obtain matrices that satisfy the defining relations of W , and thusgive rise to a representation of W (over the ring Z). Our principalobjective is to produce a W-graph for one representative of each equiv-alence class of irreducible complex representations of the Weyl groupsof types E6 and E7. Since our strategy is to do this by decomposinginduced representations, we often need to know the multiplicity withwhich an irreducible character occurs as a constituent of an inducedcharacter. The relevant information is listed in the tables at the endof this chapter. In practice we generally used Magma to compute suchthings, although for types E6 and E7 the paper [12] of J. S. Frame alsoprovides the information needed.

If χ is any character of W , and we wish to find a correspondingW-graph Γ, then, in particular, we are interested in determining, foreach J ⊆ S, the number of vertices of Γ with descent set J . As we shall


show, there is a formula for this quantity in terms of the inner products(IndW

WJ(ε), χ), where ε is the sign character (defined by the formula

ε(w) = εw = (−1)�(w)). So we give tables of these inner products forthe irreducible characters of W (D4), W (D5), W (E6) and W (E7).

Let C be the complex field, and let M = CΓ be the W-modulederived from a W-graph Γ. Let χ be the corresponding character of W ,and suppose that χ is irreducible. For each J ⊆ S define

MJ = { v ∈M | wv = ε(w)v for all w ∈WJ },the ε-isotypic component of ResW

WJ(M). Then

dim(MJ) = (ε,ResWWJ

(χ)) = (IndWWJ

(ε), χ).

Furthermore, since

MJ = { v ∈M | sv = −v for all s ∈ J },it follows that MJ =

⋂s∈J M{s}.

Lemma 4.2. For each s ∈ S the subspace M{s} of M is spanned bythe set Γ−

s = { γ ∈ Γ | s ∈ Iγ }.Proof. Let V be the subspace of M spanned by Γ−

s . It is imme-diate that V ⊆ M{s}. On the other hand the formula Eq. (1) showsthat s acts trivially on the quotient space M/V ; so M{s}/V is the zeromodule, and therefore V = M{s}. �

For each J ⊆ S we define

Γ−J = { γ ∈ Γ | J ⊆ Iγ }.

Observe that Γ−J =

⋂s∈J Γ−

s .Now let v =

∑γ∈Γ λγγ ∈ MJ , and suppose that λγ �= 0 for some

γ ∈ Γ. If s ∈ J then v ∈ M{s}, and by Lemma 4.2 it follows thatγ ∈ Γ−

s . So γ ∈ Γ−J . On the other hand, it is obvious that Γ−

J ⊆ MJ .Hence it follows that Γ−

J is a basis of MJ , and we deduce that the sizeof the set Γ−

J is given by

#Γ−J = (ε,ResW

WJ(χ)).

Corollary 4.3. For each J ⊆ S, the number of vertices γ ∈ Γwith descent set J is given by∑

{K|J⊆K⊆S}(−1)#K−#J(ε,ResW

WK(χ)).


Proof. Let nL be the number of vertices with descent set L. SinceΓ−

K is the set of all vertices whose descent sets contain K, it followsthat #Γ−

K =∑

L nL, where the sum is over all L ⊆ S such that K ⊆ L.

Now since (ε,ResWWK

(χ)) = #Γ−K ,∑

{K|J⊆K⊆S}(−1)#K−#J(ε,ResW

WK(χ)) =

∑{K|J⊆K⊆S}

(−1)#K−#J∑

{L|K⊆L⊆S}nL

=∑

{L|J⊆L⊆S}nL

∑{K|J⊆K⊆L}

(−1)#K−#J

= nJ ,

since the inner sum is 1 when L = J , and zero otherwise. �For groups W of low enough rank, it is feasible to use Theorem 4.1

and Corollary 4.3 to compute a W-graph for each irreducible characterof W . We will use this method for type D4.

2. The Weyl group of type Dn

Let x1, x2, · · · , xn generate an elementary abelian group N of rankn. That is for all i we have

xi2 = 1,

and for all i, j:xixj = xjxi.

So N is the direct product of n copies of a group of order 2, and#N = 2n. Form the group B as follows:

B = {xθ | x ∈ N, θ ∈ Sym(n) },where Sym(n) denotes the symmetric group of deree n. Multiplicationis defined as follows:

θxi = xθ(i)θ,

(xθ)(yτ) = x(θyθ−1)(θτ),

where θyθ−1 is calculated via the first equation above. Then B isisomorphic to the Weyl group of type Bn. The Weyl group of typeDn is a subgroup of index 2 in B consisting of those elements xθ suchthat x has an even number of x′is in it. The following elements are thesimple reflections of W = W (Dn):

r1 = (12), r2 = x1x2(12), r3 = (23), . . . , rn = (n− 1, n).

The correspondence with the Coxeter graph is as follows.

(n ≥ 4 vertices).r1r2

r3 r4 rn−1 rn

3. TYPE D4 59

Note that for each i,

riri−1 · · · r3(r1r2)r3r4 · · · ri = x1xi,

and x1x2, x1x3, · · · , x1xn generate the subgroup X = W ∩ N , whichhas order 2n−1 and consists of all products

xi1xi2 · · ·xir ,

where i1 < i2 < · · · < ir and r is even. The group W is isomorphic tothe semidirect product X � Sym(n).

3. Type D4

3.1. The irreducible W-graphs for D4. It is trivial to find W -graphs for 1-dimensional representations ofW . In particular, the graphconsisting of one vertex whose descent set is empty corresponds to the1-representation of W , and its dual, consisting of one vertex whosedescent set is S, corresponds to the sign representation ε. For groupscorresponding to simply laced Coxeter graphs, the Coxeter graph canbe interpreted as a W-graph for the reflection representation of W ;the descent set of the vertex corresponding to the simple reflection sis {s}. We can also write down the dual of any given W-graph. Sowe immediately find W-graphs for the characters ψ1, ψ2, ψ10 and ψ11

of W = W (D4). (See the character table in Section 5 below for thedefinitions of the ψi.) We denote the simple reflections ofW by r, s, t, u(corresponding to r1, r2, r3, r4 in the notation of last section).

ψ1

rstu

ψ2

r

s

t u

ψ10

tsu

rtu

rsu rts

ψ11

In our diagrams, all unmarked edges are understood to have weight 1.Observe that there is a surjective homomorphism from W to the

Weyl group of type A2 taking r, s, u to one of the simple reflections forA2 and t to the other. So the W-graph corresponding to the reflectionrepresentation of W (A2) gives rise to a W-graph for a two-dimensionalirreducible representation of W , the descent sets being {r, s, u} and {t}.Observe that this W -graph is self-dual.

rsu t

ψ3

There are three surjective homomorphisms from W to W (A3) that sim-ilarly allow the W-graph for the reflection representation of W (A3) to


be converted into W -graphs for irreducible three-dimensional represen-tations of W . Taking duals gives another three.

rs t u rst rsu tuψ6 ψ7

su t r sut sur trψ6 ψ7

ru t s urt urs tsψ8 ψ9

The only remaining irreducible characters ofW are ψ12 (of degree 6)and ψ13 (of degree 8). In fact ψ12 comes from the exterior square ofthe reflection representation. Following the notation of Section 1 ofChapter 1, let αr, αs, αt, αu span the real vector space V on which Wacts as a reflection group. Then W also acts on the exterior square∧2V , which is spanned by the six vectors αr ∧ αt, αr ∧ αs, αr ∧ αu,αt ∧ αs, αt ∧ αu and αs ∧ αu. Observe that

αr ∧ αt ∈ V{r,t} = { v ∈ V | rv = −v and tv = −v }which is the subspace spanned by the vertices of the W-graph whose de-scent sets contain {r, t}. So there is at least one such vertex. The sameargument obviously applies to all two-element subsets J of {r, s, t, u}.But since this representation is self-dual it also follows that there is atleast one vertex whose descent set is contained in a given two-elementsubset J . We conclude that for each of the six two-element subsets Jthere is exactly one vertex with descent set J . So the W -graph basiselements must be scalar multiples of the six basis vectors listed above.

The action of the simple reflections on this basis is as follows.

s(αr ∧ αt) = αr ∧ (αs + αt) = αr ∧ αt + αr ∧ αs;

u(αr ∧ αt) = αr ∧ (αu + αt) = αr ∧ αt + αr ∧ αu;

t(αr ∧ αs) = (αr + αt) ∧ (αs + αt) = αr ∧ αs + αr ∧ αt + αt ∧ αs;

t(αr ∧ αu) = (αr + αt) ∧ (αu + αt) = αr ∧ αu + αr ∧ αt + αt ∧ αu;

r(αt ∧ αs) = (αr + αt) ∧ αs = αr ∧ αs + αt ∧ αs;

u(αt ∧ αs) = (αt + αu) ∧ αs = αt ∧ αs − αs ∧ αu;

r(αt ∧ αu) = (αr + αt) ∧ αu = αr ∧ αu + αt ∧ αu;

s(αt ∧ αu) = (αt + αs) ∧ αu = αt ∧ αu + αs ∧ αu;

t(αs ∧ αu) = (αt + αs) ∧ (αt + αu) = αt ∧ αu − αt ∧ αs + αs ∧ αu;

u(αr ∧ αs) = αr ∧ αs; s(αr ∧ αu) = αr ∧ αu; r(αs ∧ αu) = αs ∧ αu.

Hence the desired W-graph is as follows.

3. TYPE D4 61

rt rs

ru

ts

tu su

−1

ψ12

Note that the graph is symmetric, in the sense that µ(x, y) = µ(y, x)for all vertices x and y.

For ψ13, applying Corollary 4.3 we find that there must be onevertex with descent set {r, s, u}, one with descent set {t}, and onevertex with descent set J for each two-element subset J of S. By trialand error we found that the following graph Γ satisfies the requirementsof Theorem 4.1.

st

rs

tr

su

tu

ur

t rsu

2

2 2

ψ13

Let us call the basis vectors x1, . . . , x8, corresponding (in order) tothe descent sets {u, r}, {t, u}, {s, u}, {s, t}, {r, s}, {t, r}, {t} and{r, s, u}. This graph Γ is also symmetric, the edges {x3, x8}, {x1, x8}and {x5, x8} having weight 2, the others weight 1.

Consider first the restriction to the subgroup W{r,t}. There is justone vertex, x3, with descent set ∅, and one, x6, with descent set { r, t }.

t

r

tr

t

r

t r

2

2 2

ψ13

There are two paths of type t r rt from x3 to x6; theyare x3 → x4 → x5 → x6 and x3 → x2 → x1 → x6, and both have


weight 1, where by the weight of a path we mean the product of theweights of the edges in the path. By Condition (3) of Theorem 4.1 thesum of these path weights should equal the sum of the weights of thepaths of type r t rt from x3 to x6. There is only onesuch path, namely x3 → x8 → x7 → x6, and its weight is 2.

Each r is adjacent to a unique t via a weight 1 edge and eacht is adjacent to a unique r via a weight 1 edge. Hence Conditions(1) and (2) of Theorem 4.1 are also satisfied for the pair {r, t}.

By symmetry it is clear that Conditions (1), (2) and (3) of Theo-rem 4.1 are also satisfied for the pairs {s, t} and {u, t}.

Now consider the restriction to the subgroup W{r,s}. There are noedges from either of the vertices of type r (namely, x1 and x6) toeither of the vertices of type s (x3 and x4). So Condition (5) ofTheorem 4.1 is satisfied for {r, s}. Furthermore, from each of the ver-tices of type (namely, x2 and x7) there is a unique path of type

r rs and a unique path of type s rs . These pathslead to the same rs and have the same weight (which is 2 for the pathsx2 → x3 → x8 and x2 → x1 → x8 and 1 for the paths x7 → x4 → x5

and x7 → x6 → x5). It follows that Condition (4) of Theorem 4.1 isalso satisfied for {r, s}. Furthermore, symmetry shows that the require-ments of Theorem 4.1 are also satisfied for {r, u} and {s, u}. Hence thegraph is indeed a W-graph.

Next we shall show that the 8-dimensional representation of �derived from Γ is indecomposable. Let s1, s2, s3, s4 equal r, s, t, urespectively, and write Ti for the 8 × 8 matrix representing the gener-ator Tsi

. It is sufficient to show that the only matrices that commutewith all of T1, T2, T3, T4 are scalar multiples of the identity matrix I.

Let Y = (ymn) satisfy Y Ti = TiY for all i ∈ {1, 2, 3, 4}. For each idefine

Ii = { l | si ∈ Ixl},

Ji = { 1, 2, 3, 4, 5, 6, 7, 8 }\Ii.

Thus Γ−si

= { xl | l ∈ Ii } and Γ+si

= { xl | l ∈ Ji }. In fact the sets Ji

and Ii are as follows.

J1 = {2, 3, 4, 7}, I1 = {1, 5, 6, 8},J2 = {1, 2, 6, 7}, I2 = {3, 4, 5, 8},J3 = {1, 3, 5, 8}, I3 = {2, 4, 6, 7},J4 = {4, 5, 6, 7}, I4 = {1, 2, 3, 8}.

3. TYPE D4 63

If m ∈ Ji then the m-th row of Ti is q times the m-th row of I,and if n ∈ Ii then the n-th column of Ti is −q−1 times the n-th col-umn of I. So if m ∈ Ji and n ∈ Ii then the (m,n)-entry of TiY isqymn, and the (m,n)-entry of Y Ti is −q−1ymn. Hence ymn = 0. Inthis way we find that all the entries of Y are zero, except (possibly)y27, y47, y67, y81, y83, y85 and the diagonal entries.

The matrices Ti are as follows.

T1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−q−1 1 0 0 0 0 0 00 q 0 0 0 0 0 00 0 q 0 0 0 0 00 0 0 q 0 0 0 00 0 0 1 −q−1 0 0 00 0 0 0 0 −q−1 1 00 0 0 0 0 0 q 00 0 2 0 0 0 1 −q−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

T2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q 0 0 0 0 0 0 00 q 0 0 0 0 0 00 1 −q−1 0 0 0 0 00 0 0 −q−1 0 0 1 00 0 0 0 −q−1 1 0 00 0 0 0 0 q 0 00 0 0 0 0 0 q 02 0 0 0 0 0 1 −q−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

T3 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

q 0 0 0 0 0 0 01 −q−1 1 0 0 0 0 00 0 q 0 0 0 0 00 0 1 −q−1 1 0 0 00 0 0 0 q 0 0 01 0 0 0 1 −q−1 0 00 0 0 0 0 0 −q−1 10 0 0 0 0 0 0 q

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

T4 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−q−1 0 0 0 0 1 0 00 −q−1 0 0 0 0 1 00 0 −q−1 1 0 0 0 00 0 0 q 0 0 0 00 0 0 0 q 0 0 00 0 0 0 0 q 0 00 0 0 0 0 0 q 00 0 0 0 2 0 1 −q−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

and so far we have shown that

Y =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

y11 0 0 0 0 0 0 00 y22 0 0 0 0 y27 00 0 y33 0 0 0 0 00 0 0 y44 0 0 y47 00 0 0 0 y55 0 0 00 0 0 0 0 y66 y67 00 0 0 0 0 0 y77 0

y81 0 y83 0 y85 0 0 y88

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.


The following equations are easily checked.

[T4Y ](3,7) = y47, [Y T4](3,7) = 0,

[T1Y ](1,7) = y27, [Y T1](1,7) = 0,

[T2Y ](5,7) = y67, [Y T2](5,7) = 0,

[T3Y ](7,1) = y81, [Y T3](7,1) = 0,

[T3Y ](7,3) = y83, [Y T3](7,3) = 0,

[T3Y ](7,5) = y85, [Y T3](7,5) = 0.

So Y is a diagonal matrix. Now since the (2, 1)-entry of T3 is nonzero,T3Y = Y T3 gives y11 = y22, and similarly since [T2]32, [T3]43, [T1]54,[T3]65, [T1]67 and [T4]87 are all nonzero we conclude that all the diagonalentries of Y are equal, as desired.

4. Type D5

Among its 18 characters, the pairs corresponding to the irreducibledual W- graphs are {ϕ1, ϕ2}, {ϕ3, ϕ4}, {ϕ5, ϕ6}, {ϕ8, ϕ9}, {ϕ11, ϕ12},{ϕ13, ϕ16}, {ϕ14, ϕ15}, {ϕ17, ϕ18}. The following is the list of irreducibleW-graphs corresponding to one of each pair and ϕ7 and ϕ10 which areself dual.

ϕ1

rs t u v

ϕ3

t rsu rsv tv u rst rsu

tu

rsv

tv uv

ϕ5 ϕ7

r

s

t u v

t rsu rsv stv su

rtv ru tu tv rsuv

ϕ10

ϕ8

rv r s sv

t tv t

suuru rt ru rv

rs tu tv uv

ts su sv

ϕ13

−1

ϕ14

4. TYPE D5 65

tr

ru

rv

rs

t

rsu

rsv

tv

u

tu tv

uv

ts

su

sv

tr

rs

st

ru

rus

su

ru

t

su

tu

tu

rtv

tsv

ruv

tv

tv

suv

rv

rsv

sv

2

2

2

2

2

ϕ11

ϕ18

All of the above W -graphs were initially found by hand calculationsbased on Theorem 4.1 and Corollary 4.3, used in combination with theknowledge of the restrictions of the irreducible characters from W (D5)to W (D4). Later they were checked with machine calculations similarto those used subsequently for W (E6) and W (E7).


5. Tables for types D4 and D5

5.1. Tables for type D4. The conjugacy classes and charactertable of W (D4) are well known. We number the classes C1 to C13, andfor each class we give a representative element, as well as the nameof the class in the terminology of [3]. We use the description of Weylgroups of type D given in Section 2 above.

C1 C2 C3 C4 C5 C6 C7

1 x1x2 x1x2x3x4 (12) x1x3(12) x3x4(12) (123)1 D2 A4

1 A1 D3 D2 ×A1 A2

C8 C9 C10 C11 C12 C13

x1x4(123) (1234) x1x2(1234) (12)(34) x1x3(12)(34) x1x2(12)(34)D4 A3 A3 A2

1 D4(a1) A21

Table(1): Conjugacy class representatives for W (D4).

Note that the two classes of type A3 form a single class in W (B4), asdo the two classes of type A2

1.We number the irreducible characters ψ1 to ψ13.

classes C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13

size 1 6 1 12 24 12 32 32 24 24 6 12 6ψ1 1 1 1 1 1 1 1 1 1 1 1 1 1ψ2 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 1ψ3 2 2 2 0 0 0 -1 -1 0 0 2 2 2ψ4 3 3 3 1 1 1 0 0 -1 -1 -1 -1 -1ψ5 3 3 3 -1 -1 -1 0 0 1 1 -1 -1 -1ψ6 3 -1 3 1 -1 1 0 0 -1 1 -1 -1 3ψ7 3 -1 3 -1 1 -1 0 0 1 -1 -1 -1 3ψ8 3 -1 3 1 -1 1 0 0 1 -1 3 -1 -1ψ9 3 -1 3 -1 1 -1 0 0 -1 1 3 -1 -1ψ10 4 0 -4 2 0 -2 1 -1 0 0 0 0 0ψ11 4 0 -4 -2 0 2 1 -1 0 0 0 0 0ψ12 6 -2 6 0 0 0 0 0 0 0 -2 2 -2ψ13 8 0 -8 0 0 0 -1 1 0 0 0 0 0

Table(2): Character table of W (D4)

5. TABLES FOR TYPES D4 AND D5 67

Of course the class D3 and the two classes A3 form a single orbitunder the action of the group of diagram automorphisms of W (D4),as do the class D2 and the two classes A2

1. A similar remark holds forparabolic subgroups. In the next table we refer to Wr1,r2,r3, Wr1,r3,r4

and Wr2,r3,r4 as Da3 , D

b3 and Dc

3 respectively, and Wr1,r2, Wr1,r4 andWr2,r4 as Da

2 , Db2 and Dc

2 respectively. The table gives the values of thecharacters induced from the sign characters of the various parabolicsubgroups.

Class C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13

size 1 6 1 12 24 12 32 32 24 24 6 12 6

(εDa3)W 8 4 0 -4 -2 0 2 0 0 0 0 0 0

(εDb3)W 8 0 0 -4 0 0 2 0 -2 0 4 0 0

(εDc3)W 8 0 0 -4 0 0 2 0 0 -2 0 0 4

(εA31)W 24 4 0 -6 0 -2 0 0 0 0 4 0 4

(εA2)W 32 0 0 -8 0 0 2 0 0 0 0 0 0

(εDa2)W 48 8 0 -8 0 0 0 0 0 0 0 0 0

(εDb2)W 48 0 0 -8 0 0 0 0 0 0 8 0 0

(εDc2)W 48 0 0 -8 0 0 0 0 0 0 0 0 8

(εA1)W 96 0 0 -8 0 0 0 0 0 0 0 0 0

Table(3): Induced sign characters for W (D4).

The next table lists the inner products (ψi, εW ). This information is

used to find the descent sets of the W-graph for ψi, using Corollary 4.3.

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9 ψ10 ψ11 ψ12 ψ13

(εDa3)W 0 1 0 0 1 0 0 0 0 0 1 0 0

(εDb3)W 0 1 0 0 0 0 0 0 1 0 1 0 0

(εDc3)W 0 1 0 0 0 0 1 0 0 0 1 0 0

(εA31)W 0 1 0 0 1 0 1 0 1 0 1 0 1

(εA2)W 0 1 0 0 1 0 1 0 1 0 1 1 1

(εDa2)W 0 1 1 1 2 0 1 0 1 0 2 1 2

(εDb2)W 0 1 1 0 1 0 1 1 2 0 2 1 2

(εDc2)W 0 1 1 0 1 1 2 0 1 0 2 1 2

(εA1)W 0 1 1 1 2 1 2 1 2 1 3 3 4

Table(4): Multiplicities of W (D4) irreducibles in induced sign characters


5.2. Tables for type D5.C

lass

C1

C2

C3

C4

C5

C6

Rep

rese

ntat

ive

1x

1x

2x

1x

2x

3x

4(1

2)x

1x

3(1

2)x

3x

4(1

2)W

eylgr

oup

type

1D

2A

14

A1

D3

D2×A

1

Cla

ssC

7C

8C

9C

10

C11

C12

Rep

rese

ntat

ive

x1x

3x

4x

5(1

2)(1

23)

x1x

4(1

23)

x4x

5(1

23)

(123

4)x

1x

5(1

234)

Wey

lgr

oup

type

D3×D

2A

2D

4D

2×A

2A

3D

5

Cla

ssC

13

C14

C15

C16

C17

C18

Rep

rese

ntat

ive

(123

45)

(12)

(34)

x1x

3(1

2)(3

4)x

1x

5(1

2)(3

4)(1

2)(3

45)

x1x

3(1

2)(3

45)

Wey

lgr

oup

type

A4

A2 1

D4(a

1)

D3×A

1A

1×A

2D

5(a

1)

(ψi,ϕ

j)

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

ϕ10

ϕ11

ϕ12

ϕ13

ϕ14

ϕ15

ϕ16

ϕ17

ϕ18

ψ1

10

10

00

01

00

00

00

00

00

ψ2

01

01

00

00

10

00

00

00

00

ψ3

00

00

11

00

01

00

00

00

00

ψ4

00

10

10

10

00

10

00

00

00

ψ5

00

01

01

10

00

01

00

00

00

ψ6

00

00

00

00

00

00

10

00

01

ψ7

00

00

00

00

00

00

00

01

10

ψ8

00

00

00

00

00

00

10

00

01

ψ9

00

00

00

00

00

00

00

01

10

ψ10

00

00

00

01

00

10

11

00

00

ψ11

00

00

00

00

10

01

00

11

00

ψ12

00

00

00

00

00

00

01

10

11

ψ13

00

00

00

00

01

11

00

00

11

Table(5) Table(6)

Table (5) describes the conjugacy classes of W (D5), and Table (6)gives the inner products (ψi, ϕj |W (D4) ), where the φi are the irre-

ducible characters of W (D5) and the ψi are the irreducible charactersof W (D4).

5. TABLES FOR TYPES D4 AND D5 69

clas

sC

1C

2C

3C

4C

5C

6C

7C

8C

9C

10

C11

C12

C13

C14

C15

C16

C17

C18

size

110

520

6060

2080

160

8024

024

038

460

6012

016

016

0ϕ

11

11

11

11

11

11

11

11

11

1ϕ

21

11

-1-1

-1-1

11

1-1

-11

11

1-1

-1ϕ

34

44

22

22

11

10

0-1

00

0-1

-1ϕ

44

44

-2-2

-2-2

11

10

0-1

00

01

1ϕ

55

55

11

11

-1-1

-1-1

-10

11

11

1ϕ

65

55

-1-1

-1-1

-1-1

-11

10

11

1-1

-1ϕ

76

66

00

00

00

00

01

-2-2

-20

0ϕ

85

1-3

31

-1-3

20

-21

-10

11

-10

0ϕ

95

1-3

-3-1

13

20

-2-1

10

11

-10

0ϕ

10

102

-60

00

0-2

02

00

02

2-2

00

ϕ11

153

-93

1-1

-30

00

-11

0-1

-11

00

ϕ12

153

-9-3

-11

30

00

1-1

0-1

-11

00

ϕ13

10-2

24

-20

21

-11

00

02

-20

1-1

ϕ14

10-2

22

0-2

41

-11

00

0-2

20

-11

ϕ15

10-2

2-2

02

-41

-11

00

0-2

20

1-1

ϕ16

10-2

2-4

20

-21

-11

00

02

-20

-11

ϕ17

20-4

4-2

2-2

2-1

1-1

00

00

00

1-1

ϕ18

20-4

42

-22

-2-1

1-1

00

00

00

-11

Table(7): Character table of W (D5)


C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

C15

C16

C17

C18

(εD

4)W

1060

10-1

20-2

40-1

200

320

320

0-4

800

012

012

00

00

(εA

4)W

160

0-1

600

00

320

00

-480

038

40

0-3

200

0(ε

A1×A

3)W

4012

00

-280

-240

-120

032

00

00

00

240

0-2

40-3

200

ε A2 1×A

2

W80

800

-40

00

-240

016

00

-160

00

048

00

0-3

20(ε

A1×A

2)W

160

00

-640

00

032

00

00

00

480

00

-320

0(ε

A3 1)W

240

240

0-7

200

-240

00

00

00

048

00

00

0(ε

D3)W

8024

00

-480

-480

00

640

00

00

00

00

00

(εA

3)W

800

0-4

800

00

640

00

-480

00

240

00

00

(εA

2)W

320

00

-960

00

064

00

00

00

00

00

0(ε

D2)W

480

480

0-9

600

00

00

00

00

00

00

0(ε

A2 1)W

480

00

-960

00

00

00

00

048

00

00

0(ε

A1)W

960

00

-960

00

00

00

00

00

00

00

ϕ1

ϕ2

ϕ3

ϕ4

ϕ5

ϕ6

ϕ7

ϕ8

ϕ9

ϕ10

ϕ11

ϕ12

ϕ13

ϕ14

ϕ15

ϕ16

ϕ17

ϕ18

(εD

4)W

01

01

00

00

10

00

00

00

00

(εA

4)W

01

00

00

00

10

00

00

01

00

(εA

1×A

3)W

01

01

01

00

10

01

00

01

00

(εA

2 1×A

2)W

01

01

01

00

11

01

00

02

10

(εA

1×A

2)W

01

01

01

00

21

02

00

13

21

(εA

3 1)W

01

02

12

10

22

13

00

13

31

(εD

3)W

01

02

01

10

20

02

00

11

00

(εA

3)W

01

01

00

00

20

01

00

12

10

ε A2W

01

02

01

10

31

14

01

34

42

(εD

2)W

01

13

23

30

33

36

01

34

53

(εA

2 1)W

01

02

12

10

33

25

11

35

64

(εA

1)W

01

13

23

31

45

69

34

67

119

Table(9)Induced sign characters

Table(8)Inner products with irreducibles

CHAPTER 5

W-graphs for types E6 and E7

In this chapter we describe the procedure used to construct W-graphs for the irreducible representations of W (E6) and W (E7). Forthe purposes of machine calculation it was convenient to number thesimple reflections of E7 in such a way that numbers 1 to 5 generate D5

and numbers 1 to 6 generate E6. The numbering that was chosen isshown in the diagram below.

5 4 2 3 6 7

1

Note that the generators r, s, t, u, v of D5, in the notation of Chap-ter 4, are 1, 3, 2, 4, 5 (respectively) in this notation.

1. Using Magma to induce W-graphs

In the terminology of Magma, each W-graph was represented as arecord with four fields: basering, J, I and edges. Here basering is thering in which the edge weights must lie, and can always be the ring ofintegers. The field J is the set of integers from 1 to n, where n is therank of the Coxeter group in question: that is, n is 5, 6 or 7 for D5, E6

or E7 (repectively). The field I is a sequence of subsets of J, giving thedescent sets of the vertices of the graph. Thus, the number of terms ofthe sequence is the number of vertices, the i-th term of the sequencegives the descent set of vertex number i. The field edges is a sequenceof sets of pairs: the i-th term of this sequence consists of all pairs [j,m]such that {i, j} is an edge of the W-graph, and the edge weight µ(i, j)is m.

For example, the W-graph of the exterior square of the reflectionrepresentation of E6 is stored as a file with the following lines.

J:={1,2,3,4,5,6};

graphinfo:=recformat<J,I,edges,basering>;

gammaedges:=[{[2,1],[3,1]},

{[1,1],[5,1],[6,1]},

{[1,1],[4,1],[7,1]},

{[3,1],[8,1]},

71

72 5. W-GRAPHS FOR TYPES E6 AND E7

{[9,1],[2,1]},

{[2,1],[10,-1],[9,1]},

{[3,1],[8,1],[10,1]},

{[4,1],[7,1],[11,1],[13,1]},

{[6,1],[5,1],[12,1],[14,1]},

{[7,1],[6,-1],[11,1],[14,-1]},

{[8,1],[10,1],[15,-1]},

{[9,1]},

{[8,1]},

{[9,1],[10,-1],[15,1]},

{[14,1],[11,-1]}];

IGamma:=

[

{ 1, 2 },

{ 1, 3 },

{ 1, 4 },

{ 1, 5 },

{ 1, 6 },

{ 2, 3 },

{ 2, 4 },

{ 2, 5 },

{ 2, 6 },

{ 3, 4 },

{ 3, 5 },

{ 3, 6 },

{ 4, 5 },

{ 4, 6 },

{ 5, 6 }];

wg:=rec<graphinfo|basering:=Integers(),J:=J,

I:=IGamma,edges:=gammaedges>;

The CD has such a file for each of the irreducible characters of D5,E6 and E7. For example, the file given above is named “e6rep15a.m”,and the others are named similarly. Each file includes a comment linethat specifies the corresponding character according to the terminologyused in the character tables appearing in this thesis.

The first task was to apply the algorithm of Chapter 2 to com-pute W (E6)-graphs induced from each of the irreducible W (D5)-graphsgiven in Chapter 4. The following sequence of Magma commands willinduce d5rep10a from D5 to E6.

load "reftable.m";

action:= erootAction(e6);

1. USING MAGMA TO INDUCE W-GRAPHS 73

load "induce.m";

load "d5/d5rep10a.m";

ind:= induceWGraph(wg,action,[1,2,3,4,5]);

The program “reftable.m” used here was written by R. B. Howlettfor computations with arbitrary (including infinite) Coxeter groups offinite rank. Since in this application we are only concerned with finiteCoxeter groups, standard Magma functions could be used instead. Thefunction erootAction defined in reftable.m takes as input a sequencegiving the below-diagonal entries of a Coxeter matrix. The sequencee6 in the above example is

[1,3,1,2,3,1,2,3,2,1,2,2,2,3,1,2,2,3,2,2,1]

(where the first term is the first entry in the first row of the Coxetermatrix for E6, the next two terms are the first two entries in the secondrow, and so on). The output of erootAction is a sequence of sequences,one for each simple reflection. For finite Coxeter groups, as here, the se-quence corresponding to the simple reflection s is [i1, i2, . . . , iN ], whereN is the number of positive roots, and applying s to the j-th positiveroot produces the ij-th positive root, except when the j-th positiveroot is the simple root corresponding to s. In this case ij is set equal to−j. Thus the output of erootAction is essentially just the permutationaction of the simple reflections on the root system. In the exampleabove, the variable action takes the following value:

[-1,7,3,4,5,6,2,12,13,10,11,8,9,17,18,19,14,15,16,23,24,22,20,21,28,26,27,25,29,30,31,32,33,34,36,35],[7,-2,8,9,5,6,1,3,4,16,15,12,13,14,11,10,22,18,19,20,21,17,26,27,25,23,24,30,29,28,31,32,33,35,34,36],[1,8,-3,4,5,11,12,2,14,10,6,7,17,9,15,20,13,18,23,16,21,22,19,24,25,26,31,28,29,33,27,34,30,32,35,36],[1,9,3,-4,10,6,13,14,2,5,11,17,7,8,21,16,12,24,19,20,15,22,23,18,25,29,27,28,26,32,31,30,34,33,35,36],[1,2,3,10,-5,6,7,8,16,4,11,12,19,20,15,9,23,18,13,14,25,26,17,28,21,22,30,24,29,27,33,32,31,34,35,36],[1,2,11,4,5,-6,7,15,9,10,3,18,13,21,8,16,24,12,19,25,14,27,28,17,20,30,22,23,32,26,31,29,33,34,35,36].

The function induceWGraph defined in the file “induce.m” requiresthree arguments. The second of these is a sequence of sequences likeaction above, describing the action of the simple reflections of W onthe root system. The first argument is a record describing a WJ -graph,where J is some subset of the set of simple relections. The third ar-gument is a sequence of length equal to the rank of WJ describingthe embedding of J in S. Thus if this argument is [1, 2, 3, 4, 5] (as


above), the first simple reflection of WJ is identified with the first sim-ple relection of W , the second with the second, and so on. Given ourconventions for numbering the simple roots of D5 and E6, we coulduse [3, 2, 1, 4, 5], [1, 2, 4, 3, 6] or [4, 2, 1, 3, 6] instead of [1, 2, 3, 4, 5] (butthere is no point in doing so). For inducing from E6 to E7 the thirdargument of induceWGraph should be [1, 2, 3, 4, 5, 6] or [1, 2, 4, 3, 6, 5].

The output of induceWGraph is a record describing the W -graphIndS

J (Γ), where Γ is the WJ -graph corresponding to the first argumentof induceWGraph.

2. Decomposing the induced graphs

Having computed an induced W-graph, as described in the previoussection, it is a trivial task to break it into cells, since the Magma func-tion StronglyConnectedComponents implements a suitable algorithm.Continuing the example from Section 1, after the command

ind:= induceWGraph(wg,action,[1,2,3,4,5]);

the command

cells:=getCells(ind);

returns a sequence of subsets of { i | 1 ≤ i ≤ d }, where d is the numberof vertices of the induced W-graph IndS

J (Γ) corresponding to the recordind. Integers i and j lie in the same term of cells if and only if the i-thand j-th vertices of IndS

J (Γ) lie in the same cell. The function getCells,defined in induce.m, essentially just callsStronglyConnectedComponents.In this example it turns out that there are four cells, consisting of 15,64, 81 and 110 vertices.

Next, the commands

cg1:=cellGraph(ind, cells[1]);




produce W-graph records corresponding to the cells. This is simply anexercise in discarding the vertices that are not in the relevant cell andrenumbering the ones that are left.

For all but a few of the irreducible characters of W (E6) and W (E7)there exist W-graphs occurring as cells in induced W-graphs. We nowdescribe a more general procedure for extracting submodules from W-graph modules; this procedure enabled us to find W-graphs in all theremaining cases.

Given a W-graph datum (Γ, I, µ), let M be the free Z-module withbasis Γ, and for each J ⊆ S let MJ be the free Z-module with basis

2. DECOMPOSING THE INDUCED GRAPHS 75

{ γ ∈ Γ | Iγ = J }. Define φJK : MK → MJ by

φJKγ =∑δ∈ΓJ

µ(δ, γ)δ

for each γ ∈ ΓK . Observe that if v ∈MK and s /∈ K then

(Ts − q)v =∑

{J |s∈J}φJKv,

and it follows that if (NJ)J⊆S is a family of Z-modules with NJ ⊆MJ

for each J and φJKNK ⊆ NJ for all J and K, then ⊕J⊆SANJ is an� -submodule of the � -module AΓ.

In the context of our Magma programs, the transformations φJK

above correspond to matrices. The function edgeMatrices, defined insubmod.m, computes them.

load "submod.m";

ems:=edgeMatrices(cg4);

The output of edgeMatrices is a record with fields named gensets, sizesand mats. Here gensets is ssimply a list of all the descent sets thatoccur in the W-graph in question, and sizes is a list of the same lengthas gensets, giving the numbers of vertices with the various descent sets.(Thus the sum of the terms of the sequence sizes is the dimension ofthe � -module.) The field mats is also a list of the same length asgensets. For each i, mats[i] is a list of pairs consisting of an integer jand a sizes[i]×sizes[j] matrix, which is the matrix corresponding to thetransformation φJK , where J is gensets[i] and K is gensets[j]. All thisreally is just another way of presenting the W-graph.

Choose a subset J of S that occurs as a descent set of some ver-tex of Γ, and choose an element v of the Z-module MJ . Initially, letNJ = Zv ⊆ MJ and let NL = {0} for all L �= J . Whenever L � J ,apply the transformation φLJ to v, and redefine NL to be the Z-modulegenerated by NL and φLJv. In other words, if φLJv ∈ NL then NL isleft unchanged, otherwise φLJv is appended to the list of generatorsof NL, producing a larger module. The process is repeated with v re-placed by each newly found generator of each NL, until no more newgenerators are found.

The function submod, defined in submod.m, does this.In the example above there are 40 descent sets. The command

print ems`sizes;

returns the sequence


[6,4,2,5,1,5,5,4,3,1,3,5,4,4,4,3,6,3,1,2,2,2,1,2,2,2,2,1,2,3,3,4,1,2,1,3,1,3,1,1]

and the command

print ems`gensets[40];

returns {1, 3}. We see that there is just one vertex with descentset {1, 3}. Now, for example, we could take v = γ ∈ Γ{1,3}, and ap-ply the function submod to produce a submodule of AΓ. The requiredcommand is this:

xx:= submod([1],40,ems);

Here the third argument to submod is the W-graph, presented, as de-scribed above, as a record with fields gensets, sizes and mats. Theoutput of submod will be a W-graph for a submodule, presented in thesame way. The second argument to submod is a positive integer k lessthan or equal to the length of of the sizes field of the third argument,and the first argument is an integer sequence of length sizes[k]. Thissequence should be regarded as a vector in MJ , where J is gensets[k].

In our example, the variable xx corresponds to a W-graph for theirreducible 90-dimensional representation of E6. The command

nwg:= emsToWG(xx);

converts it to to a field with records basering, J, I and edges as previouslydescribed.

The file induce.m also defines functions for producing and checkingmatrix representations of Hecke algebras and Coxeter groups, given aW-graph. For example, the command

repH:= heckeAlgRep(nwg);

creates six 90×90 matrices, one for each of the generators of the Heckealgebra of type E6. They can be printed via commands such as

print repH[1];

and so on. You can also tell Magma to check that the six matrices inquestion satisfy the 21 defining relations of the Hecke algebra via thecommand

test("e6",repH);

Magma will print “true” after checking each relation.You can also produce a representation of the Coxeter group rather

than the Hecke algebra: the command

repW:= groupRep(nwg);

3. TABLES FOR TYPE E6 77

will do this. The following sequence of Magma commands could beused to confirm that repW really is an irreducible representation of theWeyl group of type E6.

load "e6/e6rep6a.m";

refrep:= groupRep(wg);

print refrep;

we6:= sub< GL(6,Integers()) | refrep >;

e6ccl:= Classes(we6);

e6ctable:= CharacterTable(we6);

phi:= hom< we6 -> GL(90,Integers()) | repW >

chi:= [Trace(phi(e6ccl[i,3])) : i in [1..25]];

print chi;

print Index(e6ctable,chi);

It will be found that chi coincides with the 25th irreducible characterof the Weyl group, according to Magma’s character table calculation.

3. Tables for type E6

The Magma function induceWGraph mentioned in the previous sec-tion implements the W-graph induction algorithm described in Chap-ter 2. As a first step it computes the minimal coset representativesfor WJ in W and the action of the simple reflections on these. Forexample, there are 27 (= #W (E6)/#W (D5) = 51840

1920) cosets of W (D5)

in W (E6), and the 27 minimal coset representatives can be identifiedwith sequences representing reduced words in the generators. (Of allpossible reduced words for an element of W , we always use the onethat is first in the lexicographic order.) In fact these sequences are asfollows

[],

[ 6 ],

[ 6, 3 ],

[ 6, 3, 2 ],

[ 6, 3, 2, 1 ],

[ 6, 3, 2, 4 ],

[ 6, 3, 2, 1, 4 ],

[ 6, 3, 2, 4, 5 ],

[ 6, 3, 2, 1, 4, 2 ],

[ 6, 3, 2, 1, 4, 5 ],

[ 6, 3, 2, 1, 4, 2, 3 ],

[ 6, 3, 2, 1, 4, 2, 5 ],

[ 6, 3, 2, 1, 4, 2, 3, 5 ],

[ 6, 3, 2, 1, 4, 2, 3, 6 ],


[ 6, 3, 2, 1, 4, 2, 5, 4 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 6 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 6 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 1 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 6 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 1, 6 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 6, 3 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 1, 6, 3 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 1, 6, 3, 2 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 1, 6, 3, 2, 4 ],

[ 6, 3, 2, 1, 4, 2, 3, 5, 4, 2, 1, 6, 3, 2, 4, 5 ]

(where W (D5) is identified with the subgroup of W (E6) generated bythe first five simple reflections).

For programming convenience we used right cosets rather than leftcosets, despite the fact that the description presented in Chapter 2uses left cosets. The program computes a table that describes whathappens when the minimal coset represntatives are multiplied by simplereflections. For D5 in E6 the table is as follows.

[-1,-1,-1,5,4,7,6,10,-4,8,-4,-4,-4,-4,-4,

-4,-4,20,-4,18,22,21,24,23,-3,-3,-3],

[-2,-2,4,3,-1,-4,9,-4,7,12,-2,10,-2,-2,-5,

18,-2,16,21,-4,19,-4,-3,25,24,-2,-2],

[-3,3,2,-2,-2,-2,-2,-2,11,-2,9,13,12,-3,16,

15,-3,-5,-3,-5,23,24,21,22,-4,-4,-4],

[-4,-4,-4,6,7,4,5,-5,-1,-5,-1,15,16,-1,12,

13,19,-2,17,-2,-2,-2,-2,-2,26,25,-1],

[-5,-5,-5,-5,-5,8,10,6,12,7,13,9,11,17,-1,

-1,14,-1,-1,-1,-1,-1,-1,-1,-1,27,26],

[2,1,-3,-3,-3,-3,-3,-3,-3,-3,14,-3,17,11,-3,

19,13,21,16,22,18,20,-5,-5,-5,-5,-5]

For example, the fifth sequence in this list describes the effect ofappending a 5 to each of the 27 minimal coset representative sequenceslisted above, and then finding the coset representative correspondingto coset containing this word. Naming the coset representatives d1 tod27, the i-th term of the fifth sequence says what happens when di ismultiplied by s = s5. For example, the 6th term of the sequence is 8,telling us that d6s = d8. Naturally, this also means that d8s = d6, andso the 8th term of the sequence should be 6, which it is. The valuesof i for which dis is not a minimal coset representative correspond to


the terms of the sequence that are negative. In these cases we musthave dis = sjdi for some j, and the corresponding term of the sequenceis −j. Thus, for example, we have dis5 = s1di when i is 15, 16, 18, 19,20, 21, 22, 23, 24 or 25.

With this information stored, it is in principle straightforward toapply the inductive formulas given in Chapter 2 to compute the gener-alized Kazhdan-Lusztig polynomials and hence the induced W-graph.

In the remainder of this section we give various tables relating tothe characters and conjugacy classes of the Weyl group of type E6. Thetables are taken from the paper of J. S. Frame [12]. The information inthe tables can be used to help verify that the W-graphs for E6 producedby our Magma calculations are indeed correct.

Note that we have followed Frame’s convention for the charactertables of E6 and E7, writing the characters as columns rather thanrows. Irreducible characters are given names such as da, db, etc., whered is the degree, and the subscripts distinguish different characters of thesame degree. For example, the three irreducible characters of W (E6)of degree 60 are called 60p, 60n and 60s. The subscripts p, q are usedfor characters that take positive values on the class of reflections, thesubscripts n, m are used for those that take negative values on theclass of reflections, while the subscript s is used for self-dual characters(vanishing on the class of reflections).

Class C1 C2 C3 C4 C5 C6 C7 C8 C9

type 1 A21 A4

1 D4(a1) A1 × A3 A2 D4 A21 × A2 A2

2

Size 1 270 45 540 3240 240 1440 2160 480

Class C10 C11 C12 C13 C14 C15 C16 C17 C18

type A1 × A5 A32 E6(a2) E6 E6(a1) A4 A1 A3

1 A3

Size 1440 80 720 4320 5760 5184 36 540 1620

Class C19 C20 C21 C22 C23 C24 C25

type A21 × A3 D5 A1 × A2 A1 × A2

2 A5 D5(a1) A1 × A4

Size 540 6480 1440 1440 4320 4320 5184

Table(1): The types and the sizes of the 25 conjugacy classes of W (E6)


[W:P

]C

1C

2C

3C

4C

5C

6C

7C

8C

9C

10

C11

C12

C13

C14

C15

C16

C17

C18

C19

C20

C21

C22

C23

C24

C25

ε D5

27

27

73

31

93

10

00

00

02

-15

-3-5

-1-1

-30

0-1

0

ε A5

72

72

12

00

212

00

60

00

00

2-3

0-2

-40

0-6

0-2

00

ε A4∗A

1216

216

20

00

218

02

00

00

00

1-6

6-6

-40

0-6

00

0-1

ε A1∗A

2 2720

720

40

00

012

04

60

00

00

0-1

40

-12

00

0-8

-20

00

ε A4

432

432

24

00

036

00

00

00

00

2-1

20

0-8

00

-60

00

0

ε D4

270

270

66

60

36

60

00

00

00

0-9

0-6

-12

00

00

00

0

ε A3∗A

11080

1080

36

00

236

00

00

00

00

0-2

10

-6-4

00

-60

00

0

ε A2 2

1440

1440

48

00

024

00

12

00

00

00

-240

00

00

-12

00

00

ε A2∗A

2 12160

2160

56

00

018

02

00

00

00

0-3

00

-12

00

06

00

00

ε A3

2160

2160

24

00

072

00

00

00

00

0-3

60

0-8

00

00

00

0

ε A2∗A

14320

4320

48

00

036

00

00

00

00

0-4

80

00

00

-60

00

0

ε A3 1

6480

6480

72

00

00

00

00

00

00

0-5

40

-12

00

00

00

00

ε A2

8640

8640

00

00

72

00

00

00

00

0-7

20

00

00

00

00

0

ε A2 1

12960

12960

48

00

00

00

00

00

00

0-7

20

00

00

00

00

0

ε A1

25920

25920

00

00

00

00

00

00

00

-720

00

00

00

00

0

1p

6p

15

p20

p30

p64

p81

p15

q24

p60

p20

s90

s80

s60

s10

s1

n6

n15

n20

n30

n64

n81

n15

m24

n60

n

ε D5

00

01

00

00

00

00

00

01

10

10

00

00

0

ε A5

00

00

00

00

00

00

00

01

10

11

00

10

0

ε A4∗A

10

00

10

10

01

00

00

00

11

02

11

00

01

ε A1∗A

2 20

00

00

00

00

00

01

11

11

03

22

12

13

ε A4

00

00

00

01

01

00

01

01

20

32

21

10

1

ε D4

00

00

00

00

00

00

00

01

21

31

20

01

0

ε A3∗A

10

10

00

00

00

00

11

10

12

14

34

22

13

ε A2 2

00

00

00

00

00

01

12

11

21

44

43

31

4

ε A2∗A

2 10

00

00

01

00

10

23

31

12

15

46

43

26

ε A3

00

00

00

10

00

13

21

01

33

65

85

22

4

ε A2∗A

10

00

00

13

01

21

66

51

13

37

711

94

39

ε A3 1

00

00

12

61

15

19

10

92

13

38

814

12

55

13

ε A2

00

00

14

90

24

415

12

81

14

610

11

20

18

56

14

ε A2 1

00

11

13

815

24

11

421

20

16

31

16

11

13

24

24

78

21

ε A1

01

55

10

24

36

510

25

10

45

40

30

51

510

15

20

40

45

10

14

35

Table(2): Induced sign characters Table(3): Inner products with theirreducible characters


1p

6p

15

p20

p30

p64

p81

p15

q24

p60

p20

s90

s80

s60

s10

s1

n6

n15

n20

n30

n64

n81

n15

m24

n60

n

ϕ1

11

01

00

00

00

00

00

00

00

00

00

00

0

ϕ2

00

00

00

00

00

00

00

01

10

10

00

00

0

ϕ3

00

01

01

00

10

00

00

00

00

00

00

00

0

ϕ4

00

00

00

00

00

00

00

00

00

10

10

01

0

ϕ5

00

00

00

01

01

00

01

00

00

00

00

00

0

ϕ6

00

00

00

00

00

00

01

00

00

00

00

10

1

ϕ7

00

00

00

10

00

00

00

00

00

00

01

00

0

ϕ8

01

11

11

00

00

00

00

00

00

00

10

01

0

ϕ9

00

00

00

00

00

00

00

00

11

11

10

00

0

ϕ10

00

00

00

00

01

00

11

10

00

00

00

00

1

ϕ11

00

00

11

10

01

01

10

00

00

00

00

00

0

ϕ12

00

00

00

00

00

01

10

00

00

01

11

00

1

ϕ13

00

01

11

11

01

00

00

00

00

00

00

00

0

ϕ14

00

10

01

10

00

11

00

00

00

00

00

00

0

ϕ15

00

00

00

00

00

11

00

00

01

00

11

00

0

ϕ16

00

00

00

00

00

00

00

00

00

11

11

10

1

ϕ17

00

00

00

10

00

01

11

00

00

00

11

01

1

ϕ18

00

00

01

10

11

01

11

00

00

00

01

00

0

Table (4): Inner products (ϕ,χ|W (D5)) whereϕ ∈ Irr(D5) and χ ∈ Irr(E6)


1p

6p

15

p20

p30

p64

p81

p15

q24

p60

p20

s90

s80

s60

s10

s1

n6

n15

n20

n30

n64

n81

n15

m24

n60

n

C1

16

15

20

30

64

81

15

24

60

20

90

80

60

10

16

15

20

30

64

81

15

24

60

C2

12

-14

20

-33

04

-4-6

04

21

2-1

42

0-3

30

4

C3

1-2

-14

-10

09

78

-44

-6-1

612

-61

-2-1

4-1

00

97

8-4

C4

12

30

-20

-3-1

00

42

04

21

23

0-2

0-3

-10

0

C5

10

-10

00

-11

00

02

00

-21

0-1

00

0-1

10

0

C6

13

35

34

00

0-3

20

-4-6

-21

33

53

40

00

-3

C7

11

-11

-10

0-2

2-1

-20

20

01

1-1

1-1

00

-22

-1

C8

1-1

-11

-10

00

01

20

0-2

21

-1-1

1-1

00

00

1

C9

10

0-1

3-2

03

3-3

20

20

41

00

-13

-20

33

-3

C10

1-2

21

-10

01

-1-1

-20

20

01

-22

1-1

00

1-1

-1

C11

1-3

62

3-8

0-3

66

-79

-10

-31

1-3

62

3-8

0-3

66

C12

11

2-2

-10

01

22

1-3

2-3

-31

12

-2-1

00

12

2

C13

1-1

00

10

0-1

00

1-1

01

-11

-10

01

00

-10

0

C14

10

0-1

01

00

00

-10

-10

11

00

-10

10

00

0

C15

11

00

0-1

10

-10

00

00

01

10

00

-11

0-1

0

C16

14

510

10

16

95

410

00

00

0-1

-4-5

-10

-10

-16

-9-5

-4-1

0

C17

10

-32

-20

-31

42

00

00

0-1

03

-22

03

-1-4

-2

C18

12

12

00

-1-1

0-2

00

00

0-1

-2-1

-20

01

10

2

C19

1-2

12

-40

33

0-2

00

00

0-1

2-1

-24

0-3

-30

2

C20

10

-10

00

1-1

00

00

00

0-1

01

00

0-1

10

0

C21

11

-11

1-2

02

-21

00

00

0-1

-11

-1-1

20

-22

-1

C22

1-2

21

1-2

0-1

11

00

00

0-1

2-2

-1-1

20

1-1

-1

C23

10

0-1

10

01

1-1

00

00

0-1

00

1-1

00

-1-1

1

C24

11

1-1

-10

00

01

00

00

0-1

-1-1

11

00

00

-1

C25

1-1

00

01

-10

-10

00

00

0-1

10

00

-11

01

0

Table(5): Character Table of Group W (E6) [12]

4. THE CONSTRUCTION OF IRREDUCIBLE W-GRAPHS FOR TYPE E6 83

4. The construction of irreducible W-graphs for type E6

4.1. Cells in induced W-graphs. As remarked above, in mostcases a W-graph corresponding to a given irreducible character of E6

can be found as a cell in some W-graph induced from D5. The tablebelow describes exactly what happens in each case. For each irreduciblecharacter ϕi of W (D5) let Ind(ϕi) be the W-graph for E6 induced fromthe W-graph corresponding to ϕi given in Chapter 4. Bracketed sumsbelow correspond to cells. Thus, for example, inducing ϕ10 producesa graph with 3 cells, containing 60, 150 and 60 vertices (respectively).The two cells of size 60 correspond to the irreducible characters 60p and60n, while the cell of size 150 corresponds to the reducible characterwith constituents 80s, 60s and 10s.

Ind(ϕ1) = 20p + 6p + 1p;Ind(ϕ2) = 1n + 6n + 20n;Ind(ϕ3) = 20p + 64p + 24p;Ind(ϕ4) = 20n + 64n + 24n;Ind(ϕ5) = 60s + 60p + 15q;Ind(ϕ6) = 15m + 60s + 60n;Ind(ϕ7) = 81p + 81n;Ind(ϕ8) = 6p + (30p + 15p) + 20p + 64p

Ind(ϕ9) = 6n + (15n + 30n) + 20n + 64n;Ind(ϕ10) = 60p + (80s + 60s + 10s) + 60n;Ind(ϕ11) = 30p + 64p + 81p + 60p + (90s + 80s);Ind(ϕ12) = (90s + 80s) + 30n + 64n + 81n + 60n;Ind(ϕ13) = 20p + (30p + 15q) + 64p + 81p + 60p;Ind(ϕ14) = 15p + 64p + 81p + (20s + 90s);Ind(ϕ15) = (20s + 90s) + 15n + 64n + 81n;Ind(ϕ16) = 20n + (30n + 15m) + 64n + 81n + 60n;Ind(ϕ17) = 81p + (90s + 80s + 60s) + 64n + 81n + 24n + 60n;Ind(ϕ18) = 64p + 81p + 24p + 60p + (90s + 80s + 60s) + 81q.

So the following statements hold:

1p is a cell in Ind(ϕ1); 1n is a cell in Ind(ϕ2);6p is a cell in Ind(ϕ1); 6n is a cell in Ind(ϕ2);15p is a cell in Ind(ϕ14); 15n is a cell in Ind(ϕ15);15q is a cell in Ind(ϕ5); 15m is a cell in Ind(ϕ6);20p is a cell in Ind(ϕ3); 20n is a cell in Ind(ϕ4);30p is a cell in Ind(ϕ11); 30n is a cell in Ind(ϕ12);64p is a cell in Ind(ϕ14); 64n is a cell in Ind(ϕ15);81p is a cell in Ind(ϕ7); 81n is a cell in Ind(ϕ7);24p is a cell in Ind(ϕ3); 24n is a cell in Ind(ϕ4);


60p is a cell in Ind(ϕ3); 60n is a cell in Ind(ϕ3);60s is a cell in Ind(ϕ6).

Thus we may construct W-graphs for all the irreducible characters ex-cept 90s, 80s, 20s and 10s. Of course, some of the others can be foundin several different ways. For example, the W-graph corresponding tocharacter 20p can be obtained as a cell in any one of the following:Ind(ϕ1), Ind(ϕ8), Ind(ϕ13).

4.2. Special cases. For the remaining four self dual characterswe can obtain W-graphs by using the method described in Section 2above.

10s. The 10 dimensional representation (corresponding to the char-acter 10s) of E6 can be constructed easily enough by hand. It is anextension of one of the 10 dimensional irreducibles ofD5 (correspondingto the character ϕ10). The E6 W-graph is the same as the D5 W-graphwith the 6th generator of E6 added to some of the descent sets. It canalso be extracted from a cell of degree 150 in the representation of E6

induced from ϕ10. In this cell there are 3 vertices with descent set {2}.Calling the corresponding basis vectors x, y, z (labelled appropriately)the vector x+ y− z generates the 10 dimensional submodule (this wasfound by trial and error). The following sequence of Magma commandscould be used.

load "reftable.m";

action:= erootAction(e6);

load "d5/d5rep10e.m";

load "induce.m";

ind:=induceWGraph(wg,action,[1..5]);

cells:= getCells(ind);

cg:=cellGraph(ind,cells[2]);

load "submod.m";

ems:= edgeMatrices(cg);

print Index(ems`gensets,2);

print ems`sizes[13];

xx:=submod([1,1,-1],13,ems);

swg:= emsToWG(xx);

print swg;

90s. The 90 dimensional representation was constructed as follows.The W-graph corresponding to the character ϕ15 of W (D5) was inducedto E6 and the cells were found. There is a cell of degree 110, having asingle vertex with descent set {3, 5}. The submodule generated by thecorresponding basis vector gave the 90 dimensional submodule.


80s. The 80 dimensional representation was constructed as follows.The W-graph corresponding to the character ϕ10 of D5 was induced toE6 and the cells were found. There is a cell of degree 150 having asingle vertex with descent set {3, 5, 6}. The submodule generated bythe corresponding vector gave the 80 dimensional irreducible.

20s. The 20 dimensional representation corresponding to character20s can be constructed easily by hand, since it corresponds to theexterior cube of the reflection representation.

In the Coxeter diagram of E6, we denote the six basis vectors ofthe reflection module V as v1, . . . , v6. The exterior cube

∧3 V hasdimension 20; it has a basis consisting of all elements vijk = vi∧vj ∧vk,where 1 ≤ i < j < k ≤ 6. The vijk can be identified with the verticesof a W-graph, the descent set of vijk being {i, j, k}. The action of thegenerators on each such vector gives the corresponding edges. Now, forexample, since

s5v1 = v1

s5v3 = v3

s5v4 = v4 + v5

we find that

s5(v134) = v1 ∧ v3 ∧ (v4 + v5) = v1 ∧ v3 ∧ v4 + v1 ∧ v3 ∧ v5 = v134 + v135.

We conclude that there should be an edge joining v134 and v135 withµ(v135, v134) = 1. The other edges can be found similarly.

Using Magma programs, the same W-graph can also be extractedfrom the cell of degree 110 in Ind(ϕ15). There are 3 vertices withdescent set {2, 4, 5}, and if the corresponding basis vectors are denotedby x, y and z (in an appropriate order) then x − y + z generates the20 dimensional submodule.

5. The construction of irreducible W-graphs for type E7

The group W (E7) is the direct product of its derived group anda group or order 2. The character table of the derived group wascalculated by Frame [12]. These characters can be identified with thecharacters of W (E7) whose kernels contain the longest element wS.Following Frame’s notation, each of these character is given a name ofthe form da, db or dc, where d is the degree. (For example, the two ofthe degree 21a and 21b). Extending Frame’s notation in a natural way,we give the dual characters (whose kernels do not contain wS) namesof the form dx, dy or dz (with dx dual to da, etc).

Similarly to the case of E6 above, we have the following decompo-sitions of the W-graphs induced from the irreducibles of E6:


Ind(1p) = 21y + 27a + 7x + 1a;Ind(1n) = 1x + 7a + 27x + 21y;Ind(6p) = (105x + 120a) + (21a + 56x) + 27a + 7x;Ind(6n) = (105a + 120x) + (21x + 56a) + 27x + 7a

Ind(10s) = 70a + 210b + 210y + 70x;Ind(15p) = 189a + (280x + 35x) + 210a + 105x + 21a;Ind(15n) = 189x + (280a + 35a) + 210x + 105a + 21x;Ind(15q) = 216x + 280b + 105b + 189y + 15x + 35b;Ind(15m) = 216a + 280y + 105y + 189b + 15a + 35y;Ind(20p) = 189z + 168a + 210a + 189y + (120a + 105x) + 21y

+ (35b + 56x) + 27a;Ind(20n) = 189c + 168x + 210x + 189b + (120x + 105a) + 21b

+ (35y + 56a) + 27x;Ind(20s) = 189x + 35a + 336a + 336x + 189a + 35x;Ind(24p) = 105z + (420a + 84a) + 378x + 189z + 168a;Ind(24n) = 105c + (420x + 84x) + 378a + 189c + 168x;Ind(30p) = 189y + 105b + 405a + 120a + 210x + (315x + 280x) + 56x;Ind(30n) = 189b + 105y + 405x + 120x + 210x + (315a + 280a) + 56a;Ind(60p) = (70x + 280b + 315x) + (216x + 405a) + 189y + 105b + 168a

+ (512x + 512a) + 210b;Ind(60n) = (70a + 280y + 315a) + (216a + 405x) + 189b + 105y + 168x

+ (512x + 512a) + 210y;Ind(60s) = 280b + 378x + 216a + 210b + (512a + 512x) + 84x + 210y

+ 84a + 378a + 216x + 280y;Ind(64p) = (336x + 420a) + 378x + (189a + 405a) + 189y + 168a

+ (315x + 280x + 280b) + 210a + 189z + (105x + 120a);Ind(64n) = (336a + 420)x + 378a + (189x + 405x) + 189b + 168x

+ (315a + 280a + 280y) + 210x + 189c + (105a + 120x);Ind(80s) = 315a + 405a + 378x + 420a + 210b + (512a + 512x) + 210y

+ 420x + 378a + 405x + 315x;Ind(81p) = (336x + 420a) + (512a + 512x) + (336a + 420x) + 189z

+ (216x + 405a) + 105c + (280b + 280x + 315x) + 210a;Ind(81n) = (336a + 420x) + (512a + 512x) + (336x + 420a) + 189c

+ (216a + 405x) + 105z + (280y + 280a + 315a) + 210x;Ind(90s) = 280a + 189a + 405a + 378a + (336x + 420a) + 378x

+ (336a + 420x) + (512a + 512x) + 189x + 405x + 280x.

With the exception of the two characters of degree 512, every ir-reducible occurs as a cell in some induced representation. Specifically,the following statements hold.

1a is a cell of Ind(1p); 1x is a cell of Ind(1n);7a is a cell of Ind(6n) ; 7x is a cell of Ind(6p);


27a is a cell of Ind(1p); 27x is a cell of Ind(1n);21a is a cell of Ind(15p); 21x is a cell of Ind(15n);21b is a cell of Ind(20n); 21y is a cell of Ind(20p);35a is a cell of Ind(20s); 35x is a cell of Ind(20s);35b is a cell of Ind(15q); 35y is a cell of Ind(15m);105a is a cell of Ind(15n); 105x is a cell of Ind(15p);105b is a cell of Ind(15q); 105y is a cell of Ind(15m);105c is a cell of Ind(24n); 105z is a cell of Ind(24p);189a is a cell of Ind(15p); 189x is a cell of Ind(15n);189b is a cell of Ind(30n); 189y is a cell of Ind(30p);189c is a cell of Ind(24n); 189z is a cell of Ind(24p);15a is a cell of Ind(15m); 15x is a cell of Ind(15q);315a is a cell of Ind(80s); 315x is a cell of Ind(80s);405a is a cell of Ind(30p); 405x is a cell of Ind(30n);168a is a cell of Ind(20p); 168x is a cell of Ind(20n);56a is a cell of Ind(30n); 56x is a cell of Ind(30p);120a is a cell of Ind(30p); 120x is a cell of Ind(30n);210a is a cell of Ind(15p); 210x is a cell of Ind(15n);210b is a cell of Ind(60p); 210y is a cell of Ind(60n);280a is a cell of Ind(90s); 280x is a cell of Ind(90s);280b is a cell of Ind(15q); 280y is a cell of Ind(15m);336a is a cell of Ind(20s); 336x is a cell of Ind(20s);216a is a cell of Ind(15m); 216x is a cell of Ind(15q);378a is a cell of Ind(64n); 378x is a cell of Ind(64p);84a is a cell of Ind(60s); 84x is a cell of Ind(60s);420a is a cell of Ind(80s); 420x is a cell of Ind(80s);70a is a cell of Ind(10s); 70x is a cell of Ind(10s)

The 512 dimensional representations were constructed as follows.The W-graph corresponding to the character 60s of E6 was induced

to E7. There is a cell of degree 1024, having two vertices with descentset {2}. Denote the basis vectors by x and y. The submudules gen-erated by the vectors x + y and x − y gave the two 512 dimensionalsubmodules.

To conclude the chapter and this thesis, we include below sometables that help identify the irreducible W-graphs constructed via theMagma programs.


6. Tables for type E7

1a 7a 27a 21a 35a 105a 189a 21b 35b 189b 189c 15a 105b 105c 315a 405a

C1 1 7 27 21 35 105 189 21 35 189 189 15 105 105 315 405

C2 1 3 7 1 -5 5 -11 5 7 13 1 3 9 -3 3 -3

C3 1 -1 3 -3 3 1 -3 5 11 -3 21 7 -7 17 -21 -27

C4 1 3 3 5 7 5 9 1 -1 -3 -3 -1 -3 -3 -5 -3

C5 1 1 11 -1 -1 -1 1 1 1 1 -1 1 1 -1 -1 1

C6 1 4 9 6 5 15 9 6 5 9 9 0 0 0 0 0

C7 1 2 3 0 -3 1 -3 2 -1 -3 3 -2 -4 2 0 0

C8 1 0 1 -2 1 -1 1 2 1 1 1 0 0 0 0 0

C9 1 1 0 0 2 -3 0 0 2 0 0 3 3 3 0 0

C10 1 -1 0 0 0 1 0 2 2 0 0 1 -1 -1 0 0

C11 1 -2 0 3 -1 -3 0 3 -1 0 0 -3 6 6 -9 0

C12 1 2 0 3 3 1 0 -1 -1 0 0 1 2 2 3 0

C13 1 0 0 -1 1 -1 0 1 -1 0 0 -1 0 0 1 0

C14 1 1 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0

C15 1 2 2 1 0 0 -1 1 0 -1 -1 0 0 0 0 0

C16 1 -5 15 9 -5 -35 21 -11 15 -51 -39 -5 25 5 -45 45

C17 1 -1 3 -3 3 1 -3 -3 3 -3 -3 -1 1 -7 3 -3

C18 1 1 1 -1 -5 1 1 -3 1 1 -1 1 -3 -1 3 -3

C19 1 1 1 -1 -1 -1 1 -3 5 1 -5 -3 -3 3 3 -3

C20 1 -1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1

C21 1 -2 3 0 1 1 -3 -2 3 -3 3 -2 4 2 0 0

C22 1 1 0 0 -2 1 0 -2 0 0 0 1 1 -1 0 0

C23 1 -1 0 0 0 1 0 0 0 0 0 -1 1 -1 0 0

C24 1 -2 1 2 -1 -1 1 0 -1 1 1 0 0 0 0 0

C25 1 0 0 -1 0 0 1 -1 0 -1 1 0 0 0 0 0

C26 1 -1 -1 1 0 0 -1 1 0 -1 -1 0 0 0 0 0

C27 1 0 -1 0 -1 1 1 0 1 1 -1 -2 0 2 0 0

C28 1 -1 -1 1 -1 1 1 1 3 -3 1 3 -3 1 3 5

C29 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1

C30 1 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 -1

Table (6): Character table of derived group of W (E7)(Characters of odd degree)

Table (6) above gives the values of the sixteen irreducible charactersof odd degree of the derived group of W (E7); Table (7) below gives thevalues of the fourteen irreducible characters of even degree. Thesetables are taken from the paper [12] of J. S. Frame. (Frame used rowsfor classes and columns for characters.) Tables (6) and (7) togethermake up a 30 × 30 matrix A that is one quarter of the character table


of W (E7). The complete table then has the form[A AA −A

]where the thirty classes C ′

i of W (E7) that are not in the derived groupare defined by C ′

i = { gwS | g ∈ Ci }.168a 56a 120a 210a 280a 336a 216a 512a 378a 84a 420a 280b 210b 70a

C1 168 56 120 210 280 336 216 512 378 84 420 280 210 70

C2 8 8 8 2 -8 -16 8 0 2 4 -12 8 10 6

C3 8 -8 -8 2 -8 16 24 0 -6 20 4 24 -14 -10

C4 0 0 0 -2 0 0 0 0 6 4 -4 0 6 2

C5 0 0 0 -2 0 0 0 0 2 0 0 0 -2 -2

C6 6 11 15 15 10 6 -9 -16 -9 -6 0 -5 -15 -5

C7 2 1 1 -1 -2 -2 -3 0 3 2 4 -3 1 -1

C8 2 -1 -1 -1 -2 2 -1 0 -1 -2 0 -1 1 3

C9 -3 2 0 0 1 0 0 -4 0 3 3 -2 3 1

C10 1 -2 -2 2 1 -2 0 0 0 -1 1 0 1 -1

C11 6 2 -6 3 10 -6 0 8 0 3 -3 -8 -6 7

C12 2 -2 -2 -1 -2 -2 0 0 0 -1 1 0 -2 -1

C13 0 0 0 1 0 0 0 0 0 1 -1 0 0 -1

C14 0 -1 0 0 1 0 0 -1 0 0 0 1 0 1

C15 -2 1 0 0 0 1 1 2 -2 -1 0 0 0 0

C16 40 -24 40 50 -40 -16 -24 0 -30 4 20 40 10 -10

C17 8 0 0 -6 8 0 0 0 -6 4 4 0 2 -2

C18 0 -4 4 2 0 0 4 0 2 0 0 -4 -2 2

C19 0 4 -4 2 0 0 -4 0 2 0 0 4 -2 2

C20 0 0 0 0 0 0 0 0 0 0 0 0 0 0

C21 -2 -3 1 -1 2 2 -3 0 3 -2 -4 1 1 -1

C22 1 0 -2 2 -1 2 0 0 0 1 -1 -2 1 -1

C23 -1 0 0 0 -1 0 0 0 0 1 1 0 -1 1

C24 0 1 -1 -1 0 0 -1 0 -1 0 0 1 1 -1

C25 0 1 0 0 0 -1 1 0 0 -1 0 0 0 0

C26 1 1 0 0 0 1 1 -1 1 -1 0 0 0 0

C27 0 -1 1 -1 0 0 1 0 -1 0 0 -1 1 -1

C28 0 0 0 -2 0 0 0 0 -2 4 -4 0 -2 2

C29 0 0 0 0 0 0 0 0 0 0 0 0 0 0

C30 0 0 1 0 0 0 -1 1 0 0 0 0 0 0

Table (7): Character table of derived group of W (E7)(Characters of even degree)

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