generic algebras and kazhdan-lusztig theory for monomial groups

In this chapter we define induced modules and an algebra called the Hecke algebra.. In Theorem 2.2 wedescribe an important relationship between the decomposition of induced modules and t

APPROVED:

J Matthew Douglass, Major Professor Elizabeth Bator, Committee Member Douglas Brozovic, Committee Member Anne Shepler, Committee Member Nathaniel Thiem, Committee Member Neal Brand, Chair of the Department of

Mathematics

GENERIC ALGEBRAS AND KAZHDAN-LUSZTIG THEORY FOR

MONOMIAL GROUPS Shemsi I Alhaddad

Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

May 2006

UMI Number: 3214450

3214450 2006

UMI Microform Copyright

All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company

300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346

by ProQuest Information and Learning Company

Alhaddad, Shemsi I Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups Doctor of Philosophy (Mathematics), May 2006, 49 pp., references, 6 titles

The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups An important tool is the combinatorial

approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in

1979 In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the

symmetric group that is instead based on the complex reflection group G(r,1,n) Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group

I also consider possible analogues of Kazhdan-Lusztig polynomials

ACKNOWLEDGMENTS

I would like to thank my cat Joe for not eating the only extant copy of my dissertation

at the same time as he gnawed through all my library books My deepest apologies to the UNT library

I am very grateful to Matt Douglass for his constant support and encouragement, without which none of this would have been possible I am very fortunate to have had a colleague like Sam Tajima, who raised interesting questions and whose enthusiasm for

mathematics is inspiring I would also like to thank my committee members for their time and all of their wonderful advice and moral support

On a personal note, I am grateful to Helen-Marie, Joe (my brother, not the cat),

Woody, Robin and Cely for feigning interest in my work, and keeping me in touch with the world outside the math department My deepest appreciation goes to Andy, for his love and patience, and the fact that he is looking over my shoulder as I type this This thesis is

dedicated in memory of Yuba

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ii

Chapters 1 INTRODUCTION 1

2 INDUCED MODULES AND THE HECKE ALGEBRA 3

3 AN EXAMPLE OF A HECKE ALGEBRA 9

4 THE GENERIC HECKE ALGEBRA 19

5 A PARTIAL ORDER ON W H b 27

6 THE R-POLYNOMIALS 40

BIBLIOGRAPHY 49

CHAPTER 1

INTRODUCTION

In the 1800s, groups were typically regarded as subsets of the permutations of a set, or

of GL(V ) where V is a vector space In this past century, abstract groups were introduced,and group theory split into two branches: the study of abstract groups, and the study ofthe ways in which a group may be embedded in GL(V ) Representation theory entailsthe latter Representation theory is used as a conduit between abstract groups and realworld applications, including robotics, telephone network and stereo designs, and models forelementary particles In the past few decades, combinatorial representation theory has grown

to prominence Combinatorial representation theory gives group theorists explicit tools fordealing with abstract concepts This thesis takes a combinatorial approach to the study of

a specific type of Hecke algebra

The second chapter of this thesis contains background information In this chapter

we define induced modules and an algebra called the Hecke algebra In Theorem 2.2 wedescribe an important relationship between the decomposition of induced modules and thedecomposition of Hecke algebras Afterwards we describe properties of the Hecke algebra,including a depiction of the Hecke algebra as an algebra of functions that are constant onspecific cosets In this chapter we also describe a nice basis for the Hecke algebra and find

a simple formula for the structure constants associated with this basis

In Chapter 3 we analyze a specific Hecke algebra Ha Adapting the work of Chapter 2 to

Hawe define a basis for the algebra and use the structure constants to describe multiplicationrelations on the basis

In the fourth chapter, we define a Hecke algebra H, as an A-algebra with certain ators and relations, where A is the ring Z[a, q1/2, q−1/2] Then we adapt the work of Lusztig[5, Proposition 3.3] to show that H is a free A-module

gener-In Chapter 5 we define a partial order that arises from the basis of H, where H is viewed

as a free A-module This partial order agrees with the Bruhat-Chevalley order if we insistthat the indeterminate a is equal to the indeterminate q − 1 This chapter contains manyinteresting properties of H, including the fact that the basis elements are invertible Writingthe inverse of each basis element as a linear combination of basis elements, we analyze thecoefficients of the basis elements in the linear combination and find a recursive definition forthe coefficients

In the sixth chapter we adapt the work of Deodhar [3] to show that the coefficients found

in Chapter 5 are polynomial in aq−1/2

CHAPTER 2

INDUCED MODULES AND THE HECKE ALGEBRA

In this chapter we define induced modules and an algebra called the Hecke algebra InTheorem 2.2 we describe an important relationship between the decomposition of inducedmodules and the decomposition of Hecke algebras Afterwards we describe properties of theHecke algebra, including a depiction of the Hecke algebra as an algebra of functions thatare constant on specific cosets In this chapter we also describe a nice basis for the Heckealgebra and find a simple formula for the structure constants associated with this basis.Unless otherwise noted in this chapter, k is a field and V is a vector space over k, G is afinite group and H is a fixed subgroup of G

A k-representation of a group G is a group homomorphism from G to GL(V ) If V is offinite dimension n, it is common to choose a basis for V and identify GL(V ) with GLn(k).The k-group algebra, kG, associated to G is the algebra of functions from G to k, withpointwise addition and scalar multiplication, and multiplication given by the convolutionproduct

(f · g)(x) =X

y∈G

f (y)g(y−1x)

for x in G A typical function in kG looks like P

g∈Gagug, where ug is the characteristicfunction of {g}

There is an easy way to move between k-representations and kG-modules Starting with ak-representation ρ : G → GL(V ), we define an action from kG×V to V byP

g∈Gagugv =P

g∈Gagρ(g)(v) It is easy to check that this action defines a left kG-module structure on

V Therefore, each k-representation ρ : G → GL(V ) gives V a left kG-module structure

On the other hand, starting with a left kG-module V , we define a map Tg : V → V by

Tg(v) = ugv, where g is in G It is easy to check that Tgis an invertible linear transformation

Next we define ρ : G → GL(V ) by ρ(g) = Tg It is straightforward to verify that ρ

is a k-representation In this way every left kG-module V determines a k-representation

ρ : G → GL(V )

Before getting to the major ideas of this chapter, we need a few more definitions

For a ring R, a nonzero R-module M is said to be simple if M has no proper nonzerosubmodules If a nonzero R-module N can be written as a direct sum of simple R-modules,then N is called semisimple Moreover, a ring R is called semisimple if every R-module isisomorphic to a direct sum of simple R-modules

The next theorem gives conditions under which kG is a semisimple ring

Theorem 2.1 (Maschke’s Theorem) If k is a field, G is a finite group, and the characteristic

of k does not divide |G|, then the group algebra kG is a semisimple ring

From now on we assume the characteristic of k does not divide |G| By Maschke’sTheorem, kG is semisimple, and so kG-modules can be decomposed into a direct sum ofsimple kG-modules Therefore, to describe an arbitrary kG-module, it is enough to determinethe simple kG-modules A main theorem in this chapter, Theorem 2.2, uses this idea alongwith induction

The operation of induction from kH-modules to kG-modules assigns to each kH-module,

L, a left kG-module, IndGH(L), called the induced module, given by

IndGH(L) = kG ⊗kH L

Now we consider the special case where L has the form kHe, for an idempotent e in kH

In this case, it is easy to see that IndGH(L) = kG ⊗kH kHe ∼= kGe and so

EndkG(IndGH(L)) ∼= EndkG(kGe) = HomkG(kGe, kGe) ∼= (ekGe)op

The last isomorphism is given by f 7→ f (e), where f is a homomorphism from kGe tokGe From now on we only consider induced modules of the form IndGH(kHe) where e is an

idempotent in kH In this case, there are strong links between the Hecke algebra ekGe andthe decomposition of IndGH(kHe) into irreducible constituents.

A k-algebra, A, is said to be split semisimple if A is semisimple and EndA(L) = k · 1Lfor each simple left A-module L We assume from now on that kG is split semisimple

Theorem 2.2 Using the notation defined above, we have the following three statements.(i) The algebra ekGe is split semisimple

(ii) The irreducible constituents of IndGH(L) correspond to the irreducible ekGe-modules

In particular, if M is an irreducible constituent of IndGH(L), then M corresponds

to the irreducible ekGe-module eM

(iii) The multiplicity of each irreducible constituent of IndGH(L) is equal to the dimension

of the corresponding ekGe-module

Proof To prove the second statement, we define a map by M 7→ eM , and show the map

func-Proof To show that every element of eHkGeH is constant on (H, H) double cosets, it

is enough to show that the basis elements of eHkGeH are constant on double cosets Inparticular, consider eHugeH where g is in G Fix x in G Then using the convolutionproduct twice, we get

ug(z)eH(z−1y−1x)

=y∈G h∈H

of eHkGeH is constant on (H, H) double cosets

To show the converse, we fix a function f that is constant on (H, H) double cosets Firstsuppose that for each x in G, we have f (h1x) = f (x), for all h1 in H Using the convolutionproduct we have:

eHkGeH

Suppose {x1, , xn} is a complete set of double-coset representatives Let Ti denote Txi,for 1 ≤ i ≤ n Similarly, let Di denote the double coset HxiH

Theorem 2.4 The set { Ti | 1 ≤ i ≤ n } forms a basis of eHkGeH.

Proof To show that { Ti | 1 ≤ i ≤ n } is linearly independent, suppose Pn

βi|H||H|−1χDi =

nXi=1

βi|H|Ti

Since { Ti | 1 ≤ i ≤ n } forms a basis of eHkGeH, there are structure constants µ(i,j,k) sothat TiTj =P

kµ(i,j,k)Tk Now we find a formula for these constants µ(i,j,k).Fix subscripts i, j and k Using the convolution product, we find that TiTj(xk) is equal

toP

y∈GTi(y)Tj(y−1xk) Notice that Ti(y) is nonzero if and only if y is in the coset Di, and

Tj(y−1xk) is nonzero if and only if y is in the coset xkD−1j Thus,

Thus µ(i,j,k) = |H|TiTj(xk) Using equation (1) and the definition of the basis elements, weget

= |H|−1

y∈D i ∩xkD−1j

χD i(y)χD j(y−1xk)

= |H|−1|Di∩ xkD−1j |

The last equality is true because χDi(y)χDj(y−1xk) = 1, for all y in Di∩ xkD−1j

Theorem 2.5 Fix subscripts i, j and k Define µ(i,j,k) by TiTj(xk) = P

kµ(i,j,k)Tk Then

µ(i,j,k) = |H|−1|Di∩ xkD−1j |

CHAPTER 3

AN EXAMPLE OF A HECKE ALGEBRA

In this chapter we consider the special case where G is the group GLn(Fq), for a primepower q Fix relatively prime, positive integers a and b so that q − 1 = ab Then F#q =(Fq − { 0 }, ∗) is isomorphic to the direct product of a cyclic group, Fa, of order a, and acyclic group, Fb, of order b Fix a generator ζ of F#

q Then ζb generates Fa and ζa generates

Fb

Let Ha (respectively Hb) denote the group of n × n diagonal matrices whose diagonalentries are in Fa(respectively Fb) Let U denote the group of n × n upper triangular matriceswhose diagonal entries are 1 and whose strictly upper triangular entries are from Fq Thenlet Ba (respectively Bb) denote the subgroup HaU (respectively HbU ) Let B denote BbBa.The subgroup of n × n permutation matrices in G is denoted by W , and is isomorphic

to Sn The group W is generated by set S = {s1 sn−1}, where for each i,

For each w in W , the length of w, denoted `(w), is defined as the least number of factorsneeded to express w as a product of elements in S If `(w) = p, then an expression of theform w = si 1 si p is called a reduced expression for w, where si 1, , si p are in S Let w0denote the n × n matrix whose antidiagonal entries are all 1 and whose other entries are all

0 For w in W , define Uw− to be the set U ∩ (ww0)−1U ww0

Theorem 3.1 (Sharp form of Bruhat decomposition) Using the preceding notation, thegroup G can be written as the disjoint union, ˙∪x∈WUw−wB Moreover, each element of G can

be written uniquely in the form uwb with u ∈ Uw−, w ∈ W and b ∈ B

The sharp form of Bruhat decomposition gives a unique factorization of elements of G.Furthermore, since B ∼= Hb× Ha× U , we see that each element of G can be written uniquely

in the form uwhbhau0 with u in Uw−, w in W , hb in Hb, ha in Ha, and u0 in U Since Uw− is

a subset of Ba and Ba = HaU , we see that W Hb is a complete set of (Ba, Ba) double cosetrepresentatives

Recall that ζa generates Fb Therefore, each α in Fb can be written as ζam for some mwith 1 ≤ m ≤ b − 1, and so hα,i= hζam ,i To simplify future theorems, we define for each si

in S with 1 ≤ i ≤ n − 1, the set Xsi to be { hα,i | α ∈ Fb} = { hζam ,i | 1 ≤ m ≤ b − 1 } Wefurther define Xwsiw−1 to be wXsiw−1

Corollary 3.3 Fix si in S, and suppose u 6= e is in Us−

i Further suppose that the (i, i + 1)entry of u is ζc with 0 < c < q−1 Write c = am+bn where 0 ≤ m ≤ b−1 and 0 ≤ n ≤ a−1.Then siusi = u1shubhuau2, where u1 = u2 ∈ U−1

s have ζ−c in the (i, i + 1) position, hub = hζam ,i

is in Hb and hua = hζbn ,i is in Ha

Lemma 3.4 The relation u ∼ u0 if and only if hu

b = hu0

b is an equivalence relation on Us−and each equivalence class contains a elements

Proof It is straightforward to verify that ∼ is an equivalence relation on Us− To showthat each equivalence class contains a elements, fix u 6= e and u0 6= e in U−

s i with u ∼ u0.Suppose that the (i, i + 1) entry of u contains ζc, where 0 < c < q − 1 We write c = am + bnwith 0 ≤ m ≤ b − 1 and 0 ≤ n ≤ a − 1

In a similar manner, we write siu0si as u01shu0

b hu0

au02 Since u ∼ u0, we have hu

b = hu0

b , and

so there is only one choice for the power of ζ in hu0

b However, there are no restrictions onthe value of hu0

a Since there are a choices for the power of ζ in hu0

a, there are a choices for

In this chapter, let Ha denote the subalgebra eBakGeBa of kG By Theorem 2.3, we knowthat Hais the algebra of k-valued functions on G that are constant on (Ba, Ba) double cosets

As in the previous chapter, we describe a basis for Ha However, to simplify computations,

we scale the basis elements that we found in Chapter 2 for eHkGeH Before doing so, weneed a few definitions

There is a natural projection mapping π : W Hb → W For each x in W Hb, let `(x)denote `(π(x)), let qx denote q`(x), and let Dx denote the double coset BaxBa Furthermore,for each x in W Hb, let tx denote the function q−1/2x |Ba|−1χDx Notice that for each doublecoset representative x, we have tx = q−1/2x Tx, where Tx is an element of the basis of eHkGeHdescribed in the paragraph preceding Theorem 2.4 By adapting the proof of Theorem 2.4,

we see that { tx | x ∈ W Hb} forms a basis of Ha We also adapt the discussion precedingTheorem 2.5 to find that txty(z) =P

where δ0 = vδv−1 Notice that dBa∩wBav−1δ0Ba6= 0 if and only if w−1dBa∩Bav−1δ0Ba 6= 0.Since w−1d and v−1δ0 are both double coset representatives, w−1dBa∩ Bav−1δ0Ba is nonzero

if and only if w = v and d = vδv−1 This is true if and only if w = v and δ = v−1dv, which

is equivalent to wδ = dv Moreover, µ(d,v,dv) = qv−1/2qvd1/2|Ba|−1|v−1dBa ∩ Bav−1dBa| = 1.Therefore, tdtv = tdv, where d is in Hb and v is in W

(iiia) Applying equation (2) and the fact that s−1 = s, we get tvts =P

wδ∈W H bµ(v,s,wδ)twδwhere

From here there are two cases If `(ws) > `(w) then wusδ0 = wuw−1wsδ0 = u0wsδ0, where

u0 = wuw−1 is in Us− Moreover, u0wsδ0 is in BavBa if and only if wsδ0 = v, since wsδ0 is

a double coset representative This is true if and only if w = vs and δ0 = e However, if

w = vs and `(ws) > `(w), then `(v) > `(vs) This contradicts the original length assumption.Therefore, this case does not contribute to the value of µ(v,s,wδ)

Now suppose `(ws) < `(w) and u 6= e By Corollary 3.3,

wusδ0 = wssusδ0 = wsu1shubhuau2δ0

Furthermore,

wsu1shubhuau2δ0 = (ws)u1(ws)−1(ws)shubhuau2δ0 = u01whubδ0huau02,

where u01 = (ws)u1(ws)−1 and u02 = (δ0)−1u2δ0 are both in Us− Since whu

bδ0 is a double cosetrepresentative, u01whu

bδ0hu

au02 is in BavBa if and only if whu

bδ0 = v This is true if and only

if w = v and hu

b = (δ0)−1 However, if w = v and `(ws) < `(w) then `(vs) < `(v), and thiscontradicts the original length assumption Thus, this subcase does not contribute to thevalue of µ(v,s,wδ)

Next suppose `(ws) < `(w) and u = e Then wusδ0 = wsδ0 is in BavBa if and only ifwsδ0 = v This is true if and only if δ = e and w = vs In this case,

µ(v,s,vs) = qv−1/2qs−1/2q1/2vs |{ e }| = 1

Therefore, for v in W and s in S, we have tvts = tvs, when `(vs) > `(v)

(iiib) Using the same reasoning as in the proof of statement (iiia), we see that the product

tvts =P

wδ∈W H bµ(v,s,wδ)twδ, where

µ(v,s,wδ) = qv−1/2qs−1/2qwδ1/2|{ u ∈ Us− | wusδ0 ∈ BavBa}|,

and δ0 = sδs From here there are two cases

First suppose `(ws) > `(w) Using the same reasoning as in the proof of statement (iiia),

we find w = vs and δ = e Thus,

Next suppose `(ws) < `(w) and u 6= e We use Corollary 3.3 to get

wusδ0 = wssusδ0 = wsu1shubhuau2δ0

Moreover,

wsu1shubhuau2δ0 = (ws)u1(ws)−1(ws)shubhuau2δ0 = u01whubδ0huau02,

where both u01 = (ws)u1(ws)−1 and u02 = (δ0)−1u02δ0 are in Us− Since whu

bδ0 is a (Ba, Ba)double coset representative, whu

bδ0 is in BavBa if and only if whu

bδ0 = v This in turn is true

if and only if w = v and (hu

b)−1= δ0 Thus

µ(v,s,vδ) = qv−1/2qs−1/2q1/2vδ |{ u ∈ Us−| δ = s(hu

b)−1s }|

From Lemma 3.4, we see that there are a matrices u with δ = s(hub)−1s Thus, we have

µ(v,s,vδ) = q−1/2v q−1/2s q1/2vδ a = aq−1/2, where δ is a matrix that satisfies δ = s(hub)−1s for some

u in Us− Recall that hu

b is defined to be hζam ,i, for some m with 1 ≤ m ≤ b − 1 Furtherrecall that if s = si, then Xs is defined to be { hζam ,i | 1 ≤ m ≤ b − 1 } It is easy to verifythat sXss = Xs Then µ(v,s,vδ) 6= 0 if and only if δ is in Xs Thus, for each δ in Xs we have

where u0 = w−1uw is in Us− Moreover, u0w−1s is in Bav−1δ0Ba if and only if w−1s =

v−1δ0 This is true if and only if w−1s = v−1 and δ0 = e However, if w−1s = v−1 and

`(w−1s) > `(w−1) then `(v−1) > `(v−1s) This contradicts the original length assumption,since `(v−1) = `(v) and `(v−1s) = `(sv) Therefore, this case does not contribute to thevalue of µ(s,v,wδ).

Now suppose `(w−1s) < `(w−1) and u 6= e By Corollary 3.3 we have

w−1us = w−1ssus = w−1su1shubhuau2

Furthermore,

w−1su1shubhuau2 = (w−1s)u1(w−1s)−1(w−1s)shubhuau2 = u01w−1hubhuau2,

where u01 = (w−1s)u1(w−1s)−1 is in Us− Since w−1hub is a double coset representative,

u01w−1hubhuau2 is in Bav−1δ0Ba if and only if w−1hub = v−1δ0 This is true if and only if

w−1 = v−1 and hu

b = δ0 However, if w−1 = v−1 and `(w−1s) < `(w−1) then `(v−1s) < `(v−1),and this contradicts the original length assumption Thus, this subcase does not contribute

to the value of µ(s,v,wδ)

Next suppose `(w−1s) < `(w−1) and u = e Then w−1s ∈ Bav−1δ0Ba if and only if

w−1s = v−1δ0 This is true if and only if δ0 = e and w−1s = v−1 Equivalently, this is true ifand only if δ = e and w = sv In this case,

µ(s,v,sv) = qs−1/2qv−1/2q1/2sv |{ e }| = 1

Therefore, for v in W and s in S, we have tstv = tsv, when `(sv) > `(v)

(ivb) As in statement (iva), tstv =P

wδ∈W H bµ(s,v,wδ)twδ, where

µ(s,v,wδ) = qs−1/2qv−1/2qwδ1/2|{ u ∈ Us− | w−1us ∈ Bav−1δ0Ba}|

From here we have two cases First suppose `(w−1s) > `(w−1) Using the same reasoning as

in the proof of statement (iva), we get w−1s = v−1 and δ0 = e Therefore w = sv and δ = e,and since sus is in Ba, we get

µ(s,v,sv)= q−1/2s q−1/2v q1/2sv |{ u ∈ Us− | v−1sus ∈ Bav−1Ba}| = |Us−| = 1

Now suppose `(w−1s) < `(w−1) and u = e Then w−1s is in Bav−1δ0Ba if and only if

w−1s = v−1δ0 This in turn is true if and only if w−1s = v−1 and δ0 = e If w−1s = v−1and `(w−1s) < `(w−1) then `(v−1) < `(v−1s), and this contradicts the assumption that

`(v) > `(sv) Thus, this subcase does not contribute to the value of µ(s,v,wδ)

Next suppose `(w−1s) < `(w−1) and u 6= e We use Corollary 3.3 to get

b = δ0 Using the same reasoning as in (iiib), we see that δ0 is in Xs.Hence δ is in v−1Xsv Therefore wδ = vv−1dv = dv where d is in Xs Thus,

CHAPTER 4

THE GENERIC HECKE ALGEBRA

In this and all remaining chapters, assume a and q are indeterminates Let A denote

Z[a, q1/2, q−1/2] Define H to be the A-algebra with generators { ts, td | s ∈ S, d ∈ Hb} andrelations:

(i) tdtd0 = tdd0 for d and d0 in Hb,

(ii) tdts= tstsds for d in Hb and s in S,

(iii) ts its j = ts jts i for si and sj in S with |i − j| > 1,

(iv) tsitsi+1tsi = tsi+1tsitsi+1 for si in S and 1 ≤ i < n,

Corollary 4.2 Fix w in W and suppose si 1 sip is a reduced expression for w Then wecan define tw = tsi1 tsip with no ambiguity

Proof Notice that statements (iii) and (iv) above show that t : S → H satisfies the

Define twd = twtd and tdw = tdtw for w ∈ W and d ∈ Hb Using these definitions, wesee that twte = tw = tetw Also, the relation (i) gives tdte = td = tetd Therefore te is theidentity element of H

In Theorem 4.7, we show that the algebra H is a free A-module whose basis is the set{ tx | x ∈ W Hb} Before doing so, we need a few lemmas.

Lemma 4.3 Suppose s and r are in S, w is in W , `(swr) = `(w), and `(sw) = `(wr) Then

sw = wr

Proof There are two cases First suppose `(wr) > `(w), and so `(wr) > `(swr) Suppose

by way of contradiction that swr 6= w, and fix a reduced expression si1 sip for w Sinceswr 6= w and `(swr) = `(w), we have swr = s1 ˆsi spr, for some i with 1 ≤ i ≤ p, whereˆ

si indicates that si has been deleted However, this means that `(sw) < `(w), which is acontradiction Therefore, swr = w, and so sw = wr

Now suppose `(wr) < `(w), and set v = wr Then `(svr) = `(v), `(sv) = `(vr), and

`(vr) > `(v) Applying the first case, we get sv = vr, which gives sw = wr Lemma 4.4 For δ in Hb and s = sj in S, the sets δXs and sδXss are equal

Proof Label the diagonal entries of δ with δ1, , δn To show that δXs is a subset ofsδXss, fix d in Xs Label the (j, j) entry of d with α Then the (j + 1, j + 1) entry is(−1)b+1α−1 The matrix δd is given below

matrix d0 is in Xs A simple computation shows that sδd0s is equal to δd Therefore, δXs is

`(sw) = `(wr) Then rw−1Xswr = Xr

Proof By Lemma 4.3, sw = wr and so

rw−1Xswr = (wr)−1Xs(wr) = (sw)−1Xs(sw) = Xw−1 sssw = Xw−1 sw = Xr

Our primary goal in this chapter is to show that H is a free A-module whose basis is{ tx | x ∈ W Hb} In order to do this, we first consider the free A-module E with basis{ ˜tx | x ∈ W Hb}, and show that there is an algebra homomorphism that takes tx to ˜tx foreach x in W Hb Using this homomorphism, we show that { tx | x ∈ W Hb} is a basis of

H In order to define the homomorphism, we first must define and analyze the followingA-linear maps from E to E, where s is in S and d is in Hb

Lemma 4.6 PyQz = QzPy, for all y and z in S ∪ Hb.

Proof Fix x is in W Hb The proof of this lemma has nine cases, the first six of which showthat PsQr = QrPs, for s and r in S The seventh case shows that Pd1Qd2 = Qd2Pd1 for d1and d2 in Hb The eighth case shows PdQr = QrPd and the ninth case shows PsQd = QdPswhere s and r are in S and d is in Hb