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DOMINANCE AND REGULARITY IN COXETER GROUPS

A Dissertation

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

by

Tom Edgar

Matthew Dyer, Director

Graduate Program in Mathematics

Notre Dame, Indiana

July 2009

DOMINANCE AND REGULARITY IN COXETER GROUPS

Abstract

by

Tom Edgar

The main purpose of this thesis is to generalize many of the results of Dyer,

[18]. Dyer introduces a large family of finite state automata that are demonstrated

to recognize many natural subsets of Coxeter groups. In the current work, we

consider generalized length functions in Coxeter systems and show that these

length functions lead to a larger family of finite state automata, which in turn

recognize a larger class of natural subsets of Coxeter groups. By a well-known

result, these subsets (called regular subsets) will then have a rational multivariate

Poincare series.

Dominance order on the set of reflections (or positive roots) of a Coxeter system

plays a major role in the study of regular subsets. We draw the connection between

dominance in root systems and a notion of closure in root systems. We describe

some conjectures, due to Dyer, [17], and introduce a conjectural normal form for

Coxeter group elements related to closure and dominance.

In [14], Dyer introduced the notion of the imaginary cone to characterize domi-

nance. We investigate the imaginary cone in the special case of hyperbolic Coxeter

systems. We show that any infinite, irreducible, non-affine Coxeter system has

universal reflection subgroups of arbitrarily large rank. This allows us to deduce

some consequences about the growth type of Coxeter systems.

For Laura, Sue, Kristin, and Mark

ii

CONTENTS

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2: BACKGROUND MATERIAL . . . . . . . . . . . . . . . . 42.1 Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Reflection Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 The Weak Order on T and the Set of Elementary Reflections . . . 72.5 Dominance Order on T . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER 3: GENERALIZED LENGTH FUNCTIONS . . . . . . . . . . 133.1 Commutative Monoids . . . . . . . . . . . . . . . . . . . . . . . . 143.2 A General Length for Coxeter Systems . . . . . . . . . . . . . . . 153.3 Infinite Heights and Generalized Partitions for Reflections . . . . 183.4 Infinite Height and the Weak Order . . . . . . . . . . . . . . . . . 193.5 The Root Poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Monotonicity Properties in Dominance Order on T . . . . . . . . 223.7 A Partition of the Reflections . . . . . . . . . . . . . . . . . . . . 253.8 Recurrence Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 253.9 Monoids with Poset Structure . . . . . . . . . . . . . . . . . . . . 263.10 General Length and the Reflection Cocycle . . . . . . . . . . . . . 29

CHAPTER 4: FINITE STATE AUTOMATA . . . . . . . . . . . . . . . . 324.1 Regular Languages . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Canonical Automata for Coxeter Systems . . . . . . . . . . . . . . 35

iii

4.4 Some Subsets of S∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 The Boolean Algebra of Regular Sets of S∗ . . . . . . . . . . . . . 394.6 The Reflection Cocycle and Automata . . . . . . . . . . . . . . . 42

CHAPTER 5: REGULAR SUBSETS OF COXETER GROUPS . . . . . . 445.1 Boolean Algebra of Regular Sets in W . . . . . . . . . . . . . . . 445.2 Coxeter Groups are Automatic . . . . . . . . . . . . . . . . . . . . 455.3 Different Types of Regularity . . . . . . . . . . . . . . . . . . . . 455.4 p-Regularity for Different p . . . . . . . . . . . . . . . . . . . . . . 475.5 Some Properties of p-Completely Regular Sets . . . . . . . . . . . 485.6 Poincare Series and Examples of Regular Subsets of W . . . . . . 505.7 More Regular Subsets of W . . . . . . . . . . . . . . . . . . . . . 525.8 Regularity in Direct Products . . . . . . . . . . . . . . . . . . . . 565.9 Product Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . 575.10 Multivariate Product Regularity . . . . . . . . . . . . . . . . . . . 58

CHAPTER 6: REGULARITY IN T AND MIXED PRODUCTS . . . . . 636.1 Regularity of sets of Reflections . . . . . . . . . . . . . . . . . . . 636.2 Types of Regularity in T . . . . . . . . . . . . . . . . . . . . . . . 656.3 p-Complete Regularity in T . . . . . . . . . . . . . . . . . . . . . 666.4 Maps of Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . 726.5 Regularity in Mixed Products . . . . . . . . . . . . . . . . . . . . 796.6 General Regularity in Mixed Products . . . . . . . . . . . . . . . 806.7 Multivariate Poincare Series . . . . . . . . . . . . . . . . . . . . . 82

CHAPTER 7: HECKE ALGEBRA MODULES . . . . . . . . . . . . . . . 847.1 The Generic Iwahori-Hecke Algebra . . . . . . . . . . . . . . . . . 847.2 A Module for H . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.3 The Hecke Algebra Modules from hW,p,M∞ . . . . . . . . . . . . . 867.4 The 0-Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 89

CHAPTER 8: INITIAL SECTIONS OF REFLECTION ORDERS . . . . 908.1 Classification of Twisted Bruhat Orders . . . . . . . . . . . . . . . 908.2 Proof of the Classification . . . . . . . . . . . . . . . . . . . . . . 92

CHAPTER 9: CLOSURE IN THE ROOT SYSTEM . . . . . . . . . . . . 1019.1 2-closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1019.2 More types of subsets of Φ+ . . . . . . . . . . . . . . . . . . . . . 1029.3 A Conjectural Normal Form for Elements of W . . . . . . . . . . 103

iv

CHAPTER 10: HYPERBOLIC COXETER GROUPS . . . . . . . . . . . 10810.1 Coxeter Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10810.2 Symmetric Matrices and Sylvester’s Law . . . . . . . . . . . . . . 10910.3 Hyperbolic Coxeter Systems . . . . . . . . . . . . . . . . . . . . . 10910.4 Non-affine Coxeter Systems . . . . . . . . . . . . . . . . . . . . . 11010.5 A Class of Coxeter Systems . . . . . . . . . . . . . . . . . . . . . 11210.6 A Smaller Class of Coxeter Systems . . . . . . . . . . . . . . . . . 11210.7 Parabolic Subsystems of Non-affine Coxeter Systems . . . . . . . 11410.8 Characterizations of Hyperbolic Coxeter Systems . . . . . . . . . 114

CHAPTER 11: LARGE REFLECTION SUBGROUPS . . . . . . . . . . . 11611.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11611.2 A Metric on Convex Sets . . . . . . . . . . . . . . . . . . . . . . . 11711.3 Approximation of a Sphere by Polytopes . . . . . . . . . . . . . . 11811.4 Cones in Coxeter Systems . . . . . . . . . . . . . . . . . . . . . . 12011.5 Topology on Rays in Cones . . . . . . . . . . . . . . . . . . . . . 12111.6 Cones in Hyperbolic Coxeter Systems . . . . . . . . . . . . . . . . 12211.7 Universal Coxeter Systems . . . . . . . . . . . . . . . . . . . . . . 12311.8 Large Reflection Subgroups . . . . . . . . . . . . . . . . . . . . . 125

CHAPTER 12: EXPONENTIAL GROWTH IN COXETER GROUPS . . 12912.1 Growth Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13012.2 Properties of W(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13012.3 Parabolic Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . 13112.4 Proof of Proposition 12.3.3 . . . . . . . . . . . . . . . . . . . . . . 132

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

v

FIGURES

3.1 This diagram shows the idea of generalized length in B2 = 〈r, s, t |r2 = s2 = t2 = (rt)2 = 1, (rs)4 = (st)4 = 1〉. Each group ele-ment corresponds to one of the alcoves, with the yellow trianglerepresenting the identity. Each reflection corresponds to one of theaffine hyperplanes, and the color of the hyperplane distinguishes theconjugacy class. There are three conjugacy classes. The standardlength counts the number of affine hyperplanes separating w fromthe yellow alcove. The generalized length counts this same num-ber but retains additional information about the conjugacy classof each affine hyperplane separating w from the identity. For thew specified above we have l(w) = 12 while lp(w) = (3, 6, 3) wherep represents the function taking distinct values on each conjugacyclass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1 This diagram shows the tiling of the plane corresponding to B2 asin Figure 3.1. Each of the gray regions is an intersection of two halfspaces, and thus each is p-completely regular for any p. Then, thetotal gray region is a union of two p-completely regular sets and sois p-completely regular itself. Therefore, this set also has a rationalmultivariate Poincare series. . . . . . . . . . . . . . . . . . . . . . 53

8.1 This is a schematic diagram of the root system for (W,S) infinite.Φ+ consists of the roots which are non-negative linear combinationsof αr and αs. The dotted ray through the origin represents a limitline of roots. If C1 and C2 are not both satisfied by Φ+ then Ψ ispictured above with the roots in Ψ labeled by •, and those not inΨ labeled by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

vi

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor, Matthew Dyer. I am

always amazed by how much he knows and is willing to teach me, whether it be

for an hour in his office or over the computer thousands of miles away. Without

his endless patience, encouragement, and ideas, I would not have finished this

work. I would also like to thank his family, Annette and Katy, for their wonderful

hospitality and amusing conversations.

I would like to thank the members of my thesis committee, Mark Colarusso,

Sam Evens, and Michael Gekhtman, who took much of their valuable time to read

the thesis and provide thoughtful comments. I would further like to thank Sam

Evens for his conversations about research throughout my fifth year. Additionally,

I would like to thank Beth Vervelde for all the help she provided me during my

graduate career.

Furthermore, I would like to thank many of the mathematicians I have had the

pleasure of working with and learning from during my years as a student. I would

like to thank Lorelei Koss, Dave Richeson, and Barry Tesman for their excellent

teaching and for inspiring me to choose a career in mathematics. I thank Anton

Betten for his guidance in my master’s program and for teaching me a great deal

of interesting combinatorics. I would like to acknowledge and thank Brian Hall

for the many enlightening conversations about root systems and mathematics in

general. I would like to thank fellow graduate students Benjamin Jones, Don

vii

Brower, and John Wallbaum for many interesting and helpful conversations; I

would moreover like to thank all the Notre Dame graduate students for the good

times spent here.

Without my family, I would not be where I am today. I would like to express

deep gratitude to Margot, Dick, Joann, and Ray for treating me like one of their

own children during our time in Indiana. I thank my sister, Kristin, for her loving

support and willingness to just call and talk. I thank my mom, Sue, for everything

she has done for me in my life. She is an amazing woman and someone I will always

look up to. I would also like to thank my dad, Mark. He taught me how to work

hard, and he always believed in me. I have never met a better man, and I will

forever miss him.

Finally, and most of all, I thank my beautiful wife, Laura. Her love and support

have been an integral part of my success as a graduate student. She makes life

fun, and I couldn’t imagine what it would be like if she were not around. I owe

her immensely for her willingness to help me achieve this goal.

viii

SYMBOLS

W Coxeter group

S Coxeter group generators

l Standard length function on W

lp General length function on W

T Reflections

N Reflection cocycle

Φ Root system

Φ+ Positive roots

Π Simple roots

( , ) Bilinear pairing on root system

≤A Twisted Bruhat order on group

W ′ Reflection subgroup

M Monomials in Z[X]

Dominance order on reflections

≤ Weak order on T ; Order on M

`p General length function on S∗

ix

CHAPTER 1

INTRODUCTION

In this dissertation, we investigate some of the applications of dominance order

on the root system (or reflections) of a Coxeter group. Dominance was first defined

by Brink and Howlett in [5], where it is used to show that Coxeter groups are

automatic. Brink further investigates the dominance minimal roots in [4].

Despite this work, the dominance order is still poorly understood in general.

More recently, in [15] Dyer has employed the idea of using the imaginary cone

as a combinatorial tool for studying dominance in the root system. Our general

goal will be to further the understanding of the role dominance plays in different

aspects of study in Coxeter systems. The dissertation is organized as follows.

Chapter 2 introduces the basic background information necessary to develop

the results in the later chapters. These ideas will be used throughout the thesis.

The thesis is broken up into three independent parts, which are loosely related

through their connections to dominance order.

In Chapter 3, we examine general length functions on Coxeter systems. These

give rise to a class of height functions, which we demonstrate can be understood

by studying the weak order and give a nice characterization of dominance order.

These height functions allow us to define certain finite sets of reflections that parti-

tion the entire set of reflections. We begin Chapter 4 by defining regular languages

and finite state automata. We are then able to use the partition of the reflections

1

described in Chapter 3 to define a class of canonical finite automata for the Cox-

eter system. These automata generalize the automata defined by Dyer in [18]. In

Chapter 5, we use the automata defined in Chapter 4 to describe large classes of

regular subsets of Coxeter groups, i.e. sets that can be recognized by a finite state

automaton. We introduce different types of regularity; in particular, our notion

of p-complete regularity allows us to easily describe certain regular subsets of a

Coxeter group. Regularity has some interesting and important implications for

(multivariate) Poincare series of subsets of a Coxeter system, and we will discuss

these briefly. Due to the fact that many natural subsets of Coxeter systems cannot

be described as completely regular in the sense of Chapter 5, we use Chapter 6

to define an alternate type of regularity, which will allow us to demonstrate the

regularity of many subsets of the set of reflections in a Coxeter system. We end

with a characterization of regular sets inside of products W n × Tm where W is a

Coxeter system with T the set of reflections. Finally, in Chapter 7, we describe

how the finite sets that give regularity also lead to a large family of modules for

the generic Iwahori-Hecke algebra.

Next, in Chapters 8 and 9, we discuss a notion of closure in the root system

(or reflections). Chapter 8 classifies possible partial orders on a Coxeter system,

generalizing Bruhat order, using certain closed subsets of the roots. These results

appear in [20] as well. Chapter 9 focuses on discussing closure in general and other

types of subsets of roots. We outline some conjectures, posed by Dyer (see [17]),

involving closure in the root system. We also describe an application of these

conjectures related to closure of the sets defined in Chapter 4, which would lead

to a normal form for elements of the Coxeter group.

Finally, in Chapters 10-12, we investigate some properties of the imaginary

2

cone, especially in hyperbolic Coxeter systems. In Chapter 10, we provide some

characterizations of hyperbolic Coxeter systems. As a byproduct of our charac-

terizations, we obtain a simple, case-free proof of the well known fact that any

Coxeter system contains a hyperbolic subsystem as a standard parabolic subsys-

tem. Chapter 11 focuses on studying the imaginary cone in the case of hyperbolic

Coxeter systems. We use the imaginary cone to show that any Coxeter system con-

tains universal reflection subgroups of arbitrarily large rank. Furthermore, in the

hyperbolic case, the positive spans of the simple roots of the universal reflection

subgroups are shown to approximate the imaginary cone (using an appropriate

topology on the set of roots). This answers in the affirmative a question due to

Dyer [15] in the special case of hyperbolic Coxeter systems. Lastly, Chapter 12

uses the results from Chapter 11 describing large universal reflection subgroups

in any Coxeter system to extend the results of [29] regarding exponential growth

in parabolic quotients in Coxeter groups.

3

CHAPTER 2

BACKGROUND MATERIAL

We begin with a chapter introducing the basic objects and terminology used

throughout the thesis.

2.1 Coxeter Groups

Definition 2.1.1. We define a Coxeter system to be a pair (W,S) consisting of

a group W and a set of generators S ⊂ W subject only to relations of the form

(ss′)m(s,s′) = 1 where m(s, s) = 1 and m(s, s′) = m(s′, s) ≥ 2 for s 6= s′. If s and

s′ have no such relation, we say that m(s, s′) =∞.

We may sometimes abuse terminology and call (W,S), or W , a Coxeter group.

For general results about Coxeter groups, we refer the reader to [23] or [2]. In

general, S does not need to be finite. For many of the results we will state we

need S to be finite. We shall specify when this is the case.

Every element w ∈ W can be written as a product of the generators w =

s1s2 · · · sm, each si ∈ S. If m is minimal among all such expressions for w we will

say that s1s2 · · · sn is a reduced expression for w and we say that w has length n,

i.e. l(w) = n. So l : W → N denotes the standard length function on (W,S).

We let T denote the set of W -conjugates of S, T :=⋃w∈W wSw−1, and call

T the set of reflections. Then, following [19] we define N : W → P(T ) to be

4

the “reflection cocycle” given by N(w) = t ∈ T | l(tw) < l(w). We call N(w) a

cocycle because it satisfies the cocycle condition:

N(xy) = N(x) + xN(y)x−1

where + represents symmetric difference on the power set of T , A + B = (A ∪

B) \ (A ∩ B). The existence of this cocycle implies that W acts on P(T ) by

w · A = N(w) + wAw−1 for A ⊆ T .

2.2 Root Systems

Without loss of generality, we will assume that (W,S) is realized in its standard

reflection representation on a real vector space V with Π denoting the set of simple

roots and (−,−) : V × V → R representing the associated bilinear form given

by (α, β) = − cos πm(sα,sβ)

if m(sα, sβ) 6= ∞ and (α, β) ≤ −1 otherwise. Then

we denote the roots and positive roots by Φ and Φ+ respectively. Recall that

Φ = Φ+∪ − Φ+, and we may sometimes write Φ− := −Φ+.

For α ∈ Φ we let sα ∈ T represent the corresponding reflection. Recall that

the set of reflections are in bijective correspondence with the set of positive roots

under the bijection α 7→ sα : Φ+ → T . For t ∈ T , we will let αt ∈ Φ+ denote the

unique positive root α with sα = t. We recall the following two facts

l(wt) < l(w)⇔ w(αt) ∈ Φ− for t ∈ T and w ∈ W (2.2.1)

l(tw) < l(w)⇔ w−1(αt) ∈ Φ− for t ∈ T and w ∈ W (2.2.2)

5

We can then describe the W -action on Φ in terms of the length function l by

the following formula

w(±αt) = ±εαwtw−1 , ε =

1, if l(wt) < l(w)

−1, if l(wt) > l(w)(2.2.3)

for any w ∈ W and t ∈ T .

2.3 Reflection Subgroups

We call a subgroup W ′ of W a reflection subgroup of W if it is generated by

the reflections it contains, W ′ = 〈W ′ ∩ T 〉. It was shown by Dyer ([19]) and

Deodhar ([10]) independently that any reflection subgroup is also a Coxeter sys-

tem; moreover, any reflection subgroup has a canonical set of Coxeter generators

χ(W ′) = t ∈ T | N(t) ∩ W ′ = t. We say a reflection subgroup is dihedral

if it is generated by two distinct reflections or equivalently |χ(W ′)| = 2. We let

l(W ′,χ(W ′)) : W ′ → N be the length function for (W ′, χ(W ′)) and T ′ = W ′ ∩ T

be the set of reflections of W ′. Due to [19], N(W ′,χ(W ′))(x) = N(x) ∩W ′ for all

reflection subgroups W ′ where N(W ′,χ(W ′)) is the reflection cocycle for (W ′, χ(W ′)).

Any dihedral reflection subgroup, W ′ = 〈sα, sβ〉, is contained in a unique

maximal dihedral reflection subgroup, namely 〈sγ| γ ∈ Φ ∩ (Rα + Rβ)〉. We let

M = W ′ < W | W ′ is a maximal dihedral reflection subgroup

M∞ = W ′ ∈M | |W ′| =∞.

6

For any reflection t ∈ T , we get the following set:

Mt = W ′ ∈M | t ∈ W ′.

The previous remarks imply that

T \ t =⋃

W ′∈Mt

((W ′ ∩ T ) \ t) (2.3.1)

where ∪ represents a disjoint union.

Since any reflection subgroup is also a Coxeter system (W ′, χ(W ′)), we let ΦW ′ ,

Φ+W ′ , ΠW ′ ⊆ Φ be the set of roots, positive roots, and simple roots for (W ′, χ(W ′))

sitting inside the root system for (W,S). We have that ΦW ′ = α ∈ Φ| sα ∈ W ′.

For any w ∈ W , there exists a unique x ∈ W ′w with N(x)∩W ′ = ∅ (likewise a

unique x ∈ wW ′ with N(x−1)∩W ′ = ∅). According to [19], we have χ(w−1W ′w) =

x−1χ(W ′)x. Let w = yx with y ∈ W ′. Then α 7→ w−1(α) = x−1y−1(α) induces a

bijection ΦW ′∼=→ Φw−1W ′w. This bijection is obtained by composing the bijection

α → y−1(α) : ΦW ′∼=→ ΦW ′ given by the action of y−1 ∈ W ′ on the root system of

W ′ and the bijection α→ x−1(α) : ΦW ′∼=→ Φx−1W ′x = Φw−1W ′w give by the action

of x. Since N(x) ∩W ′ = ∅, the latter bijection preserves positive roots as well.

2.4 The Weak Order on T and the Set of Elementary Reflections

Much of the terminology of this section is described in terms of the root system

in [2]. We will instead use terminology associated to the set of reflections when

possible. For any reflection t ∈ T , we define the standard height to be h(t) :=

h(W,S)(t) = l(t)−12

.

At this point, we define a partial order on the set of reflections. For any

7

t, t′ ∈ T , we set t → t′ if there is some s ∈ S (it is unique if it exists) such that

t′ = sts and l(t′) = l(t) + 2 (or equivalently h(t′) = h(t) + 1). Let ≤ be the

partial order on T given as the reflexive transitive closure of →, and call (T,≤)

the reflection poset or the weak order on T . It is clear that h is the rank function

for (T,≤).

From the formula given in (2.3.1), we see that for t ∈ T we have

h(t) =∑

W ′∈Mt

h(W ′,χ(W ′))(t)

where h(W ′,χ(W ′)) is the height function for (W ′, χ(W ′)). Now, we use this charac-

terization of height to introduce the “∞-height” of a reflection as follows.

h∞(t) :=∑

W ′∈Mt|W ′|=∞

h(W ′,χ(W ′))(t). (2.4.1)

We now introduce a special subset of the reflections. Let T0 := t ∈ T | h∞(t) =

0. We call this set the set of elementary reflections. Using the bijection between

T and Φ+, this set is in bijective correspondence with the elementary roots defined

in [5]. In that paper, Brink and Howlett also demonstrate that this set is finite.

For another description of this fact, see [2].

More recently, Dyer has shown similar sets to be finite. More precisely, we

define, for any n ∈ N, Tn := t ∈ T | h∞(t) = n. For these sets, we have the

following theorem, due to Dyer, found in [14].

Theorem 2.4.1. Let (W,S) be a Coxeter system with reflections T . If S is a

finite set, then Tn is a finite set for all n ∈ N.

These sets of reflections, as well as others like it, will be defined in more

8

generality in Chapter 3, and we will make extensive use of these types of sets in

Chapter 5 in order to demonstrate nice regularity properties of Coxeter systems.

We will also use these sets to define certain modules for the generic Iwahori-Hecke

algebra associated to (W,S) in Chapter 7.

2.5 Dominance Order on T

The set of elementary reflections also has a nice characterization in terms of

another partial order on the set of reflections; it is the set of minimal elements of

this partial order. Dominance order was initially defined in [5]. The elementary

reflections and the dominance order are studied in more detail in [4]. The basic

properties of dominance can be found in [2],[4], or [5], but we will include a few

results here.

We will first define dominance on the root system associated to (W,S) and use

the canonical bijection of positive roots with reflections to transport the definition

to the reflections.

We say that α ∈ Φ dominates β ∈ Φ written β α if, for all w ∈ W ,

w(α) ∈ Φ− implies that w(β) ∈ Φ−.

Lemma 2.5.1. 1. If α, β ∈ Φ, then α dominates β in Φ if and only if for

some (respectively every) reflection subgroup W ′ of W with sα, sβ ∈ W ′, α

dominates β in ΦW ′ (with respect to W ′).

2. is a partial order on Φ.

3. Multiplication by −1 is an order-reversing bijection of (Φ,) with itself.

4. The W -action on Φ is as a group of poset automorphisms of (Φ,).

9

Proof. First, suppose that β, α ∈ Φ and that sα and sβ are contained in a reflection

subgroup W ′. According to section 2.3, we have that for any w ∈ W , there is

x ∈ wW ′ with N(x−1) ∩W ′ = ∅, and α 7→ x(α) is a bijection, which preserves

positive roots, between ΦW ′ and ΦwW ′w−1 . Let w = xy with y ∈ W ′. Suppose

that β α in ΦW ′ with respect to W ′. Then if w(α) ∈ Φ−, we have xy(α) ∈ Φ−

and so y(α) ∈ Φ−W ′ . Thus, w(β) = xy(β) ∈ x(Φ−W ′) = Φ−W ′ ⊂ Φ− and so β α

in Φ. If β α in Φ, then by definition, w(α) ∈ Φ− implies w(β) ∈ Φ− for all

w ∈ W and so this is true for all w′ ∈ W as required for 1. We now describe

the dominance order on dihedral Coxeter systems. Let (W,S) be dihedral, with

S = r, s. Then if W is finite, all the (finite) elements of Φ are incomparable

in . If W is infinite, then the dominance order is given by the coarsest partial

order satisfying the relations

· · · − αsrsrs ≺ −αsrs ≺ −αs ≺ αr ≺ αrsr ≺ αrsrsr ≺ · · ·

and

· · · − αrsrsr ≺ −αrsr ≺ −αr ≺ αs ≺ αsrs ≺ αsrsrs ≺ · · · .

In either of these cases, it is clear that (Φ,) is a partial order. For general

(not necessarily dihedral) (W,S), it is clear that is reflexive and transitive by

definition. Then, antisymmetry in general follows by using antisymmetry in a

reduction to a dihedral reflection subgroup. Namely, suppose β α β with

α 6= β in Φ, and then consider W ′ = 〈sα, sβ〉. Due to 1, we see that β α β

in ΦW ′ , and by antisymmetry in the dihedral case, we get that α = β, which

is a contradiction. Finally, 3 and 4 are simple consequences of the definition of

dominance. In particular if β α then w(β) w(α) since for any u ∈ W we have

10

uw(α) ∈ Φ− implies that uw(β) ∈ Φ−.

Then, as stated before, we use the canonical bijection between Φ+ and T to

define the notion of dominance on the set of reflections. We have that sβ sα if

and only if β α in Φ+. This is equivalent to t′ t if for all w ∈ W we have

l(wt) < l(w) implies that l(wt′) < l(w); in this case, we say that t dominates t′.

We will write (T,) when referring to the set of reflections partially ordered by

dominance.

We recall that in a poset, we say an element a covers an element b if b < a and

there is no z ∈ P with b < z < a.

Lemma 2.5.2. Let (W,S) be a Coxeter system.

1. If s, t ∈ T0 generate an infinite subgroup, then s, t is the set of canonical

Coxeter generators for the maximal dihedral reflection subgroup of (W,S)

containing s, t.

2. If α, β ∈ Φ+ are distinct elementary roots (i.e. sα and sβ are in T0) and

〈sα, sβ〉 is an infinite dihedral reflection subgroup then β covers −α in dom-

inance order.

Proof. Due to the definition of h∞ in equation (2.4.1), we know that t ∈ T0 if and

only if

0 = h∞(t) =∑

W ′∈Mt|W ′|=∞

h(W ′,χ(W ′))(t)

which implies that h(W ′,χ(W ′))(t) = 0 for all infinite dihedral reflection subgroups

W ′. Therefore, by the definition of the height, we see that t ∈ χ(W ′) for all

infinite, maximal dihedral reflection subgroups of W with t ∈ W ′ ∩ T . We see

that 1 follows from this fact. To prove 2, since 〈sα, sβ〉 is infinite then −α ≺ β

11

by the description of dominance in infinite dihedral reflection subgroups given in

2.5.1. Suppose that γ ∈ Φ and −α ≺ γ ≺ β. Since β is elementary, we cannot have

γ ∈ Φ+; similarly, since α is elementary and −γ ≺ α we cannot have −γ ∈ Φ+ so

that γ 6∈ Φ−. This contradicts the fact that Φ = Φ+∪Φ−.

Remark 2.5.3. In [14], Dyer uses the notion of the imaginary cone in a Coxeter

system (see Chapter 11) to characterize dominance. He demonstrates that every

covering α′ ≺ β′ in dominance order is in the W -orbit of a covering α ≺ β as in

part 2 of 2.5.2.

12

CHAPTER 3

GENERALIZED LENGTH FUNCTIONS

In this chapter, we begin by introducing a family of generalized length functions

on a Coxeter system. These length functions replace the standard length function

on (W,S) by taking values in Nk for some k ≥ 1 instead of simply taking values in

N. In a similar manner to section 2.4, these length functions will allow us to define

corresponding height functions and, more importantly, a class of “infinite-height”

functions on the set of reflections. These height functions lead to a partition of the

set of reflections with interesting applications. Due to Theorem 2.4.1, these sets

of reflections turn out to be finite. We will use the finiteness property to develop

many canonical automata in Chapter 4.

We proceed by interpreting these height functions in terms of the poset (T,≤)

(weak order) as defined in section 2.4. In turn, we further obtain monotonicity

properties of these length and height functions with respect to the dominance

order.

Finally, we demonstrate a basic recurrence formula for the reflection cocycle

on (W,S) in terms of this new partition of T . We end with a few results that will

be important in the following chapters. Many of the results generalize some of the

results in [18].

13

3.1 Commutative Monoids

At this point we want to define a commutative monoid M. We have two

equivalent definitions for M which we introduce below. Both definitions will be

useful for different applications, but it will be clear from context how we want to

view M since one definition will have multiplicative notation and the other will

have additive notation.

Let X be a finite set of indeterminates and Z[X] be the corresponding integral

polynomial ring. We say m ∈ Z[X] is a monomial if m =∏

x∈X x ax where ax ∈ N

for all x ∈ X. Let M := MX be the set of all monomials of Z[X], that is

M := m| m ∈ Z[X]; m is a monomial. For any m ∈ M, m =∏

x∈X x ax ,

define the degree of m to be deg(m) :=∑

x∈X ax .

If we fix an order for X = x1, ..., xk, we can also consider the commutative

monoid Nk defined by the map ζ : M → Nk given by m = xa11 xa22 · · ·x

akk 7→

(a1, ..., ak). Since ζ is an isomorphism, we will abuse notation by setting Nk =M.

The manner in which we view M will be clear depending on whether we use

additive or multiplicative notation. We note that for m ∈M, deg(m) =∑

x∈X ax

is independent of how we view M. For any monomial m = (a1, ..., ak) we let

m(i) := ai.

For X as above, let X−1 = x−1| x ∈ X. We can define M′ := m| m ∈

Z[X,X−1]; m is a monomial ∼= Zk. This set has a natural group structure. We

will also view M as a subset of M′; in the case when we view M additively,

this will explain using the notation m − n (and will explain using division when

viewing M multiplicatively).

We place a partial order ≤M on M as follows. For any m ∈ M and n ∈ M,

we say m ≤M n if and only if nm∈ M ⊂ M′ (additively this means n −m ∈

14

M ⊂ M′) . For x ∈ X, we say that m x n if nm

= x and call this a covering

relation of type x . It is clear that if mx n and m ≤M m′ ≤M n then m′ = m or

m′ = n. From this point on, we will write ≤:=≤M; this will not cause confusion

with ≤ on T as the context will make clear which ≤ is intended.

Now, by an ordered generalized partition of a monomial m we will mean a

tuple M = (m1,m2, ...), mn = 0 for n 0, with deg(mi) ≥deg(mj) for all i < j

along with∏

i mi = m. Next, we define an equivalence relation, ∼, on the set of

ordered generalized partitions in the following way. Let (m1,m2, ...) ∼ (n1,n2, ...)

if there exists a permutation of the natural numbers, σ, such that mi = nσ(i)

for all i. For (m1,m2, ...) an ordered partition, we let M = [(m1,m2, ...)] be

the equivalence class of the ordered partition. We call these equivalence classes

generated by this relation unordered generalized partitions. For any unordered

partition M = [(m1,m2, ...)], we define |M| :=∏

i mi. All the terminology of the

partial order on M can be phrased additively if necessary.

At this point, we can define a partial order on the set of unordered generalized

partitions. Let M and N be any two generalized partitions. Then we say that

M ≤ N if there exists (m1,m2, ...) ∈M and (n1,n2, ...) ∈ N such that mi ≤ ni in

M for all i. For any x ∈ X, we define the covering relation of type x by M x N

if there exists (m1,m2, ...) ∈M and (n1,n2, ...) ∈ N and j ∈ N such that mi = ni

for all i 6= j and mj x nj in M.

3.2 A General Length for Coxeter Systems

Suppose we have a fixed Coxeter system (W,S). Let M be the additive,

commutative monoid described in section 3.1 where M is generated by a finite

set X = x1, ..., xk of indeterminates. Suppose that we are given a surjective

15

function p : T → X satisfying p(t) = p(t′) if t is conjugate to t′, i.e. if there exists

v ∈ W with t = vt′v−1. For any reflection t ∈ T , we say that t is of type xi if

p(t) = xi where xi ∈ X (for simplicity we will usually say that t is of type p(t)).

Until further notice, we will viewM with additive notation and we will, for ease,

let p(t) represent ζ(p(t)).

We define a family of “length” functions in the following way. Let W ′ be any

reflection subgroup of W and w ∈ W ′ where w = r1...rn is a reduced expression

with ri ∈ χ(W ′). Then we define lp,W ′ : W ′ → M by lp,W ′(w) = lp,W ′(r1...rn) =

p(r1) + p(r2) + · · ·+ p(rn). See figure 3.1 for an example.

Similarly, for any reflection r ∈ W ′∩T with r = r1...rmrm+1...r2m+1, we define

a corresponding “height function” hp,W ′ : T → M by hp,W ′(r) = p(r1) + p(r2) +

· · ·+ p(rm). Finally, for any r ∈ W ′ ∩ T where r = r1...rmrm+1...r2m+1, we define

c(r) := p(r) = p(rm+1), which is called the center of r. Note that for t ∈ T ,

lp,W ′(t) = c(t) + 2hp,W ′(t).

From these height functions, we get the following functions on reflections as

well. For any union O ⊂M of W -orbits of maximal dihedral reflection subgroups

on M , we define

hW,p,O : t 7→∑

W ′∈MtW ′∈O

hp,W ′(t) (3.2.1)

and

lW,p,O : t 7→ c(t) + 2hW,p,O(t). (3.2.2)

In this thesis, we will only consider the case where O = M∞.

Remark 3.2.1. We note that, considering terminology from section 2.4, we have

h∞(t) = deg(hW,p,M∞(t)). Also, if M = N and q(t) = 1 for all t ∈ T , then

16

w

Figure 3.1. This diagram shows the idea of generalized length in B2 =〈r, s, t | r2 = s2 = t2 = (rt)2 = 1, (rs)4 = (st)4 = 1〉. Each group elementcorresponds to one of the alcoves, with the yellow triangle representing theidentity. Each reflection corresponds to one of the affine hyperplanes, andthe color of the hyperplane distinguishes the conjugacy class. There arethree conjugacy classes. The standard length counts the number of affinehyperplanes separating w from the yellow alcove. The generalized lengthcounts this same number but retains additional information about theconjugacy class of each affine hyperplane separating w from the identity.For the w specified above we have l(w) = 12 while lp(w) = (3, 6, 3) wherep represents the function taking distinct values on each conjugacy class.

17

h∞(t) = hW,q,M∞(t).

Lemma 3.2.2. Let (W,S) be a Coxeter system, and let ≤∅ be ordinary Bruhat

order on (W,S) (see Chapter 8). Let x, y ∈ W with x ≤∅ y. Then for all reflection

subgroups W ′ ⊆ W with x, y ∈ W ′, the following hold.

1. lp,W ′(x) ≤ lp,W ′(y), and

2. hp,W ′(x) 6> hp,W ′(y) if x, y ∈ T ∩W ′.

3. hp,W ′(x) ≤ hp,W ′(y) if x, y ∈ T ∩W ′ and c(x) = c(y).

Proof. We know that x ≤∅ y in W if and only if x ≤∅ y in W ′ for all x, y ∈ W ′ ⊆

W . The subword characterization of Bruhat order and the definition of lp,W in

terms of reduced expressions imply 1. Finally, 2 and 3 follow from 1.

3.3 Infinite Heights and Generalized Partitions for Reflections

According to equation (3.2.1) we can attach an unordered generalized partition

to each reflection t, MW,p,M∞(t), where each part, mi, is given by some distinct

hp,W ′(t) with W ′ ∈M∞.

Lemma 3.3.1. Let t ∈ T and s ∈ S with lp(sts) = lp(t) + 2p(s). Then

1. hW,p,M∞(sts) = hW,p,M∞(t) if 〈s, t〉 is finite and hW,p,M∞(sts) = hW,p,M∞(t)+

p(s) if 〈s, t〉 is infinite.

2. MW,p,M∞(sts) = MW,p,M∞(t) if 〈s, t〉 is finite. If 〈s, t〉 is infinite then

MW,p,M∞(t) p(s) MW,p,M∞(sts).

Proof. Let W ′ be a maximal dihedral reflection subgroup. According to 2.3 if

s 6∈ W ′, then N(s) ∩ W ′ = ∅ so (W ′, χ(W ′)) ∼= (sW ′s, sχ(W ′)s) and thus

18

hp,sW ′s(sts) = hp,W ′(t). If s ∈ W ′, then s ∈ χ(W ′) and sW ′s = W , and it

follows that hp,W ′(sts) = hp,W ′(t) + p(s). Now, since Msts = sW ′s| W ′ ∈Mt, if

〈s, t〉 ∈Mt ∩M∞ then we have

hW,p,M∞(sts) =∑

W ′∈MstsW ′∈M∞

hp,W ′(sts) =∑

W ′∈MtW ′∈M∞

hp,sW ′s(sts)

= p(s) +∑


hp,W ′(t) = p(s) + hW,p,M∞(t).

Otherwise we have

hW,p,M∞(sts) =∑

W ′∈MstsW ′∈M∞

hp,W ′(sts) =∑


hp,sW ′s(sts)

=∑


hp,W ′(t) = hW,p,M∞(t).

Based on these results, the lemma follows clearly.

3.4 Infinite Height and the Weak Order

Consider the Hasse diagram of the weak order (T,≤) as a directed graph with

vertex set T and directed edges (t, t′) from t to t′ for all t, t′ ∈ T with t → t′.

Furthermore, we can regard this graph as a labeled directed graph by labeling

the edges as follows. For any edge (t, t′), we know there is a unique s ∈ S with

t′ = sts. From this, we can label the edge (t, t′) with label p(s), and we call this

edge an edge of type p(s). We say that an edge (t, t′) is p(s)-short if it is labeled

with p(s) and if hW,p,M∞(t′) = hW,p,M∞(t), i.e. if 〈t, s〉 is finite. We say that the

edge is p(s)-long if it is labeled with p(s) and hW,p,M∞(t′) = hW,p,M∞(t) + p(s),

i.e. if 〈t, s〉 is infinite. Based on this, we see that for any t ≤ t′ in (T,≤), the

19

number of p(s)-long edges (xi−1, xi) in a chain t = x0 → x1 → ... → xn = t′ is

the m(j) = hW,p,M∞(t′)(j)− hW,p,M∞(t)(j) where p(s) = xj. Furthermore, for any

fixed t, then maximal number of p(s)-long edges in chains x0 → x1 → ...→ xn = t

is hW,p,M∞(t)(j) and this maximum is realized if x0 ∈ S.

Corollary 3.4.1. The map t 7→MW,p,M∞(t) is an order preserving map from the

reflection poset of (W,S) to the poset of generalized partitions.

Proof. This follows directly from part 2 of Lemma 3.3.1.

Remark 3.4.2. Notice that for t ≤ t′, we have that c(t) = c(t′) by definition of the

weak order.

3.5 The Root Poset

The reflections endowed with weak order as a poset is isomorphic to a subposet

of another poset which we briefly introduce. We call this poset the root poset

denoted (Φ,≤). Define → on Φ by α→ β if and only if there is some γ ∈ Π such

that β = sγ(α) and (γ|α) < 0, i.e. β − α ∈ R>0γ. Then, we let ≤ be the reflexive

transitive closure of →.

We can consider the Hasse diagram of (Φ,≤) as a directed graph with vertex set

Φ and edges (α, β) whenever α→ β. We label all edges by their type p(sγ) where

sγ is uniquely determined by β = sγ(α) where γ ∈ Π. We say an edge is p(sγ)-long

if it is labeled with p(sγ) and if (α|γ) ≤ −1, otherwise we call it p(sγ)-short. It

is clear that the relations (Φ+,→) and (Φ+,≤) are obtained by transporting the

relations from (T,→) and (T,≤) by means of the bijection sα 7→ α from T to Φ+.

We note that this correspondence also respects edge lengths and types, i.e. the

edge (α, β) is p(s)-long if and only if the corresponding edge (sα, sβ) is p(s)-long.

20

Using this correspondence, we translate the basic properties of the reflection

poset into corresponding properties of the root poset. In particular, we introduce

the following length function on the roots.

LW,p,M∞(α) =

lW,p,M∞(sα) if α ∈ Φ+

−lW,p,M∞(sα) if α ∈ Φ−(3.5.1)

Lemma 3.5.1. 1. For α ≤ β in Φ, the interval [α, β] is finite.

2. The map α 7→ −α is an order-reversing bijection of (Φ,≤) with itself.

3. Let α = α0 → α1 → ... → αn = β be a maximal chain in (Φ,≤). Then for

any xi ∈ X, the number of xi-long edges (αi−1, αi) in the chain (regarded

as a path from α to β in the Hasse diagram) is equal to m(i) where m =

LW,p,M∞ (β)−LW,p,M∞ (α)

2.

Proof. First, 1 follows from the corresponding fact that intervals are finite in

(T,≤), and we remark that 2 is clearly true since −α→ α for all α ∈ Π.

Now, suppose that α, β ∈ Φ+, then sα ≤ sβ in (T,≤). Let d := sα ≤ sα1 ≤

· · · ≤ sαn−1 ≤ sβ be the maximal chain from sα to sβ. Then according to 3.4,

hW,p,M∞(sβ)(i) − hW,p,M∞(sα)(i) is the number of xi-long edges in the chain. By

definition of lW,p,M∞ we have that

hW,p,M∞(sβ)− hW,p,M∞(sα) =lW,p,M∞(sβ)− c(sβ)

2−(lW,p,M∞(sα)− c(sα)

2

)

and by Remark 3.4.2 we know that since sα ≤ sβ then c(sα) = c(sβ) and the result

follows in this case. Similarly, if α, β ∈ Φ− then −α,−β ∈ Φ+ and −β ≤ −α and

the proof follows from above.

This leaves us with the case that α ∈ Φ− and β ∈ Φ+ (notice we cannot have

21

α ∈ Φ+ and β ∈ Φ−). Let αi ∈ Φ− and αi+1 ∈ Φ+, then since αi → αi+1 we

must have −αi = αi+1 (since the only positive root made negative by sγ is γ for

γ ∈ Π). In this case, we have that the edge (αi, αi+1) is p(sαi+1)-long. Note, that

since α ≤ αi+1 ≤ β we have c(sα) = c(sβ) = c(sαi+1) = p(sαi+1

) by Remark 3.4.2

and the definition of c. Then, using the previous argument, we know that the

number of xi-long edges in α→ α1 → · · · → αi is equal to the number of xi-long

edges in −αi → −αi−1 → · · · → −α which is hW,pM∞(sα)(i). Also, the number of

xi-long edges in αi+1 → αi+2 → · · · → β is hW,p,M∞(sβ)(i). Now we have the total

number of xi long edges is given by m(i) where

m : = hW,pM∞(sβ) + hW,pM∞(sα) + c(sαi+1)

=lW,p,M∞(sβ)− c(sβ)

2+lW,p,M∞(sα)− c(sα)

2+ c(sαi+1

)

=lW,p,M∞(sα) + lW,p,M∞(sβ)

2.

3.6 Monotonicity Properties in Dominance Order on T

Recall that using the canonical bijection between Φ+ and T , given by α 7→ sα,

we transport the dominance order on Φ+ to dominance order on T . We have that

sβ sα if and only if β α. This is equivalent to t′ t if for all w ∈ W we have

l(wt) < l(w) implies that l(wt′) < l(w); in this case, we say that t dominates t′.

Lemma 3.6.1. Let r, t ∈ T .

(a) t dominates r in T if and only if for some (respectively every) reflection

group W ′ of W with t, r ∈ W ′, t dominates r in dominance order on the set

W ′ ∩ T of reflections of (W ′, χ(W ′)).

22

(b) t dominates r if and only if either (1) t = r, or (2) 〈t, r〉 is infinite and

hp,W (rtr) < hp,W (t).

(c) t dominates r if and only if either (1) t = r, or (2) 〈t, r〉 is infinite and

lp,W (rt) < lp,W (t) and hp,W (r) 6> hp,W (t).

(d) If r t then t = r if and only if hp,W (r) = hp,W (t).

(e) Let t ∈ T , xi ∈ X, and m = hW,p,M∞(t). Then, m(i) = |t′ ∈ T |t′ ≺

t; p(t′) = xi|.

(f) If r t, then hW,p,M∞(r) ≤ hW,pM∞(t).

Proof. First of all, (a) follows from part 1 of Lemma 2.5.1 via the bijection between

positive roots and T . We simply transfer the structure of dominance by this

bijection and the result holds.

Next, (e) is clear for finite dihedral groups. Now, suppose we have an infinite

dihedral group (W,S) with S = r, s. Then, as in 2.5.1, we know that the

dominance order on T in this case is the coarsest partial order satisfying

s ≺ srs ≺ srsrs ≺ srsrsrs ≺ · · ·

and

r ≺ rsr ≺ rsrsr ≺ rsrsrsr ≺ · · · .

Using this description, (e) is readily checked. Then, for general (W,S), for any

23

xi ∈ X we have

hW,p,M∞(t)(i) =∑


hp,W ′(t)(i)

=∑


|t′ ∈ W ′ ∩ T | t′ ≺W ′,χ(W ′) t; p(t′) = xi|

=∑


| t′ ∈ W ′ ∩ T | t′ ≺ t; p(t′) = xi|

= |t′ ∈ T | t′ ≺ t′; p(t′) = xi|

where the first equality is the definition, the second equality follows from the

dihedral case, the third equality follows from (a), and the final equality follows

from equation (2.3.1).

Now, we can also prove (b) and (c) using this reduction to rank 2 as well. We

note that by the description for dihedral reflection groups above, (b) and (c) hold

by inspection. For the general case of (b) with arbitrary (W,S), suppose that

t, r ∈ T with r t and t 6= r. Let W ′ = 〈r, t〉. Since r t in W ′, according to [4]

we have that W ′ is infinite. The dihedral case shows that we have hp,W ′(rtr) <

hp,W ′(t). This implies that rtr <∅ t and we know c(t) = c(rtr). Therefore, Lemma

3.2.2 implies that hp,W (rtr) < hp,W (t). Conversely, if t 6= r and W ′ = 〈t, r〉 is

infinite with hp,W (rtr) < hp,W (t), then rtr <∅ t so that hp,W ′(rtr) ≤ hp,W ′(t) by

Lemma 3.2.2; from (b) in the dihedral case along with (a), we have r t. Next,

(c) follows in a similar manner. For r t ∈ T with t 6= r, let W ′ = 〈r, t〉. Then

by the dihedral case, lp,W ′(rt) < lp,W ′(t) and hp,W ′(r) 6> hp,W ′(t). Hence rt <∅ t,

and r <∅ t (by height requirement) in W ′ and thus in W as well. Therefore, by

Lemma 3.2.2, lp,W (rt) < lp,W (t) and hp,W (r) 6> hp,W (t). Conversely, if W ′ = 〈r, t〉

infinite and lp,W (rt) < lp,W (t) then rt ≤∅ t and so lp,W ′(rt) < lp,W ′(t). This implies

24

r ≤∅ t or t ≤∅ r (by inspection of infinite dihedral case). Due to the fact that

hp,W (r) 6> hp,W (t), we must have r ≤∅ t and so hp,W ′(r) 6> hp,W ′(t) by Lemma

3.2.2; consequently, r t in W ′ and thus in W by (a). Finally, (d) follows from

(b) and (f) follows from (e) directly.

3.7 A Partition of the Reflections

At this point, we define certain subsets of the set of reflections, T , of a given

Coxeter group (W,S). These sets are analogs of Tn from section 2.4. For any

monomial m ∈M, define Tm := t ∈ T | hW,p,M∞(t) = m.

Proposition 3.7.1. Suppose S is finite. Then Tm is finite for any m ∈M.

Proof. This follows from Theorem 2.4.1 since Tm ⊂ Tdeg(m) and deg(m) ∈ N.

Recall (M,≤) from section 3.1. For any m ∈M, let T≤m :=⋃

n≤m Tn.

3.8 Recurrence Formulae

We now further extend the results of [18]. Recall the reflection cocycle N :

W →P(T ). For any m ∈M and w ∈ W , we define Nm(w) := N(w) ∩ T≤m.

Proposition 3.8.1. Let w ∈ W , s ∈ S, and m ∈M.

(a) If s 6∈ Nm(w) then Nm(sw) = (s ∪ sNm(w)s) ∩ T≤m.

(b) If s ∈ Nm(w) and m ≥ p(s), then Nm−p(s)(sw) = (sNm(w)s \ s) ∩

T≤m−p(s)

Proof. Recall that N(sw) = s + sN(w)s so the containment of the right hand

side into the left hand side is clear for both parts. Now assume that s 6∈ Nm(w).

25

Then s 6∈ N(w). Let t ∈ Nm(sw) = N(sw) ∩ T≤m = (s ∪ sN(w)s) ∩ T≤m, and

then let t′ = sts. If t = s then we clearly have t = s ∈ (s ∪ sNm(s)s) ∩ T≤m. If

t 6= s, then t ∈ sN(w)s and so t′ ∈ N(w). If t = t′ then we already have assumed

t′ = t ∈ T≤m and so this proves that t′ ∈ Nm(w) so that t ∈ sNm(w)s and thus t

is in the right hand side. If 〈s, t〉 is finite or if lp(t) = lp(t′)+2p(s) then by Lemma

3.3.1 we know that hW,p,M∞(t′) ≤ hW,p,M∞(t) ≤ m by assumption that t ∈ T≤m;

thus, t′ ∈ Nm(w) and t ∈ sNm(w)s as well as the right hand side. This only

leaves the case where 〈s, t〉 is infinite and lp(t′) = lp(t) + 2p(s). Let W ′ = 〈t, s〉.

Since s ∈ S, then we know that χ(W ′) = r, s for some r ∈ T . By assumption,

s 6∈ N(t) and t 6= 1 ∈ W ′ so we must have r ∈ N(t). Therefore, we know by the

proof of Lemma 2.5 that t dominates r. We have assumed that t ∈ N(sw) and the

dominance then implies that r ∈ N(sw) as well. Also, by our first assumption, we

know that s ∈ N(sw) as well, but this forms a contradiction since we now have

〈t, s〉 = 〈r, s〉 ⊂ N(sw) (see [17]). However N(sw) is finite, and we know that

〈t, s〉 is infinite. Thus, we have a contradiction proving (a).

To prove part (b), assume that s ∈ Nm(w). Thus, we have that s ∈ N(w)

and so by symmetric difference we must have N(sw) = sN(w)s \ s. Let t ∈

Nm−p(s)(sw) = (sN(w)s \ s) ∩ T≤m−p(s). Set t′ := sts. Clearly we have that

hW,p,M∞(t′) ≤ m − p(s) + p(s) = m so t′ ∈ T≤m. Also, t′ ∈ N(w) by assumption

that t ∈ sN(w)s \ s. Thus, t′ ∈ Nm(w) and so t = st′s ∈ sNm(w)s \ s, and

we already have assumed that t ∈ T≤m−p(s).

3.9 Monoids with Poset Structure

Let (P,≤) be any poset and let M be an additively written commutative

monoid. Suppose we have a partial order ≤M on M such that 0 ≤M m for all

26

m ∈ M and m ≤ n if and only if m + k ≤ n + k for all m,n,k ∈ M. In this

case, we say ≤M is compatible with the monoid structure on M. Furthermore,

let q : P →M\ 0 be a map. Define an operator | − | : Pfin(P ) →M in the

following way, where Pfin(P ) represents all of the finite subsets of P . For finite

A ⊂ P , we have

|A| :=∑a∈A

q(a) (3.9.1)

Remark 3.9.1. Notice that if M = N with the usual total order and q(x) = 1 for

all x ∈ P then this is the usual cardinality of a finite set.

An ideal I of P is a subset of P satisfying the following property: if x ∈ I

and y ≤ x in P then y ∈ I as well. Any subset X ⊂ P is contained in a

smallest (under inclusion) ideal of P called the ideal generated by X. We say

an ideal, I, is principal if I is generated by a single element x with x ∈ P .

We say that P is lower finite if every principal ideal is finite. If we have a lower

finite poset, we define h′ : P → M by h′(x) = |z ∈ P | z < x|. For any

Γ ⊂ M define PΓ := z ∈ P | h′(z) ∈ Γ. We will consider certain cases like

Γ =≤M m := n ∈M| n ≤M m and Γ =<M m := n ∈M| n <M m

Lemma 3.9.2. Let P be a lower finite poset, I be a finite ideal of P and m ∈M.

Then

(a) |I| ≤M m iff |I ∩ P≤Mm| ≤M m.

(b) |I| = m iff |I ∩ P<Mm| = m and I ∩ P<Mm = I ∩ P≤Mm.

Proof. To prove (a), we will instead prove

|I| 6≤M m iff |I ∩ P≤Mm| 6≤M m.

27

The fact that the right hand side implies the left is clear. So assume that |I| 6≤M

m. Now, we define B := I ′ ⊆ I| I ′ is an ideal; |I ′| 6≤M m. Since I is finite, B

is clearly finite and nonempty since I ⊂ B. Let I0 ∈ B be a minimal element of B

(with respect to inclusion). We note that |∅| = 0 and due to the fact that m ≥ 0

for all m ∈M, we must have |I0| 6= ∅. Then, for any maximal element t ∈ I0, we

know that J := I0 \ t is an ideal satisfying J ( I0 ⊂ I. So by the minimality of

I0, we must have |J | ≤M m. Similarly, let K = x ∈ I0| y ∈ I0| y > x = ∅ be

the set of maximal elements in I0, then J ′ = I0 \K must also have |J ′| ≤M m.

So, for any t ∈ I0 we must have that h′(t) = |s ∈ P | s < t| ≤M |I0 \K| =

|J ′| ≤M m. Therefore, t ∈ P≤Mm. So we have shown that I0 ⊂ I ∩ P≤Mm and

thus |I ∩ P≤Mm| ≥M |I0|. If |I ∩ P≤Mm| ≤M m, then this would contradict the

fact that |I0| 6≤M m; thus, we have shown (a).

To show (b), we will use (a). First, suppose that |I| = m. This implies

that |I| 6≤M n for all n M m (where M represents the covering relation for

≤M). Using part (a), we have that |I ∩ P≤Mn| 6≤M n for all n M m. Putting

these all together we get that |I ∩ ∪kMmP≤Mk| 6≤M n for all n M m. Thus,

|I ∩P<Mm| 6≤M n for all nMm. However, we know that I ∩P<Mm ⊆ I so that

|I ∩ P<Mm| ≤M |I| = m, and thus we must have |I ∩ P<Mm| = m. Now, since

|I| = m, we have also by (a) that |I ∩P≤Mm| ≤M m. Then, by our previous work

we know that m = |I∩P<Mm| ≤M |I∩P≤Mm| ≤M m since I∩P<Mm ⊆ I∩P≤Mm.

Since ≤M is compatible with the monoid structure, this implies that |I∩P=m| = 0,

and the fact that q(t) ∈ M>0 thus implies that I ∩ P=m = ∅. Consequently, the

two sets must be equal, I ∩ P<Mm = I ∩ P≤Mm.

Finally, we prove the other direction for (b). Assume that |I ∩ P<Mm| = m

and I ∩ P<Mm = I ∩ P≤Mm. Since |I ∩ P<Mm| = m, it is evident that |I| ≥M m.

28

Suppose, for the sake of contradiction, that |I| >M m. This implies that |I| 6≤M m

and so by (a) we conclude that |I ∩ P≤Mm| 6≤M m. However, this contradicts the

assumption that I ∩ P<Mm = I ∩ P≤Mm, which implies |I ∩ P≤Mm| = m.

3.10 General Length and the Reflection Cocycle

Lemma 3.9.2 applies with P = T under dominance order by Lemma 3.6.1,M

the set of monomials with partial order as in section 3.1, and q = p. Also by

Lemma 3.6.1, we can use h′ = hW,p,M∞ and P≤Mm = T≤m defined in the previous

section as they are compatible with our earlier notation for T .

Corollary 3.10.1. Let m ∈M and x, y ∈ W .

1. lp,W (x) ≤m iff |Nm(x)| ≤m.

2. lp,W (x) = m iff |Nm(x)| = m and ∪nmNn(x) = Nm(x).

3. lp,W (xy) ≥ lp,W (x) + lp,W (y)− 2m iff |Nm(x−1) ∩Nm(y)| ≤m.

4. lp,W (xy) < lp,W (x) + lp,W (y)− 2m if |Nm(x−1) ∩Nm(y)| > m.

5. lp,W (xy) = lp,W (x) + lp,W (y) − 2m iff |Nm(x−1) ∩ Nm(y)| = m and

∪nm (Nn(x−1) ∩Nn(y)) = Nm(x−1) ∩Nm(y).

Proof. First, we note that if w = r1...rn is a reduced expression, then N(w) =

ti| i ∈ 1, ..., n where ti := r1...ri−1riri−1...r1 (see [19]). From this we see clearly

that p(ri) = p(ti) so that we have lp,W = |N(w)|. Also, we remark that N(x) is an

ideal of T under dominance order by definition, and so for any finite family (xi)i∈I

of elements of W , the intersection ∩iN(xi) is a finite ideal. From these remarks,

(1) and (2) follow directly from the previous lemma.

29

The points (3) and (5) also follow using the cocycle condition N(xy) = N(x)+

xN(y)x−1 where + denotes symmetric difference. This condition along with the

interpretation of N(x) in terms of lp,W (x) imply that

lp,W (xy) = lp,W (x) + lp,W (y)− 2|N(x−1) ∩N(y)|.

Then let I = N(x−1) ∩N(y) and the Lemma implies (3) and (5).

Finally, if |Nm(x−1) ∩Nm(y)| > m then by (3) we know lp,W (xy) 6≥ lp,W (x) +

lp,W (y) − 2m. However, we also know that |N(x−1) ∩ N(y)| = k ≥ |Nm(x−1) ∩

Nm(y)| > m due to the containment of sets. So since

lp,W (x) + lp,W (y) + 2m < lp,W (x) + lp,W (y) + 2k

we have that

lp,W (xy) = lp,W (x) + lp,W (y)− 2k < lp,W (x) + lp,W (y)− 2m

as required for (4).

We finish this section with a lemma that has the same ideas as the previous

lemma.

Lemma 3.10.2. Suppose that r, s ∈ S and x, y ∈ W with lp,W (xr) > lp,W (x) and

lp,W (ys) > lp,W (y). If ysy−1 xrx−1, then lp,W (y−1x) ≤ lp,W (x) + lp,W (y) − 2n

where n = hW,p,M∞(ysy−1).

Proof. We have s = y−1ysy−1y y−1xrx−1y since lp,W (ys) > lp,W (y). Therefore,

we know that lp,W (y−1xr) = lp,W (y−1x) + lp(r). Thus, by the exchange condition

we know that lp,W (sy−1xr) ≥ lp,W (y−1x) + lp(r)− lp(s).

30

Now, ysy−1 xrx−1 and xrx−1 ∈ N(xr) so we must have ysy−1 ∈ N(xr).

Also, ysy−1 ∈ N(ys) by definition. So

ysy−1 ∈ N(xr) ∩N(ys) = N(xr) ∩N((sy−1)−1).

Therefore, we see that if n = hW,p,M∞(ysy−1), we have |N((sy−1)−1) ∩ N(xr)| ≥

n + p(s) using Lemma 3.6.1. As in the proof of (4) from the previous lemma, this

implies that lp,W (sy−1xr) ≤ lp,W (sy−1) + lp,W (xr)− 2(n + p(s)). Putting together

the information we have, we get

lp,W (y−1x) ≤ lp,W (sy−1xr)− lp(r) + lp(s)

≤ lp,W (xr) + lp,W (sy−1)− 2(n + p(s))− lp(r) + lp(s)

≤ lp,W (x) + lp(r) + lp,W (y) + lp(s)− 2n− 2p(s)− lp(r) + lp(s)

= lp,W (x) + lp,W (y)− 2n.

31

CHAPTER 4

FINITE STATE AUTOMATA

In this chapter, we begin by introducing some standard terminology involving

finite state automata. For the basic theory of using automata in groups, we use

[21]. In [5], Brink and Howlett described a method of creating a finite state

automata that would recognize any finitely generated Coxeter system. For any

finite Coxeter system, the corresponding automata can be described by using the

Cayley graph of the group.

More recently, Dyer ([18]) has demonstrated a much larger class of finite state

automata that can recognize many different subsets of Coxeter systems as well.

We generalize the results of Dyer using generalized length functions from Chapter

3. These length functions give rise to large class of finite state automata, which

are more powerful than the family of automata introduced by Dyer. We then

describe the corresponding languages (subsets of the free monoid S∗) recognized

by our automata. These automata will be used in Chapter 5 to show that many

subsets of Coxeter systems are regular.

4.1 Regular Languages

We use this section to introduce the idea of a regular language of a monoid.

Let M be a monoid. By a (left) M-set, we mean a set A along with a function

32

M × A → A given by (m, a) 7→ ma such that 1Ma = a for all a ∈ A if 1M is the

identity of M and m1(m2a) = (m1m2)a for all a ∈ A and mi ∈ M . In this case,

we say that M acts on A.

A subset M ′ of M is called (monoid) regular if there is a left M -set A with

A finite, an element a0 ∈ A and a subset Af ⊂ A with the property that M ′ =

m ∈ M | ma0 ∈ Af. The following lemma is well known (see [21]). We include

a proof here for completeness.

Lemma 4.1.1. The family of regular subsets of M forms a Boolean subalgebra

R(M) of P(M). If M is finitely (or countably) generated, then R(M) is count-

able.

Proof. Suppose that M1 and M2 are both monoid regular subsets of M . We will

demonstrate that M c1 = M \M1, M1 ∪M2 and M1 ∩M2 are all monoid regular as

well.

Since Mi is monoid regular, there exists M -sets A1 and A2 with ai ∈ Ai and

(Ai)f ⊂ Ai such that Mi = m ∈ M | ma0 ∈ (Ai)f. Now, A1 is finite and so

A′f = A1 \ (A1)f is also finite. Then M c1 = m ∈M | ma1 ∈ A′f so M c

1 is regular.

Next, let B = A1×A2 and let m(a, a′) = (ma,ma′) be the action of M on B. Note

that B is a finite set. Then M1 ∩M2 = m ∈ M | ma1 ∈ (A1)f ;ma2 ∈ (A2)f =

m ∈M | m(a1, a2) ∈ (A1)f × (A2)f. Let Bf = (A1)f × (A2)f and then M1∩M2

is regular. Finally, M1 ∪M2 = m ∈ M | ma1 ∈ (A1)f ; or ma2 ∈ (A2)f. Let

B′f = (A1)f×A2∪A1×(A2)f ⊂ B and then M1∪M2 = m ∈M | m(a1, a2) ∈ B′f

so M1 ∪M2 is regular.

Now, let S be a set, called an alphabet. Elements of the free monoid S∗ on

S are called words. We call subsets of S∗ languages. Let N be an additively

written commutative monoid and let map L′ : S → N be a map. Then, for

33

any word sn...s1 ∈ S∗, we can define the length ` : S∗ → N by `(sn...s1) :=

L′(sn) + · · · + L′(s1). For example, if we let N = N and L′(s) = 1 for all s ∈ S,

then `(sn...s1) = n. Also, for any word w = sn...s1 ∈ S∗ we define the transpose

† : S∗ → S∗ by †(w) := w† := s1...sn.

Now, if A is a finite S∗-set, then we can describe the action by restriction of

the multiplication map S∗ × A → A to a function µ : S × A → A. Typically, µ

is called the transition function of the S∗-set A. Since the transition function µ

can be an arbitrary function, we get a bijection between functions µ : S ×A→ A

where A is a set and S∗-sets A. If S is a finite set, then we define a regular

language on S to be a monoid regular subset of S∗. Therefore, a regular language

is determined by a tuple (µ : S × A → A, a0, Af ) where A is a finite set, µ is

a function, a0 ∈ A and Af ⊆ A. The language determined by previous tuple is

L = m ∈ S∗| ma0 ∈ Af where the S∗ action is determined by µ.

4.2 Automata

We can equivalently describe the monoid regular subsets of S∗ by the termi-

nology of finite state automata. A finite state automaton for the alphabet S is a

tuple (A, a0, µ : S × A → A,Af ) such that A is a finite set with a distinguished

element a0 called the initial state, a finite subset Af ⊆ A called the accepting

states (or final states) and a transition function µ. The automaton reads a word

w = sn...s1 from right to left, one letter at a time. The machine begins in the

initial state a0; upon reading a letter si, the state moves from the current state,

a, to the state µ(si, a). We say the automaton accepts a word w = sn...s1 if after

reading w, the current state of the automaton is a member of Af . The set of words

accepted by the automaton is called the language accepted by the automaton. It

34

is clear that the language accepted by the automaton (A, a0, µ : S × A→ A,Af )

corresponds to the regular subset of S∗ associated to A viewing A as an S∗-set.

Example 4.2.1. Consider the affine Weyl group of type A1 = 〈r, s | r2 = s2 = 1〉.

We know that the set of reduced expressions in A1 is given by

e, r, s, rs, sr, rsr, srs, rsrs, srsr, ....

Below, we have a finite state automaton that recognizes the set of reduced expres-

sions of words in A1. A is the vertex set of the given graph, a0 is the blue vertex,

and Af is the set of non-red vertices. µ : S×A→ A is given by the labeled edges.

•

•

•

•

r

::tttttttttttttttttts

r

EE

s

$$JJJJJJJJJJJJJJJJJJ

r

$$JJJJJJJJJJJJJJJJJJ

s

::tttttttttttttttttt

r,see

4.3 Canonical Automata for Coxeter Systems

At this point, we want to describe some finite state automata for Coxeter

systems.

Almost all objects defined in the rest of this chapter depend on the choice of p

as in 3.2. We indicate when defining anything if it is dependent on p by including

an appropriate subscript or superscript. However, when using the terminology,

if there is no confusion about which function p we are referring to, we will leave

off the corresponding subscript or superscript for ease of notation. Also, since

(W,S) will be fixed, we let lp := lp,W . Recall, that lp takes values in M (which

35

depends on p) and that M has a partial order, which is described in section 3.1.

In this chapter, we view M additively; therefore, M ∼= Nk and M ⊂ M′ ∼= Zk

(see section 3.1).

Let (W,S) be a Coxeter system and S∗ be the free monoid on S. We denote

the natural monoid homomorphism from S∗ to W extending the identity map on

S by ν : S∗ → W . Let `p : S∗ →M be the length of w = s1 · · · sn ∈ S∗ given by

`p(s1 · · · sn) = p(s1) + · · ·+ p(sn). Note `p(w) ≥ lp,W (ν(w)).

We associate to (W,S) an S∗-set A = A(W,S). As a set we have

A(W,S,p) := A = ∞ ∪ (m, A)| m ∈M, A ⊆ T≤m.

The S∗-action can be described by its restriction to a function S × A → A

given by (s,∞) 7→ ∞ and

(s, (m, A)) 7→

∞, if p(s) 6≤m and s ∈ A(m− p(s), (sAs \ s) ∩ T≤(m−p(s))

), if p(s) ≤m and s ∈ A

(m, (sAs ∪ s) ∩ T≤m) , if s 6∈ A

We may define sub-S∗-sets An = A(W,S,p,n) of A for n ∈M defined by

An := ∞ ∪ (m, A)| m ≤ n, A ⊆ T≤m

We know that if S is finite then An is a finite S∗-set.

36

Lemma 4.3.1. Let x ∈ W , w ∈ S∗, and m ∈M. Then w · ∞ =∞ and

w · (m, Nm(x)) 7→

∞, if k 6≤m

(m− k, Nm−k(ν(w)x)) , if k ≤m

in A , where k := lp(x)+`p(w)−lp(ν(w)x)

2

Proof. This follows by induction on `p(w) since it is clearly true for each w =

s ∈ S. Then suppose that w = sw′ and the result is known for sw′. Then, by

definition of the S-action on pairs, the result is clear.

Finally, from the previous lemma, we can define

M = M(W,S,p) := ∞ ∪ (m, Nm(x))| m ∈M, x ∈ W

and M(W,S,p,n) = Mn = M ∩An for n ∈ M, which are all sub-S∗-sets of A . We

call these the canonical and n-canonical S∗-sets for (W,S) respectively.

4.4 Some Subsets of S∗

Let Aa,b := w ∈ S∗| w · a = b for any a, b ∈ M. By Lemma 4.3.1, we see

that A∞,∞ = S∗ and A∞,x = ∅ for x 6= ∞. We now intend to describe the sets

Aa,b. To do this, we introduce some subsets of S∗ in the following way.

To begin with, for any word w ∈ S∗, we define

Lp(w) :=`p(w)− lp(ν(w))

2∈M.

For any m ∈ M, we can then define S∗m := S∗p,m = w ∈ S∗| Lp(w) = m. The

elements of S∗m will be called the m-nonreduced words of (W,S). If S is finite,

37

then for any w ∈ W , there are only finitely many m-nonreduced expressions of w,

namely the elements of the set ν−1(w) ∩ S∗m.

Next, for x, y ∈ W and n ∈M′, we define more general sets

F px,y,n := w ∈ S∗| lp(yν(w)x) = lp(y) + `p(w) + lp(x)− 2n (4.4.1)

and also define, for m ∈M′

F px,y,n,m := S∗m ∩ F

px,y,n+m. (4.4.2)

With this notation, we see that S∗m = F1,1,m.

Remark 4.4.1. It is clear from the definition that F px,y,n = ∅ if n 6∈ M, i.e. if

n < 0.

Example 4.4.2. For s ∈ S, we have that the set F1,s,lp(s) is given by

w ∈ S∗|lp(sw) = lp(s) + `p(w)− 2lp(s) = `p(w)− lp(s),

which can also be described by

[ν−1(w ∈ W | lp(sw) < lp(w)) ∩ S∗0]∪[ν−1(w ∈ W | lp(sw) > lp(w)) ∩ S∗lp(s)].

Next for any x ∈ W , k ∈M′ and finite subsets A,B of T , we let

Hpx,k,A,B := w ∈ S∗| lp(ν(w)x) = `p(w)+ lp(x)−2k; N(ν(w)x)∩A = B (4.4.3)

so it is clear the Hpx,k,A,B = ∅ unless k ∈ M, B ⊆ A, and there exists a y ∈ W

such that N(y) ∩ A = B.

38

According to Lemma 4.3.1, we see that for m ∈M and x ∈ W we get

A(m,Nm(x)),∞ =⋃k 6≤m

Fx,1,k

and then for m,n ∈M,

A(m,Nm(x)),(n,B) =

∅ if n 6≤m

Hx,m−n,T≤n,B if n ≤m(4.4.4)

where both sides are empty unless B is of the form Nn(y) for some y ∈ W .

4.5 The Boolean Algebra of Regular Sets of S∗

At this point, we assume that S is finite. Then, for a, b ∈ M, we can view

Aa,b in the following way. We choose m large enough so that a, b ∈ Mm. Then

Aa,b is the regular language accepted by an automaton where Mm is the state set,

the transition function described in 4.3.1, initial state a, and single accept state

b. At this stage, we characterize the Boolean subalgebra of S∗ generated by all

Aa,b where a 6=∞, b 6=∞.

Lemma 4.5.1. The following Boolean subalgebras of S∗ are all equal:

1. The Boolean subalgebra Dp of P(S∗) generated by all sets Aa,b where a, b ∈

M \ ∞.

2. The Boolean subalgebra D ′p of P(S∗) generated by all sets Hx,k,A,B.

3. The Boolean subalgebra D ′′p of P(S∗) generated by all sets Fx,y,k,m.

4. The Boolean subalgebra D ′′′p of P(S∗) generated by all sets Fx,y,k.

39

Moreover, Dp is stable under the automorphism † : S∗ → S∗ given by †(sn....s1) =

s1...sn.

Proof. According to equation (4.4.4), we know that Dp is generated by all the sets

Hx,k,T≤n,B where x, y ∈ W , n,k ∈ M and B ⊂ T is finite. Suppose that k ∈ M,

x ∈ W , and A,B ⊆ T are finite sets. We choose an element n ∈ M such that

A ⊆ T≤n. Then

Hx,k,A,B =⋃

B′⊆T≤n

B′∩A=B

Hx,k,T≤n,B′

and so Hx,k,A,B is generated by a finite union of Aa,b for some a, b ∈ M \ ∞.

Thus Dp = D ′p.

Next, we note that for w ∈ Fx,y,k,m we get (since w ∈ S∗m)

lp(yν(w)x) = lp(y) + `p(w) + lp(x)− 2(k + m)

= lp(y) + lp(ν(w)) + 2m + lp(x)− 2(k + m)

and so lp(yν(w)x) = lp(y) + lp(ν(w)) + lp(x)− 2k. This implies that Fx,y,k,m = ∅

unless 0 ≤ k ≤ lp(y) + lp(x). This shows that S∗m = ∪0≤k≤lp(x)+lp(y)Fx,y,k,m and

that we also have

Fx,y,k =⋃

k−lp(x)−lp(y)≤m≤k

Fx,y,k−m,m

thus showing that D ′′p = D ′′′p since it is clear that Fx,y,k,m = S∗m ∩ Fx,y,k+m =

F1,1,m ∩ Fx,y,k+m.

Now, we show that D ′′′p ⊆ D ′p. For fixed x ∈ W and A ⊆ T finite, we have

S∗ =⋃k∈M

⋃B⊆A

Hx,k,A,B

Recall from section 3.10 the terminology |A| ∈ M for A ⊆ T finite. Let y ∈ W .

40

Then for w ∈ Hx,k,A,B we have

lp(yν(w)x) = lp(y) + lp(ν(w)x)− 2|N(ν(w)x) ∩N(y−1)|

= lp(x) + `p(w) + lp(y)− 2k− 2|N(ν(w)x) ∩N(y−1)|.

From this we can see that

Fx,y,m =⋃

k,n∈M:k+n=m

⋃B⊆N(y−1):|B|=n

Hx,k,N(y−1),B.

This union is finite since there are only finitely many pairs k,n inM less than m

and N(y−1) is a finite set, and so D ′′′p ⊆ D ′p.

So we now must show that D ′p ⊆ D ′′′p. Let x, y ∈ W and w ∈ S∗, then we

have

Fx,1,k ∩ Fx,y,m = w ∈ W | lp(ν(w)x) = `p(w) + lp(x)− 2k and

lp(yν(w)x) = lp(y) + `p(w) + lp(x)− 2m,(4.5.1)

and by replacing `p(w) + lp(x) = lp(ν(w)x) + 2k in the second equation (using the

first equation) we get

Fx,1,k ∩ Fx,y,m = w ∈ W | lp(ν(w)x) = `p(w) + lp(x)− 2k and

lp(yν(w)x) = lp(ν(w)x) + lp(y)− 2m + 2k.(4.5.2)

Now, we know that lp(ν(w)x) ≤ lp(y) + lp(yν(w)x) ≤ lp(y) + lp(y) + lp(ν(w)x)

consequently 0 ≤ lp(y)+lp(yν(w)x)−lp(ν(w)x) ≤ 2lp(y). Substituting appropriate

equations from (4.5.1) and (4.5.2), we note that Fx,1,k ∩ Fx,y,m is empty unless

0 ≤ 2lp(y)− 2m + 2k ≤ 2lp(y) and this implies that k ≤m ≤ k + lp(y). We also

have ∪m∈MFx,y,m = S∗. Now, let y from above be a given reflection y = t. Then

41

we have that the following sets

Lx,1,k,t := w ∈ Fx,1,k| t ∈ N(ν(w)x) = Fx,1,k ∩⋂

m: k≤m≤k+lp(t)0≤lp(t)≤2m−2k

Fx,t,m

and

L′x,1,k,t := w ∈ Fx,1,k| t 6∈ N(ν(w)x) = Fx,1,k ∩⋂

m: k≤m≤k+lp(t)lp(t)≥2m−2k∈M

Fx,t,m

are clearly in D ′′′p since these intersections are finite. If B ⊆ A, we thus get

Hx,k,A,B =

(⋂t∈B

Lx,1,k,t

)∩

⋂t∈A\B

L′x,1,k,t

,

and as we have noted before, Hx,k,A,B = ∅ if B 6⊆ A. Therefore, Hx,k,A,B is in D ′′′p ,

which is what we were trying to show.

Finally, suppose w ∈ Fx,y,m for some m. Now, we have that ν(†(w)) = ν(w)−1.

Since lp(yν(w)x) = lp(y) + `p(w) + lp(x)− 2m then we can reverse this to see that

we must also have lp(x−1ν(†(w))y−1) = lp(x

−1) + `p(†(w)) + lp(y−1)− 2m and so

†(w) ∈ Fy−1,x−1,m, which shows that Dp is stable under †.

4.6 The Reflection Cocycle and Automata

Recall that for A ⊆ T a finite subset, we write |A| =∑

t∈A p(t). Thus, we

note that lp(w) = |N(w)|. Now, we state a lemma that describes the essence

of the previous proof. The problem of determining, for all w ∈ W , the value of

lp(xw)− lp(w) for all x in some given finite subset of W is equivalent to finding, for

42

all w ∈ W , the intersection N(w)∩A for some given finite subset A of reflections.

Recall that M′ = Zk if M = Nk.

Lemma 4.6.1. (a) Let x1, ..., xn be elements of W , and l1, ..., ln ∈ M′. Let

A := ∪ni=1N(x−1i ). Then for w ∈ W , lp(xiw)− lp(w) = li for all i ∈ 1, ..., n

if and only if |(N(w) ∩ A) ∩N(x−1i )| = lp(xi)−li

2for all i ∈ 1, ..., n.

(b) Let A be a finite subset of T . Then for w ∈ W , we have

N(w) ∩ A = t ∈ A| lp(tw) = lp(w)− k for some k ∈M≤lp(t)

Proof. Recall that N(xy) = N(x) + xN(y)x−1 = x(N(x−1) + N(y))x−1. This

implies that

lp(xy) = lp(x) + lp(y)− 2|N(x−1) ∩N(y)|

where |−| is as above. Therefore, for w ∈ W , if xi ∈ W , we have lp(xiw)− lp(w) =

lp(x) − 2|N(x−1i ) ∩ N(w)|. Since N(w) ∩ N(x−1) = (N(w) ∩ A) ∩ N(x−1

i ), then

lp(xiw) − lp(w) = li if and only if lp(xi) − 2|(N(w) ∩ A) ∩ N(x−1i )| = li, and (a)

follows.

Then, (b) follows since N(w) = t ∈ T | lp(tw) < lp(w). We have lp(tw) =

lp(t) + lp(w)− 2|N(t) ∩N(w)|, we see that lp(t)− 2|N(t) ∩N(w)| = −k for some

k ∈M.

43

CHAPTER 5

REGULAR SUBSETS OF COXETER GROUPS

This chapter is devoted to the definition of various notions of regularity in

Coxeter systems. We introduce three different types of regularity. The strongest

type of regularity, p-complete regularity, will be examined in detail, and we will

describe many subsets of Coxeter systems that are p-completely regular. The set

of m-nonreduced expressions of elements of any p-completely regular subset of W

is a regular language in S∗ for all m ∈ M. This implies that the multivariate

Poincare series of p-completely regular subsets are rational, providing a natural

explanation (and strengthening) of known rationality results on Poincare series of

many natural subsets of W . We also use the results in this section to describe

regularity of subsets in products of Coxeter systems.

5.1 Boolean Algebra of Regular Sets in W

For any x, y ∈ W and m ∈M′, we can now define a subset

Wx,y,m := W px,y,m = w ∈ W | lp(ywx) = lp(y) + lp(w) + lp(x)− 2m ⊂ W.

We have that W =⋃

0≤k≤lp(x)+lp(y)

Wx,y,k. Notice that Fx,y,k,m = ν−1(Wx,y,k) ∩ S∗m

so that Dp is the Boolean subalgebra of P(S∗) generated by all intersections

ν−1(Wx,y,k) ∩ S∗m.

44

Remark 5.1.1. Again, it is clear that Wx,y,m = ∅ if m 6∈ M. This allows us to

only consider sets Wx,y,m with m ≥ 0.

Let Bp(W ) denote the Boolean subalgebra of P(W ) generated by all the sets

Wx,y,k with x, y ∈ W and k ∈M. Then Dp consists of all the sets ∪m∈Mν−1(Bm)∩

S∗m where Bm ∈ Bp(W ) for all m and Bm is either ∅ for all but finitely many

m or Bm is W for all but finitely many m. Thus, we see that Dp is uniquely

determined by the sets Bp(W ) together with the set of m-nonreduced expressions

of w, for all m ∈M and w ∈ W .

Example 5.1.2. Let J ⊆ S be a spherical subset, i.e. WJ is a finite standard

parabolic subgroup. Then, let wJ be the longest element of WJ and suppose

lp(wJ) = k. Then D(J) := w ∈ W | DL(w) = J ∈ Bp(W ) since it is the set

w ∈ W | lp(wJw) = lp(wJ) + lp(w)− 2k = lp(w)− lp(wJ) = W1,wJ ,k. This set is

known as the descent class associated to J .

5.2 Coxeter Groups are Automatic

The following result is due to Brink and Howlett, [5], in their proof that Coxeter

groups are automatic.

Theorem 5.2.1. Fix a total order of S and give S∗ the corresponding reverse

lexicographic order L. Then for w ∈ W define w as the minimal (under L)

element of the non-empty and finite set ν−1(w) ∩ S∗0. Then W = w| w ∈ W is

a regular subset of S∗.

5.3 Different Types of Regularity

Fix any map µ : W → S∗ with the following properties:

45

(i) For all w ∈ W , µ(w) ∈ S∗0

(ii) ν µ = IdW .

(iii) µ(W ) is a regular subset of S∗.

Properties (i) and (ii) require that for w ∈ W , we have µ(w) is a reduced

expression of W , and we know such a map exists due to Theorem 5.2.1.

We say that a subset A of W is µ-regular if µ(A) is a regular subset of S∗.

Definition 5.3.1. 1. A subset A of W is weakly regular if there is some map

w 7→ w : A → S∗0 such that ν(w) = w for all w ∈ A and w| w ∈ A is a

regular subset of S∗.

2. A subset of W is regular (resp. p-strongly regular) if ν−1(A)∩S∗m is a regular

subset of S∗ for m = 0 (resp. for all m ∈M).

3. A subset of W is called p-completely regular if A ∈ Bp(W ).

Example 5.3.2. Due to example 5.1.2, D(J) is p-completely regular.

Lemma 5.3.3. (a) Any p-strongly regular subset of W is regular.

(b) If A is a regular subset of W , then A is µ-regular for any map µ satisfying

conditions (i)-(iii) above.

(c) If A is µ-regular for some µ, then A is weakly regular.

Proof. Parts (a) and (c) follow directly from the definitions. To show part (b),

we assume that A is a regular subset of W . This means that ν−1(A) ∩ S∗0 is a

regular subset of S∗. Additionally, since µ satisfies (i)-(iii), we know that µ(W ) is

a regular subset of S∗ as well. Now, (b) follows since µ(A) = µ(W )∩(ν−1(A) ∩ S∗0)

which is a finite intersection of regular sets and is thus regular.

46

5.4 p-Regularity for Different p

Suppose that p : T → Xp and p′ : T → Xp′ are two different functions

satisfying the properties of section 3.2 with corresponding sets of indeterminates

Xp and Xp′ respectively. Additionally, let Mp and Mp′ be the corresponding

sets of monomials for Xp and Xp′ . Since Mp and Mp′ are the free commutative

monoids on Xp and Xp′ respectively, then for any map ϕ : Xp → Xp′ we get

a corresponding map, denoted ϕ, with ϕ : Mp → Mp′ given by the universal

property. We note that deg(m) = deg(ϕ(m)) for any m ∈Mp.

Proposition 5.4.1. Let ϕ : Xp → Xp′ be a surjective map such that ϕ(p(t)) =

p′(t) for all t ∈ T . Then the following are true:

1. Dp′ ⊂ Dp, and

2. Bp′(W ) ⊂ Bp(W ).

Proof. For q ∈ p, p′, Dq is generated by the sets F qx,y,m with x, y ∈ W and

m ∈Mq according to Lemma 4.5.1. Now, since ϕ is surjective then ϕ is surjective

as well. Let m ∈ Mp′ . Then define Q := ϕ−1(m). By surjectivity, Q 6= ∅ and

since ϕ preserves degree, |Q| <∞. Then, it follows that

F p′

x,y,m =⋃k∈Q

F px,y,k ∈ Dp

and therefore Dp′ ⊂ Dp, proving 1. Then, 2 is analogous replacing F qx,y,m by

W qx,y,m.

47

5.5 Some Properties of p-Completely Regular Sets

In this section we describe some of the basic properties of p-completely regular

sets.

Proposition 5.5.1. (a) Any p-completely regular set is p-strongly regular (and

thus also regular and µ-regular for any µ satisfying (i)-(iii) in section 5.3.

(b) For any finite subsets B,C ⊆ A of T the subset

GA,B,C := w ∈ W | N(w) ∩ A = B, N(w−1) ∩ A = C

of W is p-completely regular.

(c) If A ∈ Bp(W ) then A−1 := w−1| w ∈ A ∈ Bp(W ).

(d) Bp(W ) is closed under action by automorphisms of (W,S) that commute

with p, i.e. automorphisms satisfying θ(p(t)) = p(θ(t)) for all t ∈ T .

(e) Bp(W ) is closed under left and right multiplication by elements of W .

(f) Bp(W ) contains all finite subsets of W .

(g) Bp(W ) is countable.

Proof. Suppose that A is p-completely regular. Then as we have seen ν−1(A) ∩

S∗m ∈ Dp for all m ∈ M. But Dp contains only regular languages by Lemma

4.5.1 part (1), which proves (a). To prove (b), we let Wt := w ∈ W | lp(tw) <

lp(w) =⋃

lp(t)−c(t)2

<k≤lp(t)W1,t,k ∈ Bp(W ) and W ′

t := w ∈ W | lp(tw) > lp(w) =⋃0≤k< lp(t)+c(t)

2

W1,t,k ∈ Bp(W ) since both are finite unions. We also have sets

Xt := w ∈ W | lp(wt) < lp(w) and X ′t := w ∈ W | lp(wt) > lp(w), and both

48

are in Bp(W ) using finite unions of appropriate sets Wt,1,k. Now, since A is finite,

we get

YB :=⋃t∈B

Wt ∩⋃

t∈A\B

W ′t

YC :=⋃t∈C

Xt ∩⋃

t∈A\C

X ′t

and then GA,B,C := YB ∩ YC . In particular, this process describes an application,

in detail, of Lemma 4.6.1.

Next, (c) holds since (Wx,y,k)−1 = w−1| lp(ywx) = lp(y)+lp(w)+lp(x)−2k =

w ∈ W | lp(yw−1x) = lp(y) + lp(w) + lp(x) − 2k = w ∈ W | lp(x−1wy−1) =

lp(x−1) + lp(w) + lp(y

−1)− 2k = Wy−1,x−1,k.

Now, for any automorphism, θ, of (W,S) commuting with p, we have lp(w) =

lp(θ(w)) for all w ∈ W . Then, we see that if w ∈ Wx,y,k then lp(θ(ywx)) =

lp(ywx) = lp(y) + lp(x) + lp(w)− 2k = lp(θ(y)) + lp(θ(x)) + lp(θ(w))− 2k so that

θ(Wx,y,k) ⊂ Wθ(x),θ(y),k. Using the same argument with the automorphism θ−1

demonstrates that in face θ(Wx,y,k) = Wθ(x),θ(y),k, proving (d).

Next, (e) holds using the following observations. Let a ∈ W and b ∈ W

and x, y ∈ W . Then lp(ya−1) = lp(y) + lp(a

−1) − 2P for some P ≥ 0 and

lp(b−1x) = lp(x) + lp(b

−1)− 2Q for some Q ≥ 0. If w ∈ Wx,y,k we have lp(ywx) =

lp(y) + lp(w) + lp(x) − 2k. Then awb ∈ W satisfies lp(ya−1awbb−1x) = lp(y) +

lp(w)+ lp(x)−2k = lp(ya−1)− lp(a−1)+2P+ lp(w)− lp(b−1)+ lp(b

−1x)+2Q−2k =

lp(ya−1) + lp(awb) − 2n + lp(b

−1x) + 2P + 2Q − 2k where 0 ≤ n ≤ lp(a) + lp(b)

and awb ∈ Wb−1,a−1,n. Therefore we have

aWx,y,kb =⋃

0≤n≤lp(a)+lp(b)

(Wb−1x,ya−1,k−P−Q+n ∩Wb−1,a−1,n)

49

where we recall that Wu,v,k = ∅ if k < 0.

Finally, (f) holds since we have GS,∅,∅ = 1 ∈ Bp(W ) by (b). Then for any

w ∈ W , w1 = w ∈ Bp(W ) by (e), and so any finite union of singleton

elements of W is in Bp(W ). Then, (g) follows from the nature of the fact that

there are countably many sets Wx,y,k and we can only take finite unions and

intersections.

Example 5.5.2. For any J ⊆ S spherical, the subgroup WJ is p-completely

regular by (f). Since W = W1,1,0, then W is p-completely regular. If (W,S) is

finite, we can easily demonstrate that (W,S) is regular by letting the Cayley graph

of (W,S) be the corresponding S∗-set.

Remark 5.5.3. Given a completely regular set A ⊆ W , described explicitly as

a finite union of finite intersections of the sets W px,y,k, we could construct an

automaton on S accepting ν−1(A) ∩ S∗n for any n ∈ M using Lemma 4.5.1. In

addition, following from Lemma 5.3.3, we could construct an automaton on S

accepting µ(A) for any µ satisfying the (i)-(iii) from 5.3.3.

5.6 Poincare Series and Examples of Regular Subsets of W

The previous result leads to some results about the rationality of Poincare

series of subsets of Coxeter groups. First we recall the definition of a Poincare

series.

Definition 5.6.1. Let u be an indeterminate and consider the power series ring

Z[[u]]. For a subset A ⊆ W we define the Poincare series of A to be

P (A;W ) =∑w∈A

udeg(lp(w)).

50

If we consider the multiplicative version of M generated by X, then we can con-

sider the completion Z[[X]] of Z[X]. We define the multivariate Poincare series of

A to be

Pp(A;W ) =∑w∈A

lp(w).

The following examples are previously known in the case where lp = l is the

standard length; we use our results to show the examples are p-completely regular

for general p as well.

Example 5.6.2. 1. For any finitely generated reflection subgroup W ′ and any

standard parabolic subgroup WJ of (W,S), any double coset W ′wWJ has

a unique element of minimal length determined by the conditions N(w) ∩

χ(W ′) = ∅ and N(w−1)∩J = ∅ where χ(W ′) is the set of canonical Coxeter

generators of (W,S). In particular the set of these shortest coset represen-

tatives, denoted W ′ \W/WJ is thus given by the following:

W ′ \W/WJ =⋃

C⊂χ(W )

Gχ(W ′),∅,C ∩⋃B⊂J

GJ,B,∅

and so is p-completely regular. This applies to shortest coset representatives

W ′ \W of W and to shortest double coset representatives WK \W/WJ of

standard parabolic subgroups.

2. The weak right order ≤ on W is defined by x ≤ y if N(x) ⊆ N(y) or

equivalently if lp(x−1y) = lp(y)−lp(x). Thus, if we fix x ∈ W with lp(x) = m,

the set

W1,x−1,m = y ∈ W | lp(x−1y) = lp(y) + lp(x)− 2lp(x) = y ∈ W | y ≥ x

51

is the upper rays in weak right order, and so this set is p-completely regular.

A similar argument shoes that the upper rays in weak left order are also

p-completely regular.

3. As we have seen in the proof of Proposition 5.5.1 the sets w ∈ W | lp(tw) <

lp(w) and w ∈ W | lp(wt) < lp(w) are p-completely regular for fixed

t ∈ T . These spaces (and their complements) are known as left and right

half spaces respectively. It follows that any finite intersection and union of

left (or right) half spaces is p-completely regular. See Figure 5.1 for example.

All of the sets in the previous examples thus have rational Poincare series and

multivariate Poincare series due to a well-known result, called the transfer matrix

method (see [2] or [28]), which states that regular languages have rational Poincare

series.

5.7 More Regular Subsets of W

The sets described in this section are shown to be p-completely regular in [18]

for the case where p : T → N with p(t) = 1 for all t. We now prove that these

sets are also p-completely regular for general p.

Recall that W acts as a group of permutations on the set T ×±1 such that

for s ∈ S, ε ∈ ±1, and t ∈ T \s we have s(s, ε) = (s,−ε) and s(t, ε) = (sts, ε).

We note that as a W × ±1-set, T × ±1 is isomorphic to the standard root

system Φ, with Φ+ corresponding to T ×1 and simple roots Π corresponding to

S × 1. With this terminology, we have the following p-completely regular sets.

Proposition 5.7.1. 1. For u, v ∈ T , W (u, v) = w ∈ W | wuw−1 = v is


52

Figure 5.1. This diagram shows the tiling of the plane corresponding toB2 as in Figure 3.1. Each of the gray regions is an intersection of twohalf spaces, and thus each is p-completely regular for any p. Then, thetotal gray region is a union of two p-completely regular sets and so isp-completely regular itself. Therefore, this set also has a rational multi-variate Poincare series.

53

2. For α, β ∈ T × ±1, W (α, β) = w ∈ W | w(α) = β is p-completely

regular.

3. For finite sets Γ,∆ ⊂ T×±1, the set w ∈ W | w(Γ) = ∆ is p-completely

regular.

4. For J,K ⊆ S, w ∈ W | wJw−1 = K is p-completely regular.

5. For J,K ⊆ S, w ∈ W | w((WJ∩T )×1) = (WK∩T )×1 is p-completely

regular.

Proof. We first prove 1 for u = r and v = s with r, s ∈ S. In this case, we have

W (r, s) = w ∈ W | wrw−1 = s = w ∈ W | wr = sw. Now, W (r, s) will be

empty (and thus p-completely regular) unless r is conjugate to s. So, if r and

s are conjugate, then lp(r) = lp(s) and according to the exchange condition, we

have that W (r, s) = Z1 ∪ Z2 where

Z1 = w ∈ W | lp(wr) = lp(w)− lp(r) = lp(sw); lp(swr) = lp(w)

and

Z2 = w ∈ W | lp(wr) = lp(w) + lp(r) = lp(sw); lp(swr) = lp(w).

Then, each Zi is p-completely regular due to the following descriptions.

Z1 = Wr,1,lp(r) ∩W1,s,lp(s) ∩Wr,s,lp(r)

and

Z2 = Wr,1,0 ∩W1,s,0 ∩Wr,s,lp(r).

54

Now, in general, let u = xrx−1 and let v = ysy−1, then we know that W (r, s)

is p-completely regular by above and

yW (r, s)x−1 = ywx−1 ∈ W | wrw−1 = s

= w′ ∈ W | y−1w′xrx−1w′−1y = s

= w′ ∈ W | w′uw′−1 = v = W (u, v)

is p-completely regular since p-completely regular sets are closed under left and

right multiplication by elements of W by Proposition 5.5.1, proving 1.

Now, according to equation (2.2.3), we know that the W -action on T × ±1

is given by w(t, ε) = (wtw−1, τw,tε) where τw,t ∈ ±1 is given by τw,t = 1 if

lp(wt) > lp(w) and τw,t = −1 if lp(wt) < lp(w). Then, let α = (u, ε) and β = (v, ε′)

then

W (α, β) = W (u, v) ∩ w ∈ W | τw,uε = ε′.

Now, W (u, v) is p-completely regular by 1. If ε = ε′ then τw,u must be 1 and

so w ∈ W | τw,u = 1 = w ∈ W | lp(wu) > lp(w) is p-completely regular by

Example 3 in 5.6. Likewise if ε 6= ε′ then τw,u must be -1 and so w ∈ W | τw,u =

−2 = w ∈ W | lp(wu) < lp(w) is p-completely regular by the same example.

Thus, w ∈ W | τw,uε = ε′ is p-completely regular so that W (α, β) also is as

required.

Finally, to demonstrate 3, we note that the set described is empty unless

|Γ| = |∆|. Let α1, ..., αn = Γ and β1, ..., βn = ∆. Then we have

YΓ,∆ := w ∈ W | w(Γ) = ∆ = ∪σ∈Sn ∩ni=1 W (αi, βσ(i)),

where Sn denotes the symmetric group. Then, YΓ,∆ is p-completely regular by 2.

55

Similarly, we have 4 following from 1 in the exact same way replacing W (αi, βσ(i))

with W (si, rσ(i)). Finally, according to section 2.3, we know that w((WJ ∩ T ) ×

1) = (WK ∩ T )× 1 if and only if w(J × 1) = K × 1, so 5 follows from 3

directly using Γ = J × 1 and ∆ = K × 1 since both sets are finite.

Example 5.7.2. The centralizer of a reflection, t, is CW (t) = w ∈ W | w−1tw =

t. Due to the previous Proposition, this set is p-completely regular. So the cen-

tralizer of any finitely generated reflection subgroup is also p-completely regular.

5.8 Regularity in Direct Products

We now turn to looking at direct products of Coxeter systems. We have the

following result describing the p-completely regular subsets of a Coxeter system

in terms of the p-completely regular subsets of its irreducible components.

Lemma 5.8.1. Suppose that (W,S) is the direct product of Coxeter systems

(Wi, Si) for i = 1, ..., n. We identify W = W1 × · · · ×Wn and S = ∪Si. Then

Bp(W ) is equal to the Boolean subalgebra Bp(W1) ∗ · · · ∗Bp(Wn) of P(W ) gen-

erated by all subsets of W of the form A1 × · · · × An with Ai ∈ Bp(Wi).

Proof. This is clearly true for n = 1. Now, suppose it is true for n = k, we will

show that it is true for n = k + 1. We consider W ′ = W1 × · · · ×Wk and Wk+1

as subgroups of W . For any x, y ∈ W , we write x = x′xk+1 and y = y′yk+1 where

56

x′, y′ ∈ W ′ and xk+1, yk+1 ∈ Wk+1. Let m ∈M. Then

W px,y,m = w ∈ W | lp(ywx) = lp(y) + lp(w) + lp(x)− 2m

= w′wk+1 ∈ W ′ ×Wk+1 |

lp(y′w′x′yk+1wk+1xk+1) =

lp(y′) + lp(yk+1) + lp(w

′) + lp(wk+1) + lp(x′) + lp(xk+1)− 2m

=⋃

k1,k2∈Mk1+k2=m

W ′px′,y′,k1

× (Wk+1)pxk+1,yk+1,k2

(5.8.1)

so that Bp(W ) ⊆ Bp(W′) ∗ Bp(Wk+1). Also, equation (5.8.1) implies that

W px′,y′,k1

= W ′px′,y′,k1

×Wk+1 and W pxk+1,yk+1,k2

= W ′× (Wk+1)pxk+1,yk+1,k2. Therefore,

we have

W ′px′,y′,k1

× (Wk+1)pxk+1,yk+1,k2= W p

x′,y′,k1∩W p

xk+1,yk+1,k2

so that Bp(W′) ∗ Bp(Wk+1) ⊆ Bp(W ). However, by induction, we know that

Bp(W′) = Bp(W1) ∗ · · · ∗Bp(Wk) and the result follows.

5.9 Product Regularity

Complete regularity of a subset in a product W = W1 × · · · × Wn implies

regularity of many subsets of S∗ and so regularity of various subsets of S∗1×· · ·×S∗n.

Then we regard each S∗i as a submonoid of S∗ and each Wi as a subgroup of W ;

then we can define natural homomorphisms νi : S∗i → Wi. We say that a subset of

S∗1 ×· · ·×S∗n is product regular if it is in the Boolean algebra of subsets generated

by the products A1 × · · · × An with each Ai regular in S∗i .

Lemma 5.9.1. For any p-completely regular subset A of W1× · · · ×Wn, and any

57

n-tuple (k1, ...,kn) ∈Mn, the subset

(x1, ..., xn) ∈ (S1)∗k1× · · · × (Sn)∗kn| (ν1(x1), ..., νn(xn)) ∈ A

is product regular in (S1)∗ × · · · × (Sn)∗.

Proof. For any subset A of W , we define

A′ := x1, ..., xn) ∈ (S1)∗k1× · · · × (Sn)∗kn| (ν1(x1), ..., νn(xn)) ∈ A.

If A is a product then A = A1 × · · · × An, then we have

A′ =((S1)∗k1

∩ ν−11 (A1)

)× · · · ×

((Sn)∗kn ∩ ν

−1n (An)

).

In this, if each Ai is p-completely regular (or just p-strongly regular), then each

(Si)∗ki∩ν−1

i (Ai) is regular in (Si)∗ by Lemma 5.3.3 and thus A′ is product regular.

Now if A = ∪mBm, where the union is finite, then A′ = ∪mB′m. Now, this implies

that A′ is product regular for any p-completely regular set since any such set A is

a finite union of products of p-completely regular sets.

5.10 Multivariate Product Regularity

For any Coxeter system (W,S), define a subset of W n to be p-completely

regular if it is completely regular as a subset of the n-fold product Coxeter system

(W,S)n. We have the following result.

Lemma 5.10.1. Let n,m be natural numbers, m ∈ M, f1, ..., fm ∈ 1, ..., n,

ε1, ..., εm ∈ ±1, x0, x1, ..., xm ∈ W and A,B,C be finite subsets of W satisfying

B,C ⊆ A. Then let L be the subset of W n consisting of n-tuples satisfying the

58

following conditions:

(i) If

v := xm(wfm)εmxm−1(wfm−1)εm−1xm−2 · · ·x1(wf1)

ε1x0

and k :=∑m

i=0 lp(xi) +∑m

i=1 lp(wfi), then lp(v) = k− 2m.

(ii) N(v) ∩ A = B and N(v−1) ∩ A = C.

Then L is a p-completely regular subset of W n.

Proof. We will prove this by induction on m. To make things more tractable, we

denote L by

L = L(n,m,m, (fj), (εj), (xj), A,B,C)

to show that L depends on its parameter set.

First, suppose that m is fixed. If

L(n,m,m, (fj), (εj), (xj), ∅, ∅, ∅)

is p-completely regular, then

L(n,m,m, (fj), (εj), (xj), A,B,C)

is also p-completely regular using Lemma 4.6.1 (similar to the proof of Proposition

5.5.1 part (b)).

Now, suppose that m = 1. The set

L(1, 1,m, (1), (1), (x0, x1), ∅, ∅, ∅) = W px0,x1,m

and thus is p-completely regular by definition. Also, as in the proof of Proposition

59

5.5.1 part (c), we have

L(1, 1,m, (1), (−1), (x0, x1), ∅, ∅, ∅) = L(1, 1,m, (1), (1), (x−11 , x−1

0 ), ∅, ∅, ∅).

It follows that the sets

L(1, 1,m, (1), (ε), (x0, x1), ∅, ∅, ∅)

for ε ∈ ±1 are p-completely regular.

Let n ∈ N. Then any set

L′ = L(n, 1,m, (k), (ε), (x0, x1), ∅, ∅, ∅)

with k ∈ 1, ..., n is of the form

L′ = W k−1 × L(1, 1,m, (1), (ε), (x0, x1), ∅, ∅, ∅)×W n−k

and thus is p-completely regular. Therefore, we have shown that any set

L(n, 1,m, (k), (ε), (x0, x1), A,B,C)

is p-completely regular.

At this point, we have proved that the set L is p-completely regular if m = 1,

and if m = 0 it is clear that L is p-completely regular. So, we now move to the

induction step. We will show that for fixed m, p-complete regularity of all sets of

the form

L(n,m,m, (fj), (εj), (xj), A,B,C)

60

implies p-complete regularity of any sets of the form

L′′ := L(n,m+ 1,m, (fj), (εj), (xj), ∅, ∅, ∅).

Now, suppose that w = (w1, ..., wn) ∈ W n is an element of L′′ above. Then we

define the following two words:

vm := xm(wfm)εmxm−1(wfm−1)εm−1xm−2 · · ·x1(wf1)

ε1x0

and

vm+1 := xm+1(wfm+1)εm+1xm(wfm)εmxm−1(wfm−1)

εm−1xm−2 · · ·x1(wf1)ε1x0

and two p-lengths km :=∑m

i=0 lp(xi) +∑m

i=1 lp(wfi) and km+1 := km + lp(xm+1) +

lp(wfm+1). Let v := xm+1(wfm+1)εm+1 so that vm+1 = vvm, and let k := lp(xm+1) +

lp(wfm+1) so that km+1 = km + k. Using these equalities, we have

lp(vm+1) = lp(v) + lp(vm)− 2|N(v−1) ∩N(vm)|

= km+1 − k− km + lp(v) + lp(vm)− 2|N(v−1) ∩N(vm)|

= km+1 − 2

(k− lp(v)

2+

km − lp(vm)

2+ |N(v−1) ∩N(vm)|

).

By definition, lp(v) = lp(xm+1) + lp(wfm+1) − 2|N(x−1m+1) ∩ N((wfm+1)

εm+1)| ≤ k,

and similarly we have lp(vm) ≤ km. We know that w ∈ L′′ so we must have

lp(vm+1) = km+1 − 2m.

According to Corollary 3.10.1 part (4) we know that lp(vm+1) < km+1 − 2m if

lp(v) < k − 2m, lp(vm) < km − 2m or |Nm(v−1) ∩ Nm(vm)| > m. Otherwise, we

61

have

lp(vm+1) = km+1 − 2

(k− lp(v)

2+

km − lp(vm)

2+ |Nm(v−1) ∩Nm(vm)|

).

(5.10.1)

Suppose lp(v) = k− 2m′ and lp(vm) = km − 2m′′. Then the equation (5.10.1) is

lp(vm+1) = km+1 − 2(m′ + m′′ + |Nm(v−1) ∩Nm(vm)|

).

Since lp(vm+1) = km+1 − 2m, then we have the following:

L′′ =⋃

m′,m′′,B,CB,C⊂T≤m

m′+m′′+|B∩C|=m

Lm′,m′′,B,C

where

Lm′,m′′,B,C = L(n,m,m′, (fj)mj=1, (εj)

mj=1, (xj)

mj=0, T≤m, B, ∅)

∩ L(n, 1,m′′, (fm+1), (εm+1), (1, xm+1), T≤m, ∅, C)

both of which are p-completely regular by induction. Thus L′′ is a finite union of

p-completely regular sets and so L′′ is p-completely regular as needed.

62

CHAPTER 6

REGULARITY IN T AND MIXED PRODUCTS

The p-complete regularity in W as we have described has been useful for show-

ing that many natural subsets of W are regular, and therefore have rational

Poincare series. However, there are many natural subsets of W , in particular

subsets of T , that can be shown to not be p-complete regular in W (or even µ-

regular in W ). Nonetheless, we would like to be able to show that certain subsets

of T are regular in some sense. We will use the fact that every reflection has a

palindromic reduced expression to introduce a notion of p-complete regularity in

T . We can then extend our notion of regularity to regularity in products of T

or even mixed products of W and T . We include some examples of p-completely

regular subsets of T , and the regularity implies that these subsets have rational

(multivariate) Poincare series.

6.1 Regularity of sets of Reflections

Let X ⊂ S∗ be the set of all non-empty words in S∗. We see that X is

clearly a regular subset of S∗ being the complement of the set consisting of the

empty word. We say that a subset of X is regular if it is regular as a subset of

S∗. Define η : X → T by η(rn+1rn · · · r1) = r1 · · · rnrn+1rn · · · r1. Let `′p(x) =

2`p(x) − lp(rn+1) = 2`p(x) − p(η(x)). Let ′L be the restriction of some reverse

lexicographic order on S∗ to X.

63

Proposition 6.1.1. 1. For any m ∈M let Xm := x ∈ X| lp(η(x)) = `′p(x)−

2m. Then Xm is a regular subset of X.

2. For t ∈ T , define xt as the minimal (under the order ′L) element of the

non-empty finite set η−1(t) ∩X0. Then T = xt| t ∈ T is a regular subset

of X.

Proof. We prove 1 first. Let m ∈ M, and let x ∈ Xm. Then, we can write

x = rx′ with r ∈ S and x′ ∈ S∗k for some k ∈ M. By definition, we know that

lp(ν(x′)) = `p(x′)− 2k. Since x ∈ Xm, and substituting the previous formula we

get

lp(ν(x′)−1rν(x′)) = lp(η(x)) = `′p(x)− 2m = 2`p(x′) + lp(r)− 2m

= 2(2lp(ν(x′)) + 2k) + lp(r)− 2m

= 2lp(ν(x′)) + lp(r)− 2(m− 2k)

. (6.1.1)

Equation (6.1.1) implies that m − 2k ≥ 0 so 2k ≤ m; furthermore, (6.1.1) also

implies that ν(x′) ∈ Ar,m−2k where

Ar,m−2k := L(1, 2,m− 2k, (1, 1), (1,−1), (e, r, e), ∅, ∅, ∅)

where e ∈ W is the identity, and L is as in 5.10. According to Lemma 5.10.1

Ar,m−2k is p-completely regular. Therefore, we have x′ ∈ S∗k ∩ ν−1(Ar,m−2k) which

is regular by Lemma 5.5.1. So, for any x ∈ Xm, there is an r ∈ S and k ∈ M

with 2k ≤m such that x ∈ r(S∗k ∩ ν−1(Ar,m−2k)). Thus

Xm ⊆⋃r∈S

r

( ⋃0≤2k≤m

S∗k ∩ ν−1(Ar,m−2k)

)

64

Now, suppose that x ∈ r(S∗k ∩ ν−1(Ar,m−2k)) for some r ∈ S and k ∈ M with

2k ≤m. We have from above that

Ar,m−2k = w ∈ W | lp(w−1rw) = 2lp(w) + lp(r)− 2(m− 2k).

Thus x = rx′ with x′ ∈ ν−1(Ar,m−2k) and lp(ν(x′)) = 2`p(x′)−2k. This gives that

lp(η(x)) = lp(ν(x′)−1rν(x′)) = 2lp(ν(x′)) + lp(r)− 2(m− 2k)

= 2(`p(x′)− 2k) + lp(r)− 2(m− 2k)

= 2`p(x′) + lp(r)− 2m = 2`′p(x)− 2m,

so that x ∈ Xm. Thus we have shown

Xm =⋃r∈S

r

( ⋃0≤2k≤m

S∗k ∩ ν−1(Ar,m−2k)

),

and the right hand side is a finite union of regular sets as already shown hence

Xm is regular as well.

Part (2) follows from and unpublished result due to Bob Howlett, which is the

analog of 5.2.1 for T instead of W . The result is mentioned in [18].

6.2 Types of Regularity in T

Fix any map τ : T → X, denoted t → xt for t ∈ T , with the following

properties:

(i) For all t ∈ T , τ(t) ∈ X0.

(ii) η τ = IdT .

(iii) T := τ(T ) is a regular subset of X.

65

We know that maps, τ , satisfying (i)-(iii) above exist due to the previous propo-

sition. We say a subset A of T is τ -regular if τ(A) is regular in X.

Definition 6.2.1. A subset A of T is regular (respectively p-strongly regular) in

T if η−1(A) ∩Xm is a regular subset of X for m = 0 (respectively all m ∈M).

Lemma 6.2.2. (a) Any p-strongly regular subset of T is regular.

(b) If A is a regular subset of T then A is τ -regular for any map satisfying

(i)-(iii) above.

Proof. Part (a) is a restatement of the definition since 0 ∈ M. For part (b), we

have that for regular subset A ⊆ T and τ satisfying (i)-(iii) above, we get that

τ(A) = (η−1(A) ∩X0) ∩ τ(T ) which is an intersection of regular sets and thus

regular.

6.3 p-Complete Regularity in T

We now want to describe the analog of Bp(W ) when dealing with regular

subsets of T . To do this, we first introduce a few subsets of P(T ).

Definition 6.3.1. 1. For x ∈ W and m ∈M, we let

Rpx,m := t ∈ T | lp(xtx−1) = 2lp(x) + lp(t)− 2m.

Then, let B′′p(T ) be the Boolean subalgebra of P(T ) generated by all sets

Rpx,m for x ∈ W and m ∈M.

2. For t ∈ T and A ⊆ T , we define At := w ∈ W | w−1tw ∈ A. Then,

let B′p(T ) be the family of all subsets A of T such that for all r ∈ S,

Ar ∈ Bp(W ).

66

3. For any t ∈ T and A ⊂ T and m ∈M, we define

Apt,m := w ∈ W | w−1tw ∈ A; lp(w−1tw) = 2lp(w) + lp(t)− 2m.

We let Bp(T ) be the family of all subsets A of T such that for all r ∈ S

and m ∈ M we have Ar,m ∈ Bp(W ). We call the elements of Bp(T ) the

p-completely regular subsets of T .

Properties of these families of subsets are described in the following proposi-

tion. Again, for ease of notation, we omit the superscript p since p is understood

to be fixed.

Proposition 6.3.2. (a) If A ∈ B′p(T ) then At ∈ Bp(W ) for all t ∈ T . Like-

wise, if A ∈ Bp(T ) then At,m ∈ Bp(W ) for all t ∈ T and m ∈M.

(b) B′′p(T ),B′p(T ) ⊂ Bp(T ) are Boolean subalgebras of P(T ) closed under con-

jugation by elements of W and closed under permutations of T induced by

automorphisms, θ, of (W,S) that commute with p.

(c) B′p(T ) contains all finite subsets of T and all conjugacy classes of simple

reflections.

(d) Any set A which is p-completely regular in T is p-strongly regular in T .

Proof. Suppose A ∈ B′p(T ). Then Ar ∈ Bp(W ) for all r ∈ S. Now, let t ∈ T be

given. Then, t = u−1su for some u ∈ W and s ∈ S. Now we have

At = w ∈ W | w−1tw ∈ A = w ∈ W | w−1u−1suw ∈ A

= u−1v ∈ W | v−1sv ∈ A = u−1 · v ∈ W | v−1sv ∈ A = u−1As,

67

and u−1As ∈ Bp(W ) since As ∈ Bp(W ) and by Proposition 5.5.1 Bp(W ) is

closed under left multiplication by elements of W , thus the first part of (a) is true.

For the second part of (a), we fix u ∈ W and s ∈ S such that t = u−1su and

lp(t) = 2lp(u) + lp(s). Now, we consider the sets

Bk := u−1As,m−2k ∩W1,u,k ∈ Bp(W ).

Notice, if w ∈ Bk then w = u−1v with v ∈ As,m−2k, and so we have w−1tw =

v−1uu−1suu−1v = v−1sv ∈ A. Also, we have

lp(w−1tw) = lp(v

−1sv) = 2lp(v) + lp(s)−2(m−2k) = 2lp(uw) + lp(s)−2(m−2k),

but w ∈ W1,u,k so lp(uw) = lp(u) + lp(w)− 2k so that we get

lp(w−1tw) = 2(lp(u) + lp(w)− 2k) + lp(s)− 2(m− 2k)

= 2lp(w) + 2lp(u) + lp(s)− 2m

= 2lp(w) + lp(t)− 2m.

Therefore Bk ⊆ At,m. Similarly, At,m ∩W1,u,k ⊆ Bk. We note that Bk = ∅ unless

2k ≤m, and thus

At,m =⋃

k:0≤2k≤m

Bk.

So At,m ∈ Bp(W ) since it is a finite union of sets in Bp(W ), finishing the proof

of (a).

For the first part of (b), we notice that for A,B ∈ B′p(T ) or A,B ∈ Bp(T ) and

r ∈ S, it is clear that (A ∪B)r = Ar ∪Br, (A ∩B)r = Ar ∩Br and (Ac)r = (Ar)c

so that B′p(T ) and Bp(T ) are clearly Boolean subalgebras of P(T ) and B′′p(T )

68

is a Boolean subalgebra by definition. Now, suppose that A ∈ B′p(T ). Then

Ar ∈ Bp(W ) by definition. Also, for any r ∈ S and m ∈M, we have the set

Br,m = w ∈ W | lp(w−1rw) = 2lp(w) + lp(r)− 2m

= L(1, 2,m, (1, 1), (1,−1), (e, r, e), ∅, ∅, ∅)

following notation from section 5.10. Thus, by Lemma 5.10.1 we have Br,m ∈

Bp(W ) for all r ∈ S and m ∈ M. Finally, we see that Ar,m = Ar ∩ Br,m, and

thus Ar,m ∈ Bp(W ) for all r ∈ S and m ∈M so that A ∈ Bp(T ).

Now, we show that Rpx,m ∈ Bp(T ) for all x ∈ W and m ∈ M. For any r ∈ S

and n ∈M we have

(Rpx,m)r,n = w ∈ W | w−1rw ∈ Tx,m; lp(w

−1rw) = 2lp(w) + lp(r)− 2n

= w ∈ W | lp(xw−1rwx−1) = 2lp(x) + lp(w−1rw)− 2m;

lp(w−1rw) = 2lp(w) + lp(r)− 2n

= w ∈ W | lp(xw−1rwx−1) = 2lp(x) + 2lp(w) + lp(r)− 2(m + n);

lp(w−1rw) = 2lp(w) + lp(r)− 2n

= L(1, 2,m + n, (1, 1), (−1, 1), (x, r, x−1), ∅, ∅, ∅) ∩Br,n

using notation from the proof of Lemma 5.10.1, and by the result of that lemma,

we get that (Rpx,m)r,n ∈ Bp(W ). Thus, Rp

x,m ∈ Bp(T ) so that B′′p(T ) ⊂ Bp(T ) as

required.

Next, suppose that A ∈ B′p(T ). Then Ar ∈ Bp(W ) for all r ∈ S. Let w ∈ W

69

be fixed. Then wAw−1 ∈ B′p(T ) since for r ∈ S we have

(wAw−1)r = u ∈ W | u−1ru ∈ wAw−1 = u ∈ W | w−1u−1ruw ∈ A

= vw−1 ∈ W | v−1rv ∈ A = v ∈ W | v−1rv ∈ Aw−1 = Arw−1

is in Bp(W ) because Bp(W ) is closed under multiplication by elements of W .

For A ∈ Bp(T ) and w ∈ W , we see that for r ∈ S and m ∈M we have

(wAw−1)r,m = (wAw−1)r ∩⋃

0≤k≤lp(w)

(B(w, r, 2k) ∩Ww−1,1,k)w−1(6.3.1)

where B(w, r, 2k) = v ∈ W | lp(wv−1rvw) = 2lp(w) + 2lp(v) + lp(r)− 2(m + 2k).

Indeed, if u ∈ (wAw−1)r,m, then u ∈ (wAw−1)r and lp(u−1ru) = 2lp(u) + lp(r) −

2m. By the previous argument, u = vw−1, and then we get lp(wv−1rvw−1) =

2lp(vw−1) + lp(r)− 2m = 2(lp(v) + lp(w)− 2k) + lp(r)− 2m where k ≤ lp(w) (and

we note that this means v ∈ Ww−1,1,k). Hence, we have inclusion of the left hand

side in the right hand side of (6.3.1). However, working the equation backwards,

since k ≤ lp(w) we get inclusion the other way proving that Bp(T ) is closed under

conjugation.

Suppose that θ is an automorphism of (W,S). Then we see if A ∈ B′p(T ) we

have for any r ∈ S

θ(A)r = u ∈ W | u−1ru ∈ θ(A) = u ∈ W | θ−1(u−1ru) ∈ A

= u ∈ W | θ−1(u−1)θ−1(r)θ−1(u) ∈ A = θ(v) ∈ W | v−1θ−1(r)v ∈ A

= θ(Aθ−1(r))

and so θ(A)r ∈ Bp(W ) since Bp(W ) is closed under automorphisms. Notice

this holds for any automorphism. For A ∈ Bp(T ), and θ an automorphism of

70

(W,S) commuting with p, recall that from the proof of Proposition 5.5.1 that

θ(lp(w)) = lp(w). Then for any r ∈ S and m ∈M we have

θ(A)r,m = u ∈ W | u−1ru ∈ θ(A); lp(u−1ru) = 2lp(u) + lp(r)− 2m

= θ(A)r ∩ u ∈ W | lp(u−1ru) = 2lp(u) + lp(r)− 2m.

We know that θ(A)r ∈ Bp(W ) from above, and u ∈ W | lp(u−1ru) = 2lp(u) +

lp(r)− 2m ∈ Bp(W ) by Lemma 5.10.1. Thus, we have proved (b).

To show (c), we first note that w ∈ W | w−1rw = t ∈ Bp(W ) for all r ∈ S by

Proposition 5.7.1. Therefore t ∈ B′p(T ) for all t ∈ T , and hence any finite subset

of T is p-completely regular as well. Also, if A = wrw−1| w ∈ W is the conjugacy

class of r ∈ S, then for any s ∈ A we have As = w ∈ W | w−1sw ∈ A = W if

s is conjugate to r and As = ∅ if s is not conjugate to r. Since W ∈ Bp(W ) and

∅ ∈ Bp(W ), then As ∈ Bp(W ) for all s ∈ S and so A ∈ B′p(T ).

Finally, we show that (d) is true. We have to show that Bm := η−1(A) ∩Xm

is regular for all m ∈M if A is p-completely regular. If x ∈ η−1(A) then x = rx′

for some r ∈ S and η(x) = ν(x′)−1rν(x′). Thus, η−1(A) = ∪r∈SrB′r where

B′r = x ∈ S∗| ν(x)−1rν(x) ∈ A. Since x ∈ Xm, we also know that lp(η(x)) =

lp(ν(x′)−1rν(x′)) = 2`p(x′) + lp(r)− 2m. Therefore, we see that Bm = ∪r∈SrBr,m

where

Br,m := x′ ∈ S∗| ν(x′)−1rν(x′) ∈ A; lp(ν(x′)−1rν(x′)) = 2`p(x′) + lp(r)− 2m.

Hence, to show that Bm is regular, we need to show that each Br,m is regular (as

Bm is then a finite union of regular sets).

Now, if x′ ∈ S∗k we know that `p(x′) = lp(ν(x′)) + 2k. Therefore, we know that

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Br,m ∩ S∗k is given by the set

x′ ∈ S∗|ν(x′)−1rν(x′) ∈ A; lp(ν(x′)−1rν(x′)) = 2lp(x′) + lp(r)− 2(m− 2k),

which is equal to ν−1(Ar,m−2k) ∩ S∗k. Now, we know that since A is p-completely

regular in T then Ar,n ∈ Bp(W ) for all n ∈ M so that Ar,m−2k is p-completely

regular in W for all k. Thus, by Proposition 5.5.1 we know that ν−1(Am−2k)∩S∗k

is regular. Also, we see that Ar,m−2k = ∅ unless 0 ≤ 2k ≤m. Therefore we have

Br,m = ∪k∈M(Br,m ∩ S∗k) = ∪0≤k≤2m(Br,m ∩ S∗k)

which is a finite union and thus regular, completing the proof of (d).

6.4 Maps of Boolean Algebras

In this section, we prove some technical facts about Boolean subalgebras that

will be useful for describing regularity in mixed products of T and W .

Suppose that X1, X2, ..., Xn are sets with each Xi 6= ∅. Let X := X1×· · ·×Xn.

For each i let

πi : X → Xi

be the natural projection and let

πci : X → X1 × · · · × Xi × · · · ×Xn

be the natural projection where Xi means we omit Xi. For each i let Bi be a

Boolean subalgebra of P(Xi). Then let B := B1 ∗ · · · ∗Bn represent the Boolean

subalgebra of P(X) generated by all A1 × · · · × An with Ai ∈ Bi.

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For any Y ∈P(X1) ∗ · · · ∗P(Xn) = P(X), we define an equivalence relation

on Xi, ≡Y,i, given by a ≡Y,i b if πci(π−1i (a) ∩ Y

)= πci

(π−1i (b) ∩ Y

).

Lemma 6.4.1. Suppose X = X1 × · · · × Xn, B1 ∗ · · · ∗ Bn are as above, and

Y ∈P(X). Then the following hold.

(a) For each i, there exist finitely many equivalence classes of ≡Y,i, denoted

Y1,i, Y2,i, ..., Yni,i.

(b) Define a rectangle to be a set of the form Yj1,1×Yj2,2×· · ·×Yjn,n (where each

set in the product is one of the finitely many equivalence classes in (a)). If

a point of Y is in a rectangle, then that rectangle is a subset of Y , and thus,

Y is a union of (finitely many) rectangles.

(c) Y ∈ B1 ∗ · · · ∗ Bn if and only if Yj,i ∈ Bi for all i ∈ 1, .., n and all

j ∈ 1, ..., ni.

(d) Let Fi be a subset of Xi for each i such that Fi∩Yj,i 6= ∅ for all j ∈ 1, ..., ni.

Consider F ′ := Y ∩F of F := F1×· · ·×Fn, and let ≡F ′,i be the corresponding

equivalence relation on each Fi (in the same manner as ≡Y,i on Xi). Then

the equivalence classes for ≡F ′,i are precisely the intersections Yj,i ∩ Fi. In

particular, the rectangles for Y ∩ F that are contained in Y ∩ F are the

intersections with F of the rectangles for Y which are contained in Y .

Proof. Since Y ∈P(X), we write

Y =⋃k∈K

Ak,1 × · · · × Ak,n

where K is a finite index set. Then define Ci to be the Boolean subalgebra of

P(Xi) generated by the finite set of elements Ak,i| k ∈ K. For each k ∈ K,

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let Ark,i := Ak,i and let Ack,i = Xi \ Ak,i (the complement). Then for a sequence

ε := (ε1, ..., ε|K|) with each εi ∈ r, c we define Ai,ε := ∩k∈KAεkk,i. There are

only finitely many such sequences, and so we call the finite set Ai,ε| Ai,ε 6= ∅

the set of atoms of Ci. We necessarily have Ai,ε ∩ Ai,ε′ = ∅ for ε 6= ε′, and

each set J ∈ Ci is a finite union of atoms. Suppose now that a, b ∈ Ai,ε for

some ε. Let J ⊆ K such that εj = r for j ∈ J and εj = c for j 6∈ J . Then

Aj,i| a ∈ Aj,i = Aj,i| j ∈ J = Aj,i| b ∈ Aj,i. Therefore,

πci(π−1i (a) ∩ Y

)=⋃j∈J

Aj,1 × · · · × Aj,i × · · · × Aj,n = πci(π−1i (b) ∩ Y

);

and thus a ≡Y,i b so that each atom Ai,ε is contained in an equivalence class of

≡Y,i. Therefore, the atoms give a finer partition of Xi than ≡Y,i. Since there are

only finitely many atoms, there are only finitely many equivalence classes of ≡Y,i

proving (a).

Now, suppose that Yj1,1×Yj2,2×· · ·×Yjn,n is a rectangle and that (a1, ..., an) ∈

Y ∩ (Yj1,1 × Yj2,2 × · · · × Yjn,n). Let (b1, ..., bn) ∈ Yj1,1 × Yj2,2 × · · · × Yjn,n, as well.

Then, since a1 ≡Y,1 b1, we know that (b1, a2, ..., an) ∈ Y . Working inductively,

since ai ≡Y,i bi for all i, we have (b1, ..., bi, ai+1, ..., an) ∈ Y for all i. In particular,

(b1, ..., bn) ∈ Y . Therefore, Yj1,1×Yj2,2×· · ·×Yjn,n ⊆ Y , and so Y is a finite union

of rectangles proving (b).

Next, if Yji,i ∈ Bi for all i ∈ 1, ..., n and ji ∈ 1, ..., ni then Yj1,1 × Yj2,2 ×

· · · × Yjn,n ∈ B1 ∗ · · · ∗ Bn for all rectangles and since Y is a finite union of

rectangles by (b), we have Y ∈ B1 ∗ · · · ∗Bn. Conversely, if Y ∈ B1 ∗ · · · ∗Bn,

then from the proof of (a), each Ak,i ∈ Bi, and so each Ai,ε ∈ Bi and so Ci ⊆ Bi.

However, each Yji,i ∈ Ci ⊆ Bi as required for (c).

Finally, let Fi, F , and F ′ be as in the statement of (d). Let a ≡Y,i b and let

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a, b ∈ Fi. Then we know that

πci(π−1i (a) ∩ Y )

)= πci

(π−1i (b) ∩ Y )

).

Since both a, b ∈ Fi, this implies that

πci(π−1i (a) ∩ (Y ∩ F ))

)= πci

(π−1i (b) ∩ (Y ∩ F ))

),

and thus a ≡F ′,i b. Therefore Yj,i ∩ Fi ⊆ F ′k,i for some k.

Now, suppose that a ≡F ′,i b. Then a, b ∈ Fi and we have

πci(π−1i (a) ∩ (Y ∩ F ))

)= πci

(π−1i (b) ∩ (Y ∩ F ))

).

Let (x1, ..., a, ..., xn) ∈ π−1i (a)∩Y . Then, since Yj,m∩Fm 6= ∅, we have xm ≡Y,m x′m

with x′m ∈ Fm for all m 6= i. Therefore, we see that (x′1, ..., ai, ..., x′n) ∈ π−1

i (a) ∩

(Y ∩ F ). Since a ≡F ′,i b, then (x′1, ..., b, ..., x′n) ∈ π−1

i (b) ∩ (Y ∩ F ), and using the

equivalence xm ≡Y,m x′m in reverse, this implies that (x1, ..., b, ..., xn) ∈ π−1i (b)∩Y .

The exact same argument starting with b instead of a shows

πci(π−1i (a) ∩ Y )

)= πci

(π−1i (b) ∩ Y )

)so that a ≡Y,i b, and thus Fk,i ⊆ Yj,i ∩ F for some j. Therefore, we have equality

of equivalence classes. Equality of rectangles follows directly.

Next, we assume we additionally have another family of sets X ′1, ..., X′n. Let

X ′ := X ′1×· · ·×X ′n, and suppose that we are given a subset Hi ⊆ Hom(Xi, X′i)×

Bi. For each i, we define a Boolean algebra B′i (which depends on Hi) of P(X ′i)

consisting of sets A ⊆ X ′i such that for all (hi, Bi) ∈ Hi we have h−1i (A)∩Bi ∈ Bi.

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Then, let B′ := B′1 ∗ · · · ∗B′n be the corresponding Boolean subalgebra of P(X ′).

Define H := ((h1 × · · · × hn), (B1 × · · · ×Bn))| (hi, Bi) ∈ Hi; ∀i. Additionally,

we set

B′′ := Z ⊆ X ′| h−1(Z) ∩B ∈ B, for all (h,B) ∈ H.

Lemma 6.4.2. Suppose that for all i ∈ 1, ..., n there exists a finite subset H ′i ⊂

Hi such that ∪(hi,Bi)∈H′ihi(Bi) = Xi. Then B′′ ⊆P(X ′1) ∗ · · · ∗P(X ′n).

Proof. Define H ′ := ((h1 × · · · × hn), (B1 × · · · × Bn))| (hi, Bi) ∈ H ′i; ∀i ⊆ H.

Let Z ∈ B′′. Clearly we have

⋃(h,B)∈H′

h(h−1(Z) ∩B

)⊆ Z, (6.4.1)

and since each h−1(Z) ∩ B ∈ P(X) and h : P(X) → P(X ′), we have that the

left hand side of (6.4.1) is in P(X ′). Now, suppose that (z1, ..., zn) ∈ Z. By

assumption, we can write zi = hi(bi) for some (hi, Bi) ∈ H ′i with bi ∈ Bi. Then,

let h := h1×· · ·×hn and B := B1×· · ·×Bn so that (h,B) ∈ H ′ by construction.

Then, b := (b1, ..., bn) ∈ h−1(Z) ∩ B and so (z1, ..., zn) ∈ h (h−1(Z) ∩B) for some

(h,B) ∈ H ′. Therefore, we have

Z ⊆⋃

(h,B)∈H′h(h−1(Z) ∩B

),

and thus Z ∈P(X ′) since we have equality in (6.4.1).

Lemma 6.4.3. With notation as above, the following hold.

1. B′ ⊆ B′′.

2. Suppose that for all i ∈ 1, ..., n and for all h ∈ Hi we have that h is

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surjective; furthermore, assume that for each hi, we have

⋃Bi: (hi,Bi)∈Hi

Bi = Xi. (6.4.2)

Then B′′ ⊆ B′ (and thus B′ = B′′).

Proof. We begin by showing 1. Let C ′i ∈ B′i. Then if (hi, Bi) ∈ Hi we have

h−1i (C ′i) ∩Bi ∈ Bi. So we have that

(h1 × · · · × hn)−1(C ′i × · · · × C ′n) ∩ (B1 × · · · ×Bn)

= (h−11 (C ′1) ∩B1)× · · · × (h−1

n (C ′n) ∩Bn) ∈ B1 ∗ · · · ∗Bn = B.

Therefore, C ′1 × · · · × C ′n ∈ B′′. Since B′′ is closed under finite unions and

intersections, 1 follows.

Now, suppose that for all i and for all h ∈ Hi, we have that h is surjective.

Let Z ∈ B′′. According to Lemma 6.4.2, Z ∈P(X ′), and thus, by Lemma 6.4.1,

we have

Z =⋃k∈K

Zk,1 × · · · × Zk,n

where K is a finite set, and each Zk,1×· · ·×Zk,n is a rectangle (in the terminology

of 6.4.1). Now, let (h,B) ∈ H with h = (h1 × · · · × hn). We have,

V := h−1(Z) =⋃k∈K

h−11 (Zk,1)× · · · × h−1

n (Zk,n).

A quick calculation shows that if a ∈ Xi then

πci(π−1i (a) ∩ h−1(Z)

)= h−1

(πci(π−1i (hi(a)) ∩ Z

)). (6.4.3)

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Using equation (6.4.3) and the assumption that all hi are surjective, then if a ≡V,i

b, we get hi(a) ≡Z,i hi(b) and if a ≡Z,i b, we get a′ ≡V,i b′ for all a′ ∈ h−1i (a) and

b′ ∈ h−1i (b). Therefore, each h−1

1 (Zk,1)× · · · × h−1n (Zk,n) is a rectangle for h−1(Z)

(recalling the definition of a rectangle from 6.4.1).

Now, fix h = (h1 × · · · × hn). Consider the sets Qi := D(j)i | (hi, D

(j)i ) ∈ Hi.

By assumption

h−1(Z) ∩ (D(j1)1 × · · · ×D(jn)

n ) ∈ B

if D(ji)i ∈ Qi for all i. Then, we let Bi = ∪j∈JD(j)

i with J finite and all D(j)i ∈ Qi.

Since these unions are finite, we have

Z ′ := h−1(Z) ∩ (B1 × · · · × Bn) ∈ B.

Choose each Bi large enough so that Bi ∩ Vj,i 6= ∅ for all j ∈ 1, ..., ni (all

equivalence classes of ≡V,i); we note that this is possible by our assumption (6.4.2).

Then, we have

Z ′ =⋃k∈K

(h−11 (Zk,1) ∩ B1)× · · · × (h−1

n (Zk,n)× Bn) ∈ B.

By Lemma 6.4.1 (d), we know that (h−11 (Zk,1) ∩ B1) × · · · × (h−1

n (Zk,n) × Bn) is

a rectangle. Hence, by Lemma 6.4.1 (c), we have that each h−1i (Zk,i) ∩ Bi ∈ Bi.

Finally, since Bi ∈ Bi for all i, we know that

h−1i (Zk,i) ∩Bi = h−1

i (Zk,i) ∩ Bi ∩Bi ∈ Bi.

Therefore, for all i ∈ 1, ..., n and k ∈ K, we have Zi,k ∈ B′i and thus Z ∈ B′ as

required.

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6.5 Regularity in Mixed Products

We now extend our notions of p-complete regularity on T and W to the notion

of a p-complete regularity of products W n × Tm with n,m ∈ N.

As done previously when dealing with Cartesian products, we say that a subset

of a product B1 × · · · ×Bn with each Bi equal to T or W is p-completely regular

if it is in the Boolean subalgebra of P(B1 × · · · × Bn) generated by products

A1× · · · ×An where each Ai is p-completely regular in W if Bi = W and each Aj

is p-completely regular in T for Bj = T . To avoid trivial situations, if n = 0, we

let B1 × · · · × Bn be the singleton set and declare all subsets to be p-completely

regular. For ease of notation in the next following lemma, we assume that B =

B1× · · · ×Bn = T k ×W n−k with Bi = T for i ∈ 1, ..., k and Bi = W otherwise.

Lemma 6.5.1. Let B = B1×· · ·×Bn be as above, with each Bi = T for 1 ≤ i ≤ k

and Bi = W for k < i ≤ n. For any family t = (t1, ..., tk) in T k, we define

ft : W n → B1 × · · · ×Bn by ft(w1, ..., wn) = (u1, ..., un) where

ui :=

w−1i tiwi; if 1 ≤ i ≤ k

wi; otherwise.

Then a subset X ⊆ B1× · · · ×Bn is p-completely regular if and only if for all s =

(s1, ..., sk) ∈ Sk and (m1, ...,mk) ∈Mk, f−1s (X)∩

(Ts1,m1 × · · · × Tsk,mk

×W n−k)is p-completely regular in W n.

Proof. This lemma follows from the lemmas in Section 6.4 because the map and

sets satisfy the following. Using terminology from that section, we have H ′i :=

∪s∈Sk(fs, Ts,0), which is finite. Since each conjugacy class is p-completely regular,

and T is a finite union of conjugacy classes, we reduce to the case where we replace

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Bi = T by a conjugacy class in T so that fsi is surjective. Also, for any s ∈ Sk, we

have ∪m∈MTsi,m equal to the conjugacy class of si, so we satisfy (6.4.2). Finally,

by definition A ⊆ T is p-completely regular in T if and only if f−1s (A)∩Ts,m = As,m

is p-completely regular in W for all s ∈ S and m ∈M.

Remark 6.5.2. We see that the above proof would work for mixed products B =

B1 × · · · × Bn. We simply used the appropriate ordering for ease of notation in

the statement.

Remark 6.5.3. Let B ∼= W n−k×T k and B′ ∼= W n′−k′×T k′ be products as above, so

that B×B′ ∼= W n+n′−k−k′×T k+k′ is another such product. Let R be a p-completely

regular subset of B × B′. Then for b′ ∈ B′ the set Bb′,R := b ∈ B| (b, b′) ∈ R

is p-completely regular in B. Also b ∈ B| (b, b′) ∈ R for some b′ ∈ B′ and

b ∈ B| (b, b′) ∈ R for all b′ ∈ B′ are both p-completely regular.

6.6 General Regularity in Mixed Products

Now, we generalize Lemma 5.10.1 to the case of mixed products of copies of

W and T .

Consider B = B1×· · ·×Bn∼= W n−k×T k as above. Then, we let F be the free

group on n generators X1, ..., Xn. For any element b = (b1, ..., bn) ∈ B, we define

a group homomorphism αb : F → W determined by αb(Xi) = bi. Now, fix an

element g ∈ F , and suppose g = Xn1i1· · ·Xnm

imis a reduced word for g with ni ∈ Z

for all i. Then αb(g) = bn1i1· · · bnmim . Then, we can define a length function on F ,

dependent on b, by lbp : F →M by lbp(g) =∑m

j=1 |nj|lp(bij) where g is as above and

|nj| represents the absolute value. Now, we can see that lbp(g)− lp(αb(g)) ∈ 2M.

So for any g ∈ F and b ∈ B, we call the quantity lbp(g)− lp(αb(g)) the g-deficiency

of b.

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Theorem 6.6.1. For any m ∈M and g ∈ F , the set

Bg,m := b ∈ B| lp(αb(g)) = lbp(g)− 2m

is a p-completely regular subset of B.

Proof. Fix m ∈ M and g ∈ F with g = Xn1i1· · ·Xnm

im. Let D := i1, ..., im.

Then let C := j1, ..., jq ⊂ D be such that Bj = T for all j ∈ C and Bj = W

for all j ∈ D \ C. Then for any r ∈ Sk, we consider W rg,m := f−1

r (Bg,m). Let

Zi := W if Bi = W and Zi = Tri,ni otherwise, and let (n1, ...,nk) ∈ Mk. For

w := (w1, ..., wn) ∈ W rg,m ∩ (Z1 × · · · × Zn), we have

vw := xn1i1· · ·xnmim

where xik = wik if ik 6∈ D and xik = w−1ikrikwik if ik ∈ D. Thus, we have

Kvw :=∑

ik:ik∈D

(2|nik |lp(wik) + 1) +∑

ik:ik 6∈D

(|nik |lp(wik))

Then, since r is fixed, Lemma 5.10.1 proves that

W rg,m∩(Z1×· · ·×Zn) = (w1, ..., wn) ∈ W n| lp(vm) = Kvm−2m∩(Z1×· · ·×Zn)

is p-completely regular. Therefore, by Lemma 6.5.1, since W rg,m ∩ (Z1 × · · · ×Zn)

is p-completely regular for all r ∈ Sk and (n1, ...,nk) ∈Mk, then we have Bg,m is


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6.7 Multivariate Poincare Series

Recall the notation from 5.6. In [8], it is shown that the Poincare series for

the set of reflections, P (T ;W ), is rational. Due to the results of this chapter,

we obtain the stronger result that the multivariate Poincare series of the set of

reflections as well as any p-completely regular subset of T will be rational.

Furthermore, if we have a mixed product B := B1 × · · ·Bm+n∼= Wm × T n,

letM1, ...,Mm+n be the corresponding multiplicative monomials for each Bi gen-

erated by X1, ...,Xm+n with corresponding length functions lpi : Bi → Mi. Let

Z[[X1, ...,Xm+n]] be the completion of the polynomial ring Z[X1, ...,Xm+n]. Then

according to the transfer matrix method ([2] or [28]), for any p-completely regular

subset of A ⊆ B, we have that the multivariate Poincare series

P (A;B) =∑

(x1,...,xm+n)∈A

lp1(x1) · · · lpm+n(xm+n)

is rational. In fact, P (A;B) is a finite sum of products, each of the form f1 · · · fm+n

with each fi a rational expression in the variables Xi, i.e. fi ∈ Q(Xi).

Example 6.7.1. Let lp : W → M, lp1 : W → M1, lp2 : W → M2, and

lp3 : T →M3 be generalized length functions with M considered additively and

M1, M2, and M3 considered multiplicatively.

1. According to Theorem 6.6.1, for n ∈M,

Add(n) := (x, y) ∈ W ×W | lp(xy) = lp(x) + lp(y)− 2n

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is p-completely regular in W ×W . Therefore,

P (Add(n);W ×W ) =∑

(x,y)∈Add(n)

lp1(x)lp2(y) =∑

x∈M1;x′∈M2

ax,x′xx′

has a rational expression, where ax,x′ ∈ N is the number of pairs (x, y) ∈

W ×W with lp(xy) = lp(x) + lp(y)− 2n and lp1(x) = x and lp2(y) = x′.

2. Also by Theorem 6.6.1, for n ∈M,

Un := (x, t) ∈ W × T | lp(xtx−1) = 2lp(x) + lp(t)− 2n

is p-completely regular in W × T . It follows that

P (Un;W × T ) =∑

(x,t)∈Un

lp1(x)lp3(t)

has a rational expression. Moreover, according to Remark 6.5.3, for any

x ∈ W , the set

Un(x) := t ∈ T | lp(xtx−1) = 2lp(x) + lp(t)− 2n

is p-completely regular in T so

P (Un(x);T ) =∑

t∈Un(x)

lp3(t)

has a rational expression.

83

CHAPTER 7

HECKE ALGEBRA MODULES

The recurrence formulae given in section 3.7 look very similar to recurrence

formulae for the generic Iwahori-Hecke Algebra. In this chapter, we will further

discuss this algebra, and we introduce a large family of modules for the Iwahori-

Hecke algebra. These modules arise from the sets T≤m. In, [18], Dyer uses modules

like these to prove a weak form of Lusztig’s conjecture about the boundedness

of the a-function. There is some hope that deeper regularity properties of the

modules described in this section may lead to more results about the a-function.

Throughout this chapter, we fix an arbitrary Coxeter system, (W,S), with

reflections T , reflection cocycle N : W → P(T ), and with generalized length

function lp,W where p is a surjective function from T to the set of indeterminates

X as in section 3.2. In addition to X, let V be another set of indeterminates

equipped with a surjective function q : T → V satisfying q(t) = q(t′) if t is

conjugate to t′. For ease of notation, we denote p(r) = pr and q(r) = qr for any

r ∈ S.

7.1 The Generic Iwahori-Hecke Algebra

Let R denote the integral polynomial ring R = Z[X, V ] in indeterminates

X ∪ V . We consider the generic Iwahori-Hecke algebra H of (W,S) over R with

84

generators (as a unital R-algebra) tr for r ∈ S subject to the braid relations of

(W,S)

trts · · ·︸︷︷︸m(r,s) factors

= tstr · · ·︸︷︷︸m(r,s) factors

and the quadratic relations t2r = pr+qrtr. As an R-module, H is free with R-basis

tww∈W . The multiplication is determined by the following formulae: t1 = IdH

and

trtw =

trw if r 6∈ N(w)

prtrw + qrtw if r ∈ N(w).

By these formulae, it is clear that the set tw| w ∈ W is a spanning set for

H as a Z[X, V ]-module. In fact, this set is also a Z[X, V ]-basis for H (see [26]).

7.2 A Module for H

We regard R as a free Z[V ] module with basis corresponding to M := MX ,

i.e. monomials in Z[X]. We will denote monomials inMX by boldface letters, e.g.

m. For any m ∈M, we denote the dual basis element of R† := HomZ[V ](R,Z[V ])

by m∗ so that m∗(n) = 0 if n 6= m and m∗(m) = 1.

R† is an R-module with the standard R-structure given by (rf)(r′) = f(rr′)

for f ∈ R† and r, r′ ∈ R. Then, we note that

prm∗ =

(

mpr

)∗if pr ≤m

0 otherwise

Let R′ be the R-submodule of R† spanned by the dual basis elements ofMX .

Then R′ is free as a Z[V ]-module with basis MX .

Next, we can form the H -module H ′ := H ⊗R R′. For ease, we denote the

85

element tw ⊗m∗ of H ′ by tw,m for m ∈ M and w ∈ W . These elements form a

Z[V ]-basis of H ′. We thus have

prtw,m =

tw,mpr

if pr ≤m

0 otherwise

for all r ∈ S and

trtw,m =

trw,m if r 6∈ N(w)

trw,mpr

+ qrtw,m if r ∈ N(w) and pr ≤m

qrtw,m if r ∈ N(w) and pr 6≤m

for all r ∈ S.

For each n ∈M, the Z[V ]-span of the elements tw,m with w ∈ W and m ≤ n

is a H -submodule, called H ′n of H ′. Provided S is finite, then H ′

n has finite

rank as a Z[V ]-module. For n ≥ m > 0, multiplication by m induces a short

exact sequence of H -modules:

0 −→∑k<m

H ′k −→H ′

nm−→H ′

nm−→ 0

7.3 The Hecke Algebra Modules from hW,p,M∞

For m ∈ M, let Qm := N(w) ∩ T≤m| w ∈ W. Define a free Z[V ]-module

H ′′ with Z[V ]-basis given by formal symbols tA,m with m ∈ M and A ∈ Qm.

For n ∈ M, let H ′′n be the Z[V ]-submodule spanned by the basis elements tA,m

with m ≤ n and A ∈ Qm.

Proposition 7.3.1. 1. There is a unique H -module structure on H ′′ such

that the Z[V ]-module map ρ : H ′ −→ H ′′ that is determined by ρ(tw,m) =

86

tN(w)∩T≤m,m is an H -module epimorphism.

2. For m ∈ M, H ′′m = ρ(H ′

m) is a submodule of H ′′. So H ′′m is a submodule

of H ′′n if m ≤ n.

Proof. We define Z[V ]-linear endomorphisms p′r and θr on H ′′ for r ∈ S by

p′r(tA,m) =

tA∩T≤mpr,mpr

if pr ≤m

0 otherwise

and

θr(tA,m) =

tA′,m if r 6∈ N(w)

tA′′r ,mpr + qrtA,m if r ∈ N(w) and pr ≤m

qrtA,m if r ∈ N(w) and pr 6≤m

where

A′ := (rAr ∪ r) ∩ T≤m and A′′r := (rAr \ r) ∩ T≤mpr

Now suppose that tw,m ∈H ′. Then on one hand we have

ρ(pr(tw,m)) =

ρ(tw,mpr

) if pr ≤m

ρ(0) otherwise

=

tN(w)∩T≤mpr,mpr

if pr ≤m

0 otherwise

and on the other hand we have that

p′r(ρ(tw,m)) = p′r(tN(w)∩T≤m,m) =

tN(w)∩T≤mpr,mpr

if pr ≤m

0 otherwise

.

Thus, we see that p′r(ρ(tw,n)) = ρ(pr(tw,n)) and extending by linearity we get

p′r(ρ(h)) = ρ(pr(h)) for all h ∈H ′.

Next, let tw,m ∈H ′ and tr ∈H . Again we compute the following:

87

ρ(tr(tw,m)) =

ρ(trw,m) if r 6∈ N(w)

ρ(trw,mpr

+ qrtw,m) if r ∈ N(w) and pr ≤m

ρ(qrtw,m) if r ∈ N(w) and pr 6≤m

=

tN(rw)∩T≤m,m if r 6∈ N(w)

tN(rw)∩T≤mpr,mpr

+ qrtN(w)∩T≤m,m if r ∈ N(w) and pr ≤m

qrtN(w)∩T≤m,m if r ∈ N(w) and pr 6≤m

by linearity of ρ on Z[V ]. Also,

θr(ρ(tw,m)) = θr(tN(w)∩T≤m,m) =

tB′,m if r 6∈ N(w)

tB′′r ,mpr + qrtB,m if r ∈ N(w) and pr ≤m

qrtB,m if r ∈ N(w) and pr 6≤m

where B = N(w) ∩ T≤m. Then B′ = (rBr ∪ r) ∩ T≤m = N(rw) ∩ T≤m since

r 6∈ N(w) then N(rw) = r∪ rN(w)r. Finally we have B′′r = rBr \r∩T≤mpr

=

N(rw)∩ T≤mpr

since r ∈ N(w) then N(rw) = rN(w)r \ r. These all follow from

Proposition 3.8.1. Therefore we have shown that θr(ρ(tw,n)) = ρ(tr(tw,n)) for all

tw,n and so extending by linearity we get θr(ρ(h)) = ρ(tr(h)) for all h ∈H ′.

Now, ρ is clearly an epimorphism,, and so we thus have an H -module structure

on H ′′ in which pr acts by p′r for all r ∈ S, qr acts via the natural Z[V ]-module

structure on H ′′ for all r ∈ S, and tr acts by θr for all r ∈ S. Hence, 1 is true.

Then 2 follows from the definition of ρ.

88

7.4 The 0-Hecke Algebra

Now, we regard Z[V ] as the R-module R/XR. Then for any m ∈ M we

define H ′′m := H ′′

m/(∑

n<m H ′′n ), which is annihilated by X and so may naturally

regarded as a H := H /XH module. We see that H is a free Z[V ]-module

with Z[V ]-basis given by the images in the quotient of elements tw for w ∈ W .

By abuse of notation, we will denote these elements still as tw. We also see that

H ′′m has a free Z[V ]-basis consisting of the images of elements tA,m for A ∈ Qm.

The Z[V ]-algebra structure of H is determined by

trtw =

trw, if r 6∈ N(w)

prtw, if r ∈ N(w)

for r ∈ S and w ∈ W , and the H -module structure on H ′′m is determined by

trtA,m =

t(rAr∪r)∩T≤m,m if r 6∈ N(w)

prtA,m if r ∈ N(w)

for r ∈ S and A ∈ Qm.

89

CHAPTER 8

INITIAL SECTIONS OF REFLECTION ORDERS

In this chapter, we recall the definition due to Dyer (c.f. [11], [12], or [13])

of reflection orders, which are certain total orders of the set of reflections T ,

and of their initial sections, which are subsets of T that are known to lead to

partial orders (twisted Bruhat orders) on W that are similar to Bruhat order.

In particular, using the initial section ∅ ⊆ T we get the Bruhat order on W , and

using T ⊆ T , we get the reverse Bruhat order on W . In this chapter, we determine

all subsets of T that give rise to partial orders on W in the same manner. The

subsets of T that have this property are those which are “locally” (at each dihedral

reflection subgroup) initial sections; it has been conjectured by Dyer ([12]) that

these sets coincide with the initial sections. Reflection orders are closely related to

a notion of closure in the positive root system. In Chapter 9, we will examine the

relationship between closure in root systems and dominance order on the positive

roots. For general references about the properties of reflection orders and initial

sections, see [2], [11],[12], and [13].

8.1 Classification of Twisted Bruhat Orders

Let (W,S) be a Coxeter system. Recall that W acts on P(T ) by w · A =

N(w) + wAw−1 for any A ⊆ T and w ∈ W , where + represents symmetric

90

difference. Now, for any A ⊆ T , we follow [11] by defining a directed graph Ω(W,A)

with the vertex set of Ω(W,A) equal to W and edge set E(W,A) = (tw, w)| t ∈ w ·A.

We also have the following equivalent definition of E(W,A).

E(W,A) = (tw, w)| t ∈ N(w) + wAw−1 = (tw, w)| w−1tw ∈ N(w−1) + A

= (wt′, w)| t′ ∈ N(w−1) + A,

and we will find it convenient to use this description, E(W,A) = (wt, w)| t ∈

N(w−1) +A. For x ∈ W , we have that t ∈ w ·A if and only if t ∈ (wx−1) · x ·A,

and thus the map w 7→ wx−1 defines an isomorphism Ω(W,A)∼= Ω(W,x·A). In

addition, for A ⊆ T we can define a length function lA : W → Z in the following

way:

lA(v, w) = l(wv−1)− 2∣∣∣[N(vw−1) ∩ v · A]

∣∣∣ ∈ Z

and then set lA(w) = lA(1, w). We can define a pre-order ≤A for any A ⊆ T given

by the following: v ≤A w if and only if there exist t1, ..., tn ∈ T with w = vt1...tn

such that ti 6∈ [N((vt1...ti−1)−1) + A] for all i = 1, ..., n.

Remark 8.1.1. The multivariate analog of such single variable length functions,

lA, similar to that of Chapter 3, has not been studied but should be of interest.

Definition 8.1.2. Following [12], a total order on the set T ′ in a dihedral Coxeter

system, (W ′, S ′) with S = r, s, is called a dihedral reflection order if it is either

≺R or ≺′R where ≺R and ≺′R are defined by r ≺R rsr ≺R · · · ≺R srs ≺R s and

s ≺′R srs ≺′R · · · ≺′R rsr ≺′R r. Then, for any Coxeter system (W,S) we call

a total order ≺R on T a reflection order if the restriction ≺R∣∣W ′∩T is a dihedral

reflection order for all dihedral reflection subgroups W ′ of (W,S) with respect to

the canonical generators χ(W ′).

91

Remark 8.1.3. We can also define a reflection order in terms of the root system Φ

associated to the Coxeter system. This definition can be found in [2], and we will

discuss this definition in more detail in Chapter 9.

Recall from [12] that an initial section of a reflection order is a subset A ⊆ T

such that there is a reflection order ≺R with the property that a ≺R b for all

a ∈ A and b ∈ T \ A. It is shown in [11] that ≤A is a partial order of W if A is

an initial section of a reflection order. Our main result in this chapter, Theorem

8.1.4 below, describes all subsets A of T for which ≤A is a partial order.

Let A(W,S) be the set of initial sections of reflection orders of T . Now, we

define A(W,S) = A ⊆ T | A ∩W ′ ∈ A(W ′,χ(W ′)) ∀W ′ ⊆ W dihedral. It has been

conjectured by Dyer that A = A. We now come to the main result:

Theorem 8.1.4. Let (W,S) be any Coxeter system. The following are equivalent:

1. Ω(W,A) is acyclic.

2. ≤A is a partial order.

3. Ω(W,A) has no cycle of length four.

4. A ∈ A.

5. lA(xt) < lA(x) for all x ∈ W and t ∈ N(x−1) + A.

8.2 Proof of the Classification

In the following proofs, for any positive root α ∈ Φ+, let tα ∈ T be the

corresponding reflection, and for any reflection t ∈ T , let αt ∈ Φ+ be the cor-

responding positive root. To begin with we investigate the dihedral case. Sup-

pose (W,S) is dihedral, i.e. S = r, s. There is a bijection between subsets

92

A ⊆ T and subsets Ψ ⊂ Φ such that Ψ ∪ −Ψ = Φ and Ψ ∩ −Ψ = ∅ given by

A = AΨ = tα| α ∈ Ψ ∩ Φ+. We note that A−Ψ = AΨ + T .

Lemma 8.2.1. For any AΨ ⊆ T , w · AΨ = Aw(Ψ).

Proof. It is enough to show that this is true for r ∈ S. Now we know that

r · AΨ = r+ rAΨr = r+ rtαr| α ∈ Ψ ∩ Φ+ = r+ tr(α)| α ∈ Ψ ∩ Φ+ =

r + tα| α ∈ r(Ψ) ∩ r(Φ+). If αr ∈ Ψ then −αr ∈ r(Ψ) ∩ r(Φ+) and so r ∈

tα| α ∈ r(Ψ)∩r(Φ+). This implies that r ·AΨ = tα| α ∈ r(Ψ)∩Φ+. If αr 6∈ Ψ

then r 6∈ tα| α ∈ r(Ψ)∩r(Φ+). But αr ∈ r(Ψ) so r+tα| α ∈ r(Ψ)∩r(Φ+) =

tα| α ∈ r(Ψ) ∩ Φ+. Thus, in both cases r · AΨ = tα| α ∈ r(Ψ) ∩ Φ+.

Since we are considering (W,S) dihedral, we choose an orientation of the plane

spanned by Φ. Then for any root α ∈ Φ we can define the neighbor of α, denoted

nbr(α), to be the root lying directly next to α if we traverse the root system

clockwise. Recall that S = r, s. Interchanging r and s if necessary, we assume

without loss of generality that αr = nbr(−αs). We note that this implies that

nbr(αs) 6∈ Φ+.

Lemma 8.2.2. Let (W,S) be dihedral. Suppose α ∈ Φ+ and γ := nbr(α) ∈ Φ+.

Then tαtγ = sr.

Proof. Let (rsr...)n denote an element with n simple reflections listed (we note

that l((rsr...)n) is not necessarily n). With this terminology, t ∈ T can be written

t = (rsr...)2i+1 or t = (srs...)2i+1 for some i ≥ 0. Under the chosen orientation

for Φ+, if tα = (rsr...)2i+1 then we can write tγ = (rsr...)2i+3 so that tαtγ =

(rsr...)2i+1(rsr...)2i+3 = sr. Otherwise, if tα = (srs...)2i+1 (i ≥ 1) then tγ =

(srs...)2i−1 and so tαtγ = sr.

93

Now, given A = AΨ, we introduce two conditions that a positive system, Γ+,

of Φ can have:

C1: There are roots α, nbr(α) ∈ Γ+ such that α 6∈ Ψ and nbr(α) ∈ Ψ.

C2: There are roots β, nbr(β) ∈ Γ+ such that β ∈ Ψ with nbr(β) 6∈ Ψ.

Lemma 8.2.3. Let (W,S) be dihedral and let A = AΨ ⊆ T .

1. If the positive system r(Φ+) has condition C1, then there exists a path 1→

x→ sr in Ω(W,A).

2. If the positive system r(Φ+) has condition C2, then there exists a path sr →

x→ 1 in Ω(W,A).

Proof. Replacing A by A+ T reverses the orientation of edges in Ω(W,A) and so 2

follows from 1.

We now prove 1. There are two cases to consider. If α 6= αs, then both α

and γ := nbr(α) ∈ Φ+. So tα 6∈ A and tγ ∈ A. Also, since tα 6∈ r, s, it

follows that tγ ∈ N(tα). So we have that tγ 6∈ N(tα) + A. Thus, we have a path

1 → tα → tαtγ = sr, where the last equality follows from Lemma 8.2.2. Now, if

α = αs then nbr(α) = −αr. Since −αr ∈ Ψ we know that αr 6∈ Ψ which implies

r 6∈ A. Also, s 6∈ A and clearly r 6∈ N(s). Together, we see that there is a path

1→ s→ sr in Ω(W,A).

Lemma 8.2.4. For (W,S) dihedral, let A = AΨ ⊆ T . If there are no 4-cycles in

Ω(W,A) then there is no positive system, Γ+ ⊂ Φ satisfying C1 and C2.

Proof. Suppose there is a positive system Γ+ satisfying C1 and C2. It is clear that

−Γ+ also satisfies C1 and C2. Since Γ+ satisfies both C1 and C2, then w(Γ+)

94

will also satisfy both conditions (w with even length respects both conditions and

w with odd length interchanges the conditions). Thus, we can find a w ∈ W such

that r(Φ+) = w(Γ+) or r(Φ+) = w(−Γ+). So we have that r(Φ+) satisfies C1 and

C2 with respect to Aw(Ψ) = w ·AΨ. Since Ω(W,A) is isomorphic to Ω(W,w·A), we can

assume without loss of generality that r(Φ+) satisfies C1 and C2 with respect to

AΨ. Thus, Lemma 8.2.3 implies that we have a path sr → u → 1 → v → sr in

Ω(W,A).

Lemma 8.2.5. Let (W,S) be dihedral and A = AΨ ⊆ T . If there is no positive

system, Γ+, of Φ satisfying C1 and C2, then A ∈ A(W,S).

Proof. Let A 6∈ A(W,S). Recall that the only two reflection orders on (W,S) are

≺R and ≺′R described in Definition 8.1.2. Since A is not an initial section of either

of these orders, without loss of generality (replacing A with T \ A if necessary)

we can find t0 ∈ A and t1, t2 ∈ T \ A such that t1 ≺R t0 and t2 ≺′R t0, i.e.

t2 ≺R t0 ≺R t1.

Now, if (W,S) is finite, we can list all of the reflections and thus can find

t′, t′′ ∈ T such that t2 R t′ ≺R t0 R t′′ ≺R t1 with αt′ 6∈ Ψ and nbr(αt′) ∈ Ψ,

and αt′′ ∈ Ψ and nbr(αt′′) 6∈ Ψ. Thus Φ+ satisfies C1 and C2.

If (W,S) is infinite, then T = Tr ∪ Ts (disjoint union) where Tr = t ∈ T | r ∈

N(t) and Ts = t ∈ T | s ∈ N(t). Suppose Φ+ does not satisfy C1 and C2.

We cannot have t1, t2, t0 ∈ Tu for some u ∈ r, s since this would imply, by

the reasoning for (W,S) finite, that Φ+ satisfies C1 and C2. So, we can assume

that t2, t0 ∈ Tr and t1 ∈ Ts (the case t2 ∈ Tr and t0, t1 ∈ Ts is exactly similar).

Additionally, since C1 and C2 aren’t both satisfied by Φ+, we may assume the

following conditions hold (see Figure 8.1):

1. αt0 = nbr(αt2),

95

2. t ∈ T \ A if t R t2,

3. t ∈ T \ A if t ∈ Ts and t R t1,

4. t ∈ A if t ∈ Tr and t0 R t,

5. t ∈ A if t1 ≺R t.

Now, consider the positive system Γ+ with simple roots αt0 and −nbr(αt0). Then

Γ+ satisfies C1 using the roots αt2 and αt0 = nbr(αt2). Also, by above, αt1 6∈ Ψ

and nbr(αt1) ∈ Ψ (note that even if t1 = s this is true since nbr(αs) = −αr ∈ Ψ

because by above αr 6∈ Ψ). Using this, we see that Γ+ satisfies C2 using the roots

−αt1 and nbr(−αt1) = −nbr(αt1) (again see Figure 8.1).

With the dihedral case taken care of, we proceed to the general case. For the

remainder of the chapter, we assume that (W,S) is a general Coxeter system.

Proposition 8.2.6. For all w ∈ W and A ∈ A(W,S), w · A ∈ A(W,S).

Proof. It suffices to check the condition for w = r ∈ S. Let W ′ be a dihedral

reflection subgroup. If r ∈ W ′, then (r ·A) ∩W ′ = N(W ′,χ(W ′))(r) + r(A ∩W ′)r ∈

A(W ′,χ(W ′)) by [12]. Now, if r 6∈ W ′, then (r · A) ∩W ′ = r(A ∩ rW ′r)r. However,

conjugation by r defines an isomorphism (W ′, χ(W ′)) ∼= (rW ′r, rχ(W ′)r) in this

case, and by assumption A∩rW ′r ∈ A(rW ′r,rχ(W ′)r), thus (r ·A)∩W ′ ∈ A(W ′,χ(W ′)).

Before, we prove Theorem 8.1.4, we provide one more important proposition,

which is proven in [11], but we include a proof here as well.

Proposition 8.2.7. Let A ∈ A(W,S), x ∈ W , t ∈ T . Then lA(x, xt) > 0 iff

t 6∈ N(x−1) + A.

96

. . .

. . .

. . .. . .. .

.

. . .

. . . . . .

αs

αt2 αt1

αt0

αr

Figure 8.1. This is a schematic diagram of the root system for (W,S) infi-nite. Φ+ consists of the roots which are non-negative linear combinationsof αr and αs. The dotted ray through the origin represents a limit lineof roots. If C1 and C2 are not both satisfied by Φ+ then Ψ is picturedabove with the roots in Ψ labeled by •, and those not in Ψ labeled by .

97

Proof. We begin by proving the statement for all A ∈ A(W,S) and t ∈ T with x = 1,

i.e. lA(1, t) = lA(t) > 0 if and only if t 6∈ A. If (W,S) is dihedral with S = r, s,

then we see that, without loss of generality, t = (rsr...)2n+1 with l(t) = 2n + 1.

Then, N(t) = r, rsr, ..., (rsr...)2n+1, ..., (rsr...)4n+1. Since A ∈ A(W,S), then if

(rsr...)2i+1 ∈ A either (rsr...)2j+1 ∈ A for all j ≤ i or (rsr...)2j+1 ∈ A for all

i ≤ j ≤ 2n. In particular |N(t) ∩ A| ≥ n + 1 if and only if t = (rsr...)2n+1 ∈ A.

Therefore, in this case lA(t) > 0 if and only if t 6∈ A.

Suppose that (W,S) is not dihedral. Then if W ′ is any dihedral reflection

subgroup containing t, since A ∈ A(W,S) then A′ := A ∩W ′ ∈ A(W ′,χ(W ′)), and so

we have lA′(t) makes sense in the Coxeter system (W ′, χ(W ′)). Now, by equation

2.3.1, we get that

N(t) \ t =⋃

W ′∈Mt

N(W ′,χ(W ′))(t) \ t

and

N(t) \ t ∩ A =⋃

W ′∈Mt

(N(W ′,χ(W ′))(t) \ t

)∩ A,

which implies that

l(t)− 1 = |N(t) \ t| =∑

W ′∈Mt

|N(W ′,χ(W ′))(t) \ t| =∑

W ′∈Mt

l(W ′,χ(W ′))(t)− 1

and

lA(t)− 1 =∑

W ′∈Mt

(lA∩W ′(t)− 1). (8.2.1)

Now, t 6∈ A, if and only if t 6∈ A ∩W ′ for all W ′ ∈ Mt. This occurs if and only

if lA∩W ′(t) > 0 for all W ′ ∈Mt and then by equation (8.2.1) is true if and only if

we get lA(t) > 0 as required.

98

Now, if x ∈ W ,

lA(x, xt) = l(xtx−1)− 2|N(xtx−1) ∩ x · A|

= l(xtx−1)− 2|N(xtx−1) ∩ 1 · (x · A)| = lx·A(1, xtx−1)

so lA(x, xt) = lx·A(xtx−1). Therefore, lA(x, xt) > 0 if and only if lx·A(xtx−1) > 0.

Also, t 6∈ N(x−1) + A if and only if xtx−1 6∈ xN(x−1)x−1 + xAx−1 = N(x) +

xAx−1 = x · A. So t 6∈ N(x−1) + A if and only if xtx−1 6∈ x · A.

Putting together the previous statements, we have that the statement of the

proposition is equivalent to the statement lx·A(xtx−1) > 0 if and only if t 6∈ x ·A.

Following Proposition 8.2.6, we let A′ = x · A ∈ A(W,S) and t′ = xtx−1, then this

case is equivalent to lA′(t′) > 0 if and only if t′ 6∈ A′ which is true due to the x = 1

case above.

At this point, we can use the previous lemmas and propositions to prove The-

orem 8.1.4:

Proof. (1)⇔(2)⇒(3) is clear.

(3)⇒(4): According to section 2.3, we know that for any dihedral subgroup W ′

of W , N(W ′,χ(W ′))(w) = N(W,S)(w) ∩W ′. This implies Ω(W ′,A∩W ′) is a subgraph of

Ω(W,A). So, if A 6∈ A(W,S), Lemma 8.2.4 and Lemma 8.2.5 imply that there exists

some dihedral subgroup W ′ of W with Ω(W ′,A∩W ′) containing a cycle with four

edges.

(4)⇒(5): Suppose that A ∈ A(W,S), x ∈ W and t ∈ N(x−1)+A. Then Proposition

8.2.7 implies that lA(x, xt) < 0. But by [11], lA(1, x) + lA(x, xt) = lA(1, xt) and

this implies lA(xt)− lA(x) = lA(x, xt) < 0.

(5)⇒(1): Suppose Ω(W,A) has a cycle. This means that xt1...tn = x for some n > 0

(all ti ∈ T and with ti 6∈ N((xt1...ti−1)−1) + A for all i). By assumption, lA(x) <

99

lA(xt1) < lA(xt1t2) < ... < lA(xt1t2...tn) = lA(x) which is a contradiction.

Remark 8.2.8. In case (W,S) is finite, say of order m, we can give a much simpler

proof of the equivalence of (1), (2), (4), and (5). By the proof above, we have

(4)⇒(5)⇒(1)⇔(2). If (4) fails, then w ·A 6= T for all w ∈ W (or else A = w−1 ·T

which is an initial section). So we can choose t 6∈ w · A. It follows that we can

recursively choose t1, ..., tm such that t1 6∈ A and ti 6∈ ti−1 · · · t1 ·A for i = 2, ...,m.

This gives us the following path in Ω(W,A): 1→ t1 → . . .→ tm · · · t1. However, this

path has m + 1 elements and W has m elements, so there must be two elements

in the path that are the same thus creating a cycle.

100

CHAPTER 9

CLOSURE IN THE ROOT SYSTEM

This chapter recalls a notion of closure on the positive roots that has been

studied by Dyer [16]. A conjectural relation, due to Dyer, between these closures

and the sets T≤m (m ∈ N), as defined in section 2.4 and generalized in Chapter

3, would lead to a new and potentially interesting normal form for elements of

Coxeter groups, via known special cases of other conjectures of Dyer ([17]) involv-

ing initial sections and these closures. The results of the previous chapter afford

the possibility of bringing the study of twisted Bruhat orders to bear on these

conjectures.

9.1 2-closure

Let (W,S) be a fixed Coxeter system. Recall that we denote the roots and

positive roots by Φ and Φ+ respectively. We begin by describing a notion of

closure on the subsets of positive roots, Φ+. This closure is similar to the notion

of closure in the crystallographic root system of a finite Weyl group as described

in [3].

Definition 9.1.1. Let Γ ⊆ Φ+. We say that Γ is 2-closed if for all α, β ∈ Γ, we

have (R≥0α + R≥0β) ∩ Φ+ ⊆ Γ.

101

Remark 9.1.2. Let TΓ ⊆ T be the subset of reflections corresponding to Γ via the

canonical bijection. Then, 2-closure is the same as the following. TΓ is 2-closed

if for all tα, tβ ∈ TΓ with W ′ = 〈tα, tβ〉, t ∈ W ′ ∩ T | tα ≺R t ≺R tβ ∪ t ∈

W ′ ∩ T | tβ ≺R t ≺R tα ⊂ TΓ for all reflection orders ≺R on T (see Chapter 8).

Definition 9.1.3. We say a subset Γ ⊆ Φ+ is biclosed if Γ and Φ+ \ Γ are both

2-closed.

In particular, initial sections of reflection orders are biclosed subsets. Moreover,

A(W,S) as defined in Chapter 8 is the set of all subsets of T corresponding to

biclosed subsets of Φ+ under the natural bijection Φ+ ↔ T .

9.2 More types of subsets of Φ+

We introduce three more types of subsets of Φ+. These notions are due to

Dyer, and further information about them can be found in [17].

Definition 9.2.1. Let Γ ⊆ Φ+.

1. We say Γ is balanced if for all α ∈ Γ and W ′ ∈ Msα , then β ∈ Φ+W ′ such

that l(W ′,χ(W ′))(sβ) < l(W ′,χ(W ′))(sα) implies that β ∈ Γ.

2. We say Γ is bipedal if for all α ∈ Γ and W ′ ∈Msα with α 6∈ ΠW ′ , ΠW ′ ⊂ Γ.

3. We say Γ is unipedal if for all α ∈ Γ and W ′ ∈ Msα , then ΠW ′ = β, γ

implies that either β ∈ Γ or γ ∈ Γ.

The following proposition is then clear from the definitions.

Proposition 9.2.2. Suppose Γ ⊆ Φ+.

1. If Γ is balanced then Γ is bipedal.

102

2. If Γ is bipedal, then Γ is unipedal.

3. If Φ+ \ Γ is closed, then Γ is unipedal.

4. If Γ ⊆ Φ+ is bipedal and ∆ ⊆ Φ+ is unipedal, then Γ ∩∆ is unipedal.

For any subset Γ ⊆ Φ+, we let Γ denote the 2-closure of Γ, that is the smallest

2-closed set containing Γ. The following is conjectured by Dyer in [17].

Conjecture 9.2.3. If Γ ⊆ Φ+ is unipedal, then Γ is biclosed.

This would strengthen the following conjecture, also in [17], parts of which

were raised in [13, Remark 2.1.4].

Conjecture 9.2.4. Let A := Γ ⊆ Φ+| Γ is biclosed. Then A is a complete

lattice with∨i∈I Γi =

⋃i∈I Γi for an arbitrary family Γii∈I ⊂ A.

The last conjecture extends the known fact that W under weak order is a meet

semi-lattice (see [2] for example). In fact, [17] proves the following:

Theorem 9.2.5. Let Φw := Φ+ ∩ w(−Φ+). If Γ ⊆ Φw for some w ∈ W and Γ is

unipedal, then Γ = Φx ⊆ Φw for some x ∈ W .

The previous theorem provides a new formula for the join (when defined) of

elements ofW in weak order. For x, y, z ∈ W , x∨y = z if and only if Φx ∪ Φy = Φz.

There is a similar (proven) formula for meet, which we do not give.

9.3 A Conjectural Normal Form for Elements of W

Our conjectural normal form for elements of W depends on the following con-

jecture of Dyer in [17].

103

Conjecture 9.3.1. Let m ∈ N and let Φ+≤m = α ∈ Φ+| h∞(sα) ≤ m be the

subset of positive roots corresponding to T≤m. Then Φ+≤m is balanced.

In fact, we will only need the weaker conjecture that Φ+≤m is bipedal (see part

1 of Proposition 9.2.2). Then we have the following theorem.

Theorem 9.3.2. Let m ∈ N such that Φ+≤m is bipedal.

1. For any x ∈ W , there is a unique x′ ∈ W such that Nm(x′) = Nm(x),

and any y ∈ W with Nm(y) ⊇ Nm(x) can be written y = x′y′′ with l(y) =

l(x′) + l(y′′).

2. Any 1 6= w ∈ W can be uniquely written as w = w1 · · ·wn for certain

wi ∈ W \1 with i = 1, ..., n and n ≥ 1 such that l(w) = l(w1) + · · ·+ l(wn)

and wi = (wi · · ·wn)′ for each i where x 7→ x′ is as in 1.

Proof. Let x ∈ W . Then Φx is biclosed and thus unipedal by part 3 of Proposition

9.2.2. By assumption, Φ+≤m is bipedal and so Φx ∩ Φ+

≤m is unipedal by part 4 of

9.2.2. Therefore, according to Theorem 9.2.5, we see that Φx ∩ Φ+≤m = Φx′ for

some x′ ∈ W . If y ∈ W with Nm(y) ⊇ Nm(x), then Φx ∩ Φ+≤m ⊆ Φy ∩ Φ+

≤m ⊆ Φy.

Therefore we get that

Φx′ = Φx ∩ Φ+≤m ⊆ Φy = Φy,

and so y = x′y′′ with l(y) = l(x′) + l(y′′) due to well known properties of the weak

order on (W,S), which can be found in [2]. This proves 1, and 2 follows from 1

inductively.

The next proposition shows that the previous theorem holds in many basic

Coxeter systems.

Proposition 9.3.3. Let (W,S) be a Coxeter system, and let m ∈ N.

104

1. If (W,S) is finite or affine, then Φ+≤m is bipedal.

2. If (W,S) is right-angled, that is m(s, s′) ∈ 1, 2,∞ for all (s, s′) ∈ S × S,

then Φ+≤m is bipedal.

Proof. If (W,S) is finite, then Φ+≤m = Φ+ for all m ∈ N, and so the result is

trivial. Suppose that (W,S) is affine. Then there is a well-known description of

the root system of (W,S) in terms of the root system, Ψ, of the associated finite

Weyl group, (W , S). We remind the reader of this now (see [24]). We let U

denote the span of Ψ, and let ∆ be the simple roots for Ψ. We define V := U+Rδ

to be a vector space with U as a subspace of codimension one. Then, we have

Φ = α + nδ| α ∈ Ψ; n ∈ Z,

Φ+ := α + nδ| α ∈ Ψ+; n ≥ 0 ∪ −α + nδ| α ∈ Ψ+; n ≥ 1,

and Π = ∆∪δ− α is the set of simple roots for Φ where α is the highest root for

Ψ. It is well known that every reflection, sα+nδ (α ∈ Ψ), is contained in a unique

infinite maximal dihedral reflection subgroup, namely the group generated by

sα, sδ−α. It follows from inspection in the dihedral case then that for α ∈ Ψ+ we

get h∞(sα+nδ) = h∞(s−α+(n+1)δ) = n. Now, let m ∈ N, and suppose α+eδ ∈ Φ≤m.

Let sα+eδ ∈ W ′ be a dihedral reflection subgroup with ΠW ′ = β + nδ, γ + kδ.

Then, we have α + eδ = a(β + nδ) + b(γ + kδ) with a, b ∈ Z>0. Therefore,

e = an + bk, and so an + bk ≤ m with a, b ∈ Z>0. This implies that n ≤ m

and k ≤ m, and thus h∞(β + nδ) ≤ n ≤ m and h∞(γ + kδ) ≤ k ≤ m. Hence

ΠW ′ ⊂ Φ≤m as required.

To show 2, suppose that (W,S) is right-angled. We know that every dihedral

reflection subgroup either has order 4 or ∞. Then for any t ∈ T and W ′ ∈ Mt

105

with |W ′| = 4, we must have t ∈ χ(W ′) and so h(W ′,χ(W ′))(t) = 0. Therefore, we

must have

h∞(t) =∑

W ′∈Mt|W ′|=∞

h(W ′,χ(W ′))(t) =∑

W ′∈Mt

h(W ′,χ(W ′)) = h(t).

Now suppose α ∈ Φ+≤m and W ′ ∈ Msα . Let β, γ = ΠW ′ . Then, by previous

statement and the fact that sβ ≤∅ sα in Bruhat order, we get that

h∞(sβ) = h(sβ) ≤ h(sα) = h∞(sα) ≤ m,

and so β ∈ Φ+≤m, and similarly for γ as required.

This normal form described by Theorem 9.3.2 could be called the m-automatic

normal form for reasons we now describe. Suppose m ∈ N is fixed. Let p : T → N

be given by p(t) = 1 for all t ∈ T . Then lp = l, and we can construct the m-

canonical automata from Chapter 5, Mm. If we input a reduced expression for w ∈

W in to the automata starting at (m,Nm(1)), we end on the node corresponding

to (m,Nm(w)). According to the conjecture, there is a unique minimal element

x ∈ W with (m,Nm(x)) = (m,Nm(w)). This element is w1. Then, we repeat this

process with w−11 w and so on until we are finished. Since this normal form comes

from the m-canonical automata, the name is justified.

Remark 9.3.4. Let p be as in Chapter 3 with associated setM. For any m ∈M,

we can define Φ+≤m := α ∈ Φ+| hW,p,M∞(sα) ≤m. Following Conjecture 9.3.1, it

is reasonable to conjecture that Φ+≤m is balanced (or weaker, bipedal). We include

the following example to show that this is in fact not the case in general.

Example 9.3.5. Let (W,S) be the affine Weyl group of type B2. That is S =

106

r, s, t with m(r, s) = m(s, t) = 4 and m(r, t) = 2. Due to well known facts,

there are 8 elementary reflections. We list this set below.

T0 = r, rsr, srs, s, sts, tst, t, rtstr.

Now, there are three conjugacy classes of reflections. Define p : T → N3 by

p(u) = (1, 0, 0) if u is conjugate to r, p(u) = (0, 1, 0) if u is conjugate to s, and

p(u) = (0, 0, 1) if u is conjugate to t; let lp : W → N3 and hW,p,M∞ : T → N3 be

the corresponding length and infinite height functions.

Next, let Φ+≤(0,1,0) be as defined in the previous remark. Define

W ′ := 〈rstsr, rsrtstrsr, srstsrs, s〉

so that χ(W ′) = rstsr, s. Then the reflections of W ′ have the following infinite

heights: hW,p,M∞(s) = (0, 0, 0), hW,p,M∞(rstsr) = (1, 0, 0), hW,p,M∞(rsrtstrsr) =

(0, 1, 0), and hW,p,M∞(srstsrs) = (1, 0, 0). Then clearly, αrsrtstrsr ∈ Φ+≤(0,1,0) and

rsrtstrsr ∈ W ′, but αrstsr ∈ ΠW ′ and αrstsr 6∈ Φ+≤(0,1,0). Therefore Φ+

(0,1,0) is not

bipedal. We note that (according to Proposition 9.3.3) this example does not

provide a counterexample to Conjecture 9.3.1.

107

CHAPTER 10

HYPERBOLIC COXETER GROUPS

The standard classification of finite reflection groups (or finite Coxeter sys-

tems) and affine Coxeter systems can be found in [23] or [24]. Affine Coxeter

systems are in some natural sense the smallest infinite Coxeter groups (e.g. they

have only polynomial growth, see Chapter 12). In this chapter, we will consider

the “next” class of infinite Coxeter systems known as hyperbolic Coxeter systems.

There is a very well known, case-by-case, classification of hyperbolic Coxeter sys-

tems, which can be found in [23, chapter 6]. In this chapter, we discuss several

characterizations of hyperbolic Coxeter systems by properties of their Coxeter

graphs. According to [6], it is well known that any irreducible, infinite, non-affine

Coxeter system contains a standard parabolic subsystem that is a hyperbolic Cox-

eter system; however, we could not find a proof in the literature, and the proof of

the characterizations we include allows us to provide a case-free proof of this fact.

For the remainder of this chapter, we will assume that S is a finite set.

10.1 Coxeter Graphs

Fix a Coxeter system, (W,S). We let Γ := Γ(W,S) be the associated Coxeter

graph; that is, Γ is the undirected, labeled graph with vertex set S and edges

(si, sj) whenever m(si, sj) ≥ 3 and each edge labeled by the corresponding order

108

m(si, sj). To this graph, we associate the matrix of the bilinear form defined in

section 2.2. That is, A is a symmetric matrix with Ai,j := Asi,sj = (αi, αj). We

say that (W,S) is irreducible if Γ is connected. For J ⊆ S, let ΓJ be the full

subgraph of Γ with vertices corresponding to J . We may abuse notation and say

that s ∈ S − J is connected to J if s is connected to ΓJ .

10.2 Symmetric Matrices and Sylvester’s Law

For any real symmetric matrix over a finite dimensional vector space, we define

the inertia of A to be the triple (a, b, c) where a represents the number of positive

eigenvalues of A, b represents the number of negative eigenvalues of A, and c is the

number of zero eigenvalues of A. We will abuse notation and say that a Coxeter

system has inertia (a, b, c) if the corresponding matrix A has inertia (a, b, c). We

say that an n×n symmetric matrix A is positive definite if it has inertia (n, 0, 0).

We say that a n × n symmetric matrix is positive semidefinite if it has inertia

(a, 0, c), i.e. it has only non-negative eigenvalues. If A is positive definite or

positive semi-definite then we say A is of positive type. We state the following

fact of real symmetric matrices (see for example [22]):

Proposition 10.2.1 (Sylvester’s law of inertia). If A is an n× n real symmetric

matrix with inertia (a, b, c), then A is congruent to the diagonal matrix D =

Ia ⊕−Ib ⊕ 0c.

10.3 Hyperbolic Coxeter Systems

Let (W,S) be an irreducible Coxeter system with associated graph Γ (so Γ

is connected) and matrix A (representing the bilinear form as in 2.2). Suppose

S is finite and |S| = n. Following [23], we know that W is finite if and only

109

if A is positive definite, that is, if A has inertia (n, 0, 0). Also, we know that

W is affine if and only if the inertia is (n − 1, 0, 1). We define an irreducible

Coxeter system to be hyperbolic if it has inertia (n − 1, 1, 0) and (v, v) < 0

for all v ∈ C := λ ∈ V | (λ, αs) > 0 ∀s ∈ S. Also, we have the following

characterization of hyperbolic Coxeter systems:

Proposition 10.3.1. [23, Prop. 6.8] Let (W,S) be an irreducible Coxeter system,

with graph Γ and associated bilinear form (−,−) with matrix A. Then (W,S) is

hyperbolic if and only if the following two conditions are satisfied:

1. (−,−) (or equivalently A) is non-degenerate, but not positive definite.

2. For each s ∈ S, the Coxeter graph Γ′, obtained by removing s from Γ, is of

positive type.

10.4 Non-affine Coxeter Systems

At this point, we prove a quick technical lemma that will be needed in what is to

follow. From this point on, we may use (W,S), W and Γ = Γ(W,S) interchangeably

to represent the corresponding Coxeter system. There is no confusion in doing this

as every finite rank (W,S) has a corresponding Coxeter graph and every simple

finite, undirected, edge-labeled graph (labels in N≥3 ∪ ∞) gives rise to a finite

rank Coxeter system. We say a graph Γ is of finite (resp. affine) type if the

corresponding Coxeter system is finite (resp. affine).

Lemma 10.4.1. Suppose that (W,S) is an infinite, irreducible, non-affine Coxeter

system. Let |S| = n. Suppose that there exists s ∈ S such that the standard

parabolic subsystem (WS\s, S \ s) is a connected affine Coxeter system. Then

the inertia of (W,S) is (n− 1, 1, 0).

110

Proof. Let (W,S) be as in the statement. Thus, there exists s ∈ S such that

S \ s is of connected affine type. By removing another simple reflection s′ we

must get a finite Coxeter system of rank n− 2, and so (W,S) must have at least

n− 2 positive eigenvalues.

Now, we order S so that S = s1, ..., sn−1, sn where sn = s. Then, we let A′

be the (n − 1) × (n − 1) leading principal minor of the matrix A associated to

(W,S) and Γ. Since A′ is the matrix for (WS\s, S \ s) which is of affine type

of rank n − 1, it has inertia (n − 2, 0, 1) and there exists an isotropic root called

δ. Additionally, we know that δ =∑n−1

i=1 kiαsi with ki > 0 for all i (c.f. [24]). By

Proposition 10.2.1, we know that there exists an invertible matrix C such that

CtrA′C = In−2 ⊕ 01. We can lift this to a matrix D = C ⊕ I1 so that

B := DtrAD =

In−2

0

.

.

.

0

a1

.

.

.

an−2

0 · · · 0 0 c

a1 · · · an−2 c 1

where c = (αs, δ) and ai ∈ R for all i. Since (W,S) is irreducible, Γ is con-

nected and so there exists some i 6= n such that (αsn , αsi) < 0. Thus by the

characterization of δ above we see that (αsn , δ) 6= 0, i.e. c 6= 0.

Next, we want to find the determinant of B. To do this, we expand the matrix

along the (n−1)’st row noting that we have to multiply c by −1 due to its position

in the matrix. We thus get that det(B) = −c · det(B′) where

B′ :=

In−20

.

.

.

0

a1 · · · an−2 c

111

Now, we can find the determinant of B′ easily by expanding along the (n− 1)’st

column of B′ (where we will multiply c by +1 due to its position in the matrix.

This gives us that det(B′) = c det(In−2). Putting the results together we see that

det(B) = −c det(B′) = −c2 det(In−2) = −c2.

Since B has at least n−2 positive eigenvalues and it has negative determinant (due

to the fact that c 6= 0), we know that exactly one of the remaining eigenvalues must

be negative. Indeed, if both were negative or both were positive, the determinant

of B would be positive. Thus B has inertia (n−1, 1, 0). Using Proposition 10.2.1,

we can conclude that A must have inertia (n− 1, 1, 0) as well.

10.5 A Class of Coxeter Systems

We define a class of Coxeter systems as follows.

Definition 10.5.1. Let H be the class of all irreducible, finite rank, infinite,

non-affine Coxeter systems, (W,S), with Coxeter graph Γ such that every proper

subgraph Γ′ ⊂ Γ is the disjoint union of graphs that are Coxeter graphs of only

finite or affine type.

Lemma 10.5.2. Every hyperbolic Coxeter system is in H.

Proof. By Proposition 10.3.1 part (2) we see that any hyperbolic Coxeter system

must be in H. In particular, the hyperbolic Coxeter systems are precisely those

in H with inertia (n− 1, 1, 0).

10.6 A Smaller Class of Coxeter Systems

We now define a certain subclass of H that will be interesting to study.

112

Definition 10.6.1. Let h ⊆ H be the class of all Coxeter systems, (W,S), in H

that have Γ satisfying exactly one of the following (it will be clear that a graph

cannot satisfy both):

(h1) There exists s ∈ S such that Γ − s is a disjoint union of Coxeter graphs

of finite type.

(h2) Γ− s is a connected Coxeter graph of affine type for all s ∈ S.

Example 10.6.2. Notice that any infinite rank 3 Coxeter system can be described

by one of the two following graphs (note: we color a vertex if we intend to refer

to it below):

(a)

•

•

m

k////l

(b)

• •m k

with l ≥ m ≥ k ≥ 3. For arbitrary m and k, if we remove vertex from graph

(b) we will be left with the Coxeter graph for a dihedral Coxeter system of order

4. Thus, graph (b) is clearly of type (h1) (provided it is infinite and non-affine).

For graph (a), if k 6= ∞, then removing vertex leaves us with the Coxeter

graph for a finite dihedral Coxeter system, and thus (a) is of type (h1) (provided

it is not affine). If k =∞ then we necessarily have m =∞ and l =∞. Therefore,

removing any vertex from (a) gives us the graph • •∞ , which is clearly of affine

type. Thus any infinite, non-affine irreducible rank 3 Coxeter system is in h.

Lemma 10.6.3. Any Coxeter system in h is hyperbolic.

Proof. Let (W,S) be in h ⊆ H and let |S| = n, i.e. the rank of (W,S) is n. Let Γ

be the Coxeter graph of (W,S) and let A be the matrix associated to (−,−) (or

similarly to Γ).

113

Case 1: (W,S) is of type (h1). Then, since there exists s ∈ S such that (WS−s, S−

s) is of finite type and so the associated bilinear form and matrix A′ must

have inertia (n−1, 0, 0). Thus A must have at least n−1 positive eigenvalues.

If A has inertia (n, 0, 0) (resp. (n − 1, 0, 1)) then (W,S) would be of finite

type (resp. affine type) and thus (W,S) 6∈ H, which is a contradiction (since

h ⊆ H by definition). Therefore A must have inertia (n − 1, 1, 0) forcing

(W,S) to be hyperbolic by Proposition 10.3.1.

Case 2: (W,S) is of type (h2). Then since S−s is of connected affine type Lemma

10.4.1 implies that the inertia of (W,S) is (n− 1, 1, 0). In particular (−,−)

is non-degenerate. Furthermore, since for each r ∈ S we have Γ− r is of

positive type, Proposition 10.3.1 implies that (W,S) is hyperbolic.

10.7 Parabolic Subsystems of Non-affine Coxeter Systems

Lemma 10.7.1. Any infinite, irreducible, finite rank, non-affine Coxeter system

(W,S) contains a standard parabolic subsystem in the class H.

Proof. Let (W ′, S ′) be a minimal rank, irreducible, infinite, non-affine parabolic

subsystem of (W,S), which must exist. Then since (W ′, S ′) is minimal rank, if we

remove any vertex s then (WS′\s, S′ \ s) cannot contain an infinite, non-affine

parabolic subsystem. Thus, (WS′\s, S′ \ s) must be of finite or affine type.

10.8 Characterizations of Hyperbolic Coxeter Systems

Lemma 10.8.1. Let (W,S) be in the class H. Then (W,S) is in the class h.

114

Proof. Suppose that (W,S) is not of type (h1) and let Γ := ΓS. Then because

(W,S) is in class H, we know that (WS\s, S \ s) is of affine type for every

s ∈ S. Suppose that s ∈ S and ΓS\s is not connected. Let Γ0,Γ1, ...,Γn be

the connected components (so n ≥ 1). Now, since ΓS\s is of affine type we can

assume without loss of generality that Γ0 is of affine type. Then Γ′ = Γ0 ∪ s is

irreducible (since (W,S) is irreducible) and is properly contained in Γ0∪s∪s1

where s1 ∈ Γ1. Thus Γ′ is properly contained in Γ. However, Γ′ cannot be of finite

or affine type or else its subgraph Γ0 would be finite, contradicting that Γ0 is

affine. Therefore if ΓS\s is not connected then Γ properly contains Γ′ which is

infinite and non-affine, contradicting the assumption that (W,S) is in H. Thus,

we have shown that if (W,S) is not of type (h1), then for every s ∈ S, Γ \ s is

of connected affine type, i.e. (W,S) is of type (h2).

Theorem 10.8.2. The following hold:

1. The classes h and H both coincide with the class of hyperbolic Coxeter sys-

tems.

2. Any finite rank, irreducible, infinite, non-affine Coxeter system contains a

standard parabolic subsystem of hyperbolic type.

Proof. By Lemma 10.5.2, we know that all hyperbolic Coxeter systems are con-

tained in H. Then, by Lemma 10.8.1 we know that H is contained in h (so clearly

h and H are equal). Finally, by Lemma 10.6.3 we know that every element of h

is a hyperbolic Coxeter system. Thus we have equality throughout and 1 follows.

For 2, we use part 1 of the theorem just proved along with Lemma 10.7.1.

115

CHAPTER 11

LARGE REFLECTION SUBGROUPS

By a universal Coxeter system, we mean one with no braid relations (the un-

derlying group is a free product of cyclic groups of order 2). The main result

of this chapter is that all irreducible, non-affine Coxeter systems have universal

reflection subgroups of arbitrarily large rank. To prove this, we use the imagi-

nary cone, which was introduced by Dyer ([14]) in order to prove a conjectured

characterization of coverings in dominance order (see Remark 2.5.3). Dyer raised

the question of whether the imaginary cone of an infinite, irreducible, non-affine

Coxeter system can be approximated by imaginary cones of universal reflection

subgroups. We show this is the case for hyperbolic Coxeter groups, and then our

main result follows from Chapter 10. Many of the auxiliary results in this chapter

can be found in [15]. Since we will define a few different cones, we need some

terminology regarding convexity and polyhedral cones. For a reference on such

terms, consult [1].

11.1 Convex Sets

In this section, we describe our terminology involved with convex sets. Let V

be a real vector space.

Definition 11.1.1. We say that x is a convex combination of x1, ...xn ∈ V if there

exists λ1, ..., λn ∈ R such that

116

1. x = λ1x1 + · · ·+ λnxn

2. λ1 + · · ·+ λn = 1

3. λi ≥ 0 for all i.

For any set M we call the set of all convex combinations of elements of M the

convex hull of M and we denote it by conv(M).

For any two elements x, y ∈ V , we denote

[x, y] := λx+ (1− λ)y| 0 ≤ λ ≤ 1.

We say a set C ⊂ V is convex if, for all x, y ∈ C (x 6= y), [x, y] ⊂ C.

11.2 A Metric on Convex Sets

Assume that V is finite dimensional, so that V ∼= Rn. Then V has the usual

distance function as a metric d : V × V → R. Let Y ⊆ V and x ∈ V . Define

d(x, Y ) = infy∈Y

d(x, y) (11.2.1)

Then for any X ⊆ V and Y ⊆ V , we define

d(X, Y ) = supx∈X

d(x, Y ) (11.2.2)

Definition 11.2.1. Let K := X ⊆ V | X is compact.

The following proposition can be found in [25, §21 VII].

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Proposition 11.2.2. Consider the distance function defined in (11.2.2). We

create a new distance function on K given by dK(X, Y ) = maxd(X, Y ), d(Y,X).

Then dK defines a metric on K.

The metric defined above is known as the Hausdorff metric (or Hausdorff

distance).

11.3 Approximation of a Sphere by Polytopes

Now, suppose that Sd is the d-sphere sitting inside of Rd+1. We want to show

that Sd can be approximated by the boundary of a convex hull of a finite number

of points. For any set A ⊆ Rn, let ∂A denote the boundary of A. To make the

previous statement more concrete, we have the following lemma.

Lemma 11.3.1. There exists a sequence of compact, convex sets Pm with Pm =

conv(x1, ..., xm) where x1, ..., xm ⊂ Sd and m <∞ such that dK(∂Pm, Sd)→ 0

as m→∞, i.e. ∂Pm → Sd in the metric space K.

Proof. Let ε > 0 be given. Now, for any point z ∈ Rd+1, define Ba(z) to be the

open ball of radius a centered at z. Then we define

U := B ε2(z)| z ∈ Sd.

Clearly Sd ⊂⋃U∈U U . Since Sd is compact, we can find a finite open subcover of

U call it U ′ := U1, ..., Um (where m = mε is dependent on ε).

Now, each Ui is a ball of radius ε/2 centered at xi ∈ Sd. Thus, we have a finite

set of points x1, ..., xm ⊂ Sd. Let Pm := conv(x1, ..., xm). Let ∂Pm denote the

boundary of Pm.

118

Suppose y ∈ ∂Pm. Now, there exists a minimal set I := xi1 , ..., xik ⊂

x1, ..., xm such that y ∈ conv(xi1 , ..., xik = conv(I) ⊂ ∂Pm. Then, we can

choose a supporting affine hyperplane H of Pm such that I ⊂ Pm ∩ H. Let H+

and H− represent the positive and negative half-spaces associated to H. Suppose,

without loss of generality, that Pm ⊂ H−. Let H ′ := int(H+) and let u be an

outer unit normal vector of H (u points into H+). By assumption, Pm ∩H ′ = ∅

and so ∂Pm ∩H ′ = ∅ and x1, ..., xm ∩H ′ = ∅.

Next, for the sake of contradiction, we suppose that d(y, Sd) ≥ ε. Then let

L = H + ε2u be the affine hyperplane parallel to H with d(H,L) = ε

2and L ⊂ H ′.

It follows that Sd ∩ L+ 6= ∅, where L+ is the positive half-space associated to L.

Indeed, if Sd∩L+ = ∅, then Sd ⊆ L− and so d(Pm∩H,Sd) ≤ d(H,L). This would

imply that

d(y, Sd) ≤ supx∈Pm∩H

d(x, Sd) = d(Pm ∩H,Sd) ≤ d(H,L) =ε

2< ε,

which is not true by assumption. Thus, we let z ∈ Sd ∩ L+. Then B ε2(z) ⊂ H ′

by construction. However, z ∈ Sd so z ∈ Ui for some Ui ∈ U ′ so xi ∈ B ε2(z) ⊂ H ′

which is a contradiction. Therefore, d(y, Sd) < ε.

Since y is arbitrary, we can conclude that d(∂Pm, Sd) < ε. Now, suppose

y ∈ Sd, then y ∈ Ui for some i. Thus,

d(y, ∂Pm) = infz∈∂Pm

d(y, z) ≤ d(y, xi) <ε

2.

Again, since y is arbitrary, we can conclude that d(Sd, ∂Pm) < ε2. Thus we have

determined that dK(∂Pm, Sd) = maxd(Sd, ∂Pm), d(∂Pm, S

d) < max ε2, ε = ε

as desired.

119

11.4 Cones in Coxeter Systems

Recall, for any reflection subgroup W ′, we let χ(W ′) be the set of canonical

generators of W ′ as a Coxeter system. Then, according to section 2.3, we have a

corresponding set of roots, positive roots, and simple roots for the Coxeter system

(W ′, χ(W ′)) given by

ΦW ′ = α ∈ Φ| sα ∈ W ′, Φ+W ′ := ΦW ′ ∩ Φ+, ΠW ′ = α ∈ Φ+| sα ∈ χ(W ′)

respectively.

We now introduce several cones that appear inside of the vector space V in-

troduced in section 2.2. For any W ′ a reflection subgroup of W , we define the

fundamental chamber for W ′ on the vector space V by

CW ′ = v ∈ V | (v, α) ≥ 0 for all α ∈ ΠW ′.

Then we denote the Tits cone of W ′ by XW ′ where this is defined by

XW ′ =⋃w∈W ′

w(CW ′).

In addition to these we define the following:

KW ′ := R≥0ΠW ′ ∩ −CW ′ , ZW ′ :=⋃w∈W ′

w(KW ′)

where ZW ′ is known as the imaginary cone of W ′ on V (we only define it if S is

finite). The definition of the imaginary cone here is taken from [15], which models

it loosely on one characterization of imaginary cones of Kac-Moody Lie algebras

120

([24]). The results we use are analogs for general W of results proved by Kac for

imaginary cones of Kac-Moody Lie algebras (for which, however, the proofs use

imaginary roots, which have no counterpart here). We will refer to CW ,XW ,KW

and ZW simply by C ,X ,K and Z .

11.5 Topology on Rays in Cones

By a ray through v in V we mean a set λv| λ ∈ R≥0. We let R denote the

set of rays of V . We can give R a topology and a metric in the following way. Let

B be compact convex body containing 0 in V , and let ∂B denote the boundary of

B. Define the map ϕ : R → ∂B by ϕ(x) = x∩∂B. It is clear that ϕ is a bijection.

We thus declare ϕ to be a homeomorphism between R and ∂B, and this gives R a

topology and a metric corresponding to ∂B in V , with the topology independent

of B. We also introduce the following subsets of R. Let R+ := R≥0α| α ∈ Φ+

and let R0 := R+ \R+. We say a ray, R≥0α, is positive if (α, α) > 0 and isotropic

if (α, α) = 0. We recall some facts about these sets (both proven in [15]).

Proposition 11.5.1. 1. R+ consists of positive rays and is discrete in the

subspace topology.

2. R0 consists of isotropic rays and is closed in R.

3. R0 is the set of limit rays of R+.

In addition, we have the following theorem.

Theorem 11.5.2. The cone Z is the convex closure of⋃r∈R0

r ∪ 0.

This theorem has the following immediate corollary.

121

Corollary 11.5.3. If W ′ is a finitely generated reflection subgroup of W , then

ZW ′ ⊆ ZW .

Remark 11.5.4. A much stronger fact, proven in [15], which we will not use, is

that in the situation of the corollary above we actually have ZW ′ ⊆ ZW .

11.6 Cones in Hyperbolic Coxeter Systems

Recall that we say (W,S) is a hyperbolic Coxeter system if the bilinear form

(−,−) on V has inertia (n− 1, 1, 0) and every proper standard parabolic Coxeter

system is of finite or affine type (see Chapter 10).

Suppose we fix a hyperbolic Coxeter system (W,S). We know that V has

a basis x0, ..., xn such that (xi, xj) = δi,j, (xi, xi) = 1 for all i ∈ 1, ..., n, and

(x0, x0) = −1. Then, according to [15], we have (after replacing x0 by −x0 if

necessary) that

Z = L

where L = ∑n

i=0 λixi| λ0 ≥√∑n

i=1 λ2i .

Now, consider the setup as in section 11.5. Let RΠ ⊂ R be the set of rays

spanned by non-zero elements of R≥0Π. Since (W,S) is hyperbolic, the bilinear

form is non-degenerate, and thus the interior of C is non-empty (see [15]). Take

ρ ∈ int(C ) so that (ρ, α) > 0 for all α ∈ Π. Therefore, we may define a map

τ : R≥0Π \ 0 → H := v ∈ V | (v, ρ) = 1

given by τ(v) = v/(ρ, v). Then the image of τ is the set P := v ∈ V | (ρ, v) = 1,

which is a convex polytope in the affine hyperplane H with points (ρ, α)−1α for

α ∈ Π as vertices. From τ , we introduce a map on R as follows. For r ∈ R, we

122

let τ(r) = τ(x) for all x ∈ r \ 0 (note τ is independent of choice of x ∈ r \ 0).

With terminology from section 11.5 we may take B so that P ⊂ ∂B.

Now, we can see that Z ∩H is a disk. Also, by Theorem 11.5.2 we see that

Z ∩H = conv(⋃

r∈R0r)∩H.

Lemma 11.6.1. R0 is the set of rays in the boundary of Z .

Proof. First, a ray r ∈ R0 is isotropic by Proposition 11.5.1, and thus it is in

±∂(L )) = ±∂(Z ) due to the description of L above. However, r must also be

in Z , so r ∈ ∂(Z ).

For the reverse implication, the statements above imply it is enough to show

that the result is true inside of H, that is the convex hull of limit points of

(images of) roots in H includes the entire boundary sphere of(Z = L

)∩ H.

We let the boundary of Z ∩ H be denoted by S := ∂τ(Z ), and we let A :=

τ(conv

(⋃r∈R0

r))

. Then A is the convex hull of the points⋃r∈R0

τ(r). But then

the statement is clear since S is a sphere and is contained in A since Z ∩ H =

conv(⋃

r∈R0r)∩H. So every point in S must also be in A or else we would not have

that the boundary of A is a sphere (which is true since (W,S) is hyperbolic).

11.7 Universal Coxeter Systems

Suppose we have a hyperbolic Coxeter system (W,S) as in section 11.6. Then

with the same setup there, we know that Z = L and intersecting with H we get

that S = ∂τ(Z ) is a sphere in H.

Lemma 11.7.1. Suppose that x, y ∈ S. Then there exist αx, αy ∈ Φ+ such that

τ(αx), τ(αy) ∈ τ (R+) are sufficiently close to x and y respectively; furthermore,

αx and αy satisfy (αx, αy) < −1. In particular, if S ′ := sαx , sαy then (W ′, S ′) is

an indefinite infinite dihedral reflection subgroup of (W,S) and χ(W ′) = S ′.

123

Proof. Since x, y ∈ S then x and y are limit points of τ(R+) and therefore we can

find (images of) positive roots arbitrarily close to x and y. Thus, we pick positive

roots αx ∈ Φ+ and αy ∈ Φ+ such that τ(αx) and τ(αy) are close enough to x

and y respectively to ensure that aτ(αx) + bτ(αy)| a, b ∈ R>0 ∩ (Z ∩H) 6= ∅.

Again, this is possible since S is a sphere and x and y are two points (x 6= y)

on the sphere and so the relative interior of the line connecting x and y must be

included in the relative interior of the disk bounded by the sphere which is Z ∩H.

Now, we pick a point in the interior of the imaginary cone, v = aτ(αx) + bτ(αy) ∈

int(Z ∩H) ⊂ int(Z ) with a, b ∈ R>0. Then (v, v) < 0 necessarily and so (−,−)

restricted to Spanαx, αy cannot be of finite type or affine type. Thus, (W ′, S ′)

cannot be a finite dihedral subgroup. Therefore, we see that |(αx, αy)| > 1. Now

if (αx, αy) > 1 then by [4, Proposition 2.2] (and without loss of generality) we

must have αx dominates αy. However, this is impossible since, by investigation

of dominance in infinite dihedral groups ([14]), we know that αx can dominate αy

only if [αx, αy] ∩ ZW ′ = ∅, which is not the case here. Thus, (αx, αy) < −1 and

so S ′ = sαx , sαy is the canonical set of generators for the infinite dihedral group

generated by S ′, (W ′, S ′).

Now, by Lemma 11.3.1 recall that for arbitrary m we can find m points

x1, ..., xm ⊂ S such that Pm := convx1, ..., xm along with the property that

dK(Pm,S) → 0 as m → ∞. Then, by Lemma 11.7.1 above, since m is finite, we

can find αxi ∈ Φ+ with τ(αxi) sufficiently close to xi such that S ′i,j := sαxi , sαxj

is the set of canonical generators for the (infinite) dihedral group it generates,

(W ′i,j, S

′i,j). Now, let S ′ := sαx1 , ..., sαxm and W ′ be the reflection subgroup

generated by S ′.

Proposition 11.7.2. Then (W ′, S ′) is a universal Coxeter system with χ(W ′) =

124

S ′.

Proof. Consider the Coxeter system (W ′, S ′). Then by [19, Proposition 3.5], S ′ =

χ(W ′) if and only if s, s′ = χ(〈s, s′〉) for all s, s′ ∈ S ′ and this follows from

Lemma 11.7.1. Finally, since m(s, s′) = ∞ for all s 6= s′ ∈ S ′ then (W ′, S ′) is a

universal Coxeter system.

11.8 Large Reflection Subgroups

With the terminology as in section 11.6, we have H containing P . Thus, we

can view τ(Z ) as a subset of P . Additionally, we can view any subset of RΠ as a

subset of P . Thus, we will abuse notation by defining dK on any compact subset

of RΠ in the following way: for X, Y ⊂ RΠ, dK(X, Y ) = dK(τ(X), τ(Y )). We put

the previous results together to obtain the following result.

Proposition 11.8.1. Suppose (W,S) is hyperbolic. There exists a sequence of fi-

nite rank, universal reflection subgroups, (W ′m, S

′m), of W with dK(R≥0ΠW ′m ,Z )→

0 as m→∞ and dK(Z W ′m ,Z )→ 0 as m→∞.

Proof. Let ε > 0 be given. Then we can pick m large enough so that dK(Pm,S) <

ε/2 by Lemma 11.3.1 where Pm = convx1, ..., xm for some x1, ..., xm ⊂ S.

Thus, dK(Pm,Z ∩ H) < ε/2. According to Lemma 11.7.2 and the discussion

before it, we can choose αx1 , ..., αxm ⊂ Φ+ so that τ(αxi) is sufficiently close to

xi (in particular so that dK(τ(αxi), xi) < ε/2) with the property that (W ′m, S

′m) is

a universal Coxeter system where S ′m := sαx1 , ..., sαxm and W ′m = 〈S ′m〉. Now, let

Qm = convτ(αx1), ..., τ(αxm). Then Qm = R≥0ΠW ′m ∩H and dK(Qm, Pm) < ε/2

125

since dK(τ(αxi), xi) < ε/2 (by assumption above). Therefore, we see that

dK(R≥0ΠW ′ ,Z ) = dK(Qm,Z ∩H)

≤ dK(Qm, Pm) + dK(Pm,Z ∩H)

= dK(Qm, Pm) + dK(Pm,S) < ε/2 + ε/2 = ε

as desired. It remains to show that we can take dK(ZW ′m ,Z )→ 0 also.

We temporarily fix i 6= j. Then since (αxi , αxj) < −1, we have

Z 〈sαxi ,sαxj 〉= z ∈ R≥0αxi +R≥0αxj | (z, z) ≤ 0

= R≥0βxi,xj + R≥0βxj ,xi

(which is a two-dimensional cone) for some isotropic linearly independent vectors

βxi,xj and βxj ,xi . We choose our previous notation so that βxi,xj ∈ R≥0αxi +

R≥0βxj ,xi and βxj ,xi ∈ R≥0βxi,xj + R≥0αxj .

Then, from [14], since 〈sαxi , sαxj 〉 is dihedral, we have

Z 〈sαxi ,sαxj 〉= Z ∩

(R≥0αxi + R≥0αxj

).

Thus, τ(βxi,xj) and τ(βxj ,xi) are the points given by the intersection

[τ(αxi), τ(αxj)] ∩ S = τ(βxi,xj), τ(βxj ,xi).

This implies that we can choose αxi and αxj so that all of the following hold:

dK(τ(αxi), xi) < ε/4 dK(τ(αxj), xj) < ε/4

dK(τ(βxi,xj), xi) < ε/4 dK(τ(βxj ,xi), xj) < ε/4.(11.8.1)

126

Next, by Corollary 11.5.3, we know that

Z 〈sαxi ,sαxj 〉⊆ Z W ′m ⊆ Z

for all pairs i 6= j. Each of these are convex cones, and so we also have

∑i 6=j

R≥0

(R≥0βxi,xj + R≥0βxj ,xi

)⊆ Z W ′m ⊆ Z . (11.8.2)

Now, let ε > 0. We choose m large enough and αx1 , ..., αxm sufficiently close to

x1, ..., xm respectively so that (11.8.1) holds and so that dK(R≥0ΠW ′m ,Z ) < ε/2

where W ′m = 〈sαx1 , ..., sαxm 〉 (which is possible by the first part of the proposition).

Then (11.8.1) implies that dK(αxi , βxi,xj) < ε/2 for all j 6= i so that

dK([τ(βxi,xj), τ(βxj ,xi)], [τ(αxi), τ(αxj)]) < ε/2

and thus

dK

(∑i 6=j


),m∑k=1

R≥0αxk

)<ε

2.

Together, since∑m

k=1 R≥0αxk = R≥0ΠW ′m , we get that

dK

(∑i 6=j


),Z

)≤

dK

(∑i 6=j


),m∑k=1

R≥0αxk

)+ dK(R≥0ΠW ′m ,Z ) <

ε

2+ε

2= ε.

Then, equation (11.8.2) implies that dK(Z W ′m ,Z ) < ε as well, finishing the proof.

The next theorem follows from the previous propositions and Chapter 10.

127

Theorem 11.8.2. Let (W,S) be a finite rank, irreducible, infinite, non-affine

Coxeter system. Then, for any m ∈ N, W contains a reflection subgroup W ′ with

|χ(W ′)| = m such that (W ′, χ(W ′)) is a universal Coxeter system.

Proof. Let m ∈ N. By Theorem 10.8.2, (W,S) contains a hyperbolic parabolic

subsystem (WJ , J). By Proposition 11.7.2, we have that (WJ , J) contains a uni-

versal Coxeter system (W ′, S ′) of rank m and thus (W,S) also contains (W ′, S ′)

proving the theorem.

128

CHAPTER 12

EXPONENTIAL GROWTH IN COXETER GROUPS

For any finitely generated group, we can define the growth function of the group

by determining the number of elements of the group of all lengths (with respect

to the generating set). In [6], de la Harpe demonstrates that irreducible, infinite,

non-affine Coxeter systems have what is known as exponential growth by showing

that any such Coxeter system must contain a free non-abelian subgroup. Also, in

[27], Margulis and Vinberg prove the stronger result that such a W must contain

a finite index subgroup which surjects onto a non-abelian free group. Finite and

affine Coxeter systems are known to have polynomial growth.

Even more recently, in [29], Viswanath uses an alternate method to show that

any irreducible, infinite, non-affine simply-laced Coxeter system has exponential

growth. He also obtains the stronger result that for such a system (W,S) and

any J ( S the quotient W/WJ also has exponential growth. His method involves

finding a particular universal rank 3 reflection subgroup inside of (W,S). In [29,

Remark 3], the author notes that it would be interesting to know if this result

holds in the non-simply laced case as well. We intend to answer this question

using our results from the previous two chapters.

129

12.1 Growth Types

To begin with, we introduce the basic notions of growth types in finitely gen-

erated groups. Let (W,S) be a finitely generated Coxeter system. We follow [7]

or [29] for general terminology and results.

Definition 12.1.1. Let a := (ak)k≥0 be a sequence of non-decreasing natural

numbers. We define the exponential growth rate of the sequence to be ω(a) :=

lim supk→∞ a1/kk .

For each k ∈ N, we define a subset W≤k := w ∈ W | l(w) ≤ k. Then, for any

subset F ⊆ W with 1 ∈ F , we define F≤k := F ∩W≤k. Finally, for any k ∈ N and

F ⊆ W , we define the number γF,k := |F≤k| and the sequence γF := (γF,k)k≥0.

Then, let ω(F ) := ω(γF ) = lim supk→∞ γ1/kF,k .

Then, we say that a subset F has exponential growth if ω(F ) > 1 and subex-

ponential growth otherwise. If a subset F has subexponential growth, and there is

C ∈ R>0 and d ∈ Z≥0 with γF,k ≤ Ckd for all k ≥ 0, then we say F has polynomial

growth. If F is of subexponential growth and not of polynomial growth, F has

intermediate growth.

We note that if P (F,W ) =∑

w∈F ul(w) =

∑k≥0 aku

k is the Poincare series of

F , then the coefficients ak = γF,k − γF,k−1 for k ≥ 1 and a0 = γF,0.

12.2 Properties of W(3)

The proof that quotients have exponential growth requires some properties of

the universal Coxeter system W(3) = 〈a, b, c| a2 = b2 = c2 = 1〉 with Coxeter

diagram

• •

•

∞

∞////∞

130

These properties are all known and listed in [29], but we remind the reader here

for completeness.

W(3) has the following properties:

1. The Poincare series of W(3) is P (W(3)) = 1+q1−2q

.

2. Each w ∈ W(3) has a unique reduced expression.

3. The conjugacy classes of a, b, and c are all distinct.

4. For x 6= y ∈ W(3)/(W(3))a =: Wa(3) , we know that xax−1 6= yay−1. Also,

P (Wa(3) ,W(3)) =

P (W(3))

1+q= 1

1−2qso there are exactly 2k elements x ∈ W a

(3)

with l(x) = k. For each such x, we have l(xax−1) ≤ 2l(x) + 1 = 2k + 1.

5. Property 4 implies that for k ≥ 0 there are more than 2k elements t ∈

xax−1| x ∈ W a(3) with l(t) ≤ 2k + 1.

6. For any reflection t ∈ xax−1| x ∈ W a(3) , properties 2 and 3 imply that a

appears in any reduced expression for t.

12.3 Parabolic Quotients

We now prove a generalization of [29, Theorem 1].

Theorem 12.3.1. Let (W,S) be an irreducible, infinite, non-affine Coxeter sys-

tem. Then for all J ( S, W J = W/WJ has exponential growth.

Our proof is similar to the one found in [29], however, we need to make some

changes so that proof works for all Coxeter systems (as opposed to only simply-

laced Coxeter systems).

We first recall an important result due to Deodhar, [9].

131

Theorem 12.3.2. Let (W,S) be a Coxeter system, T the set of reflections, and

J ⊆ S. If t1, t2 ∈ T \ WJ with t1 6= t2, then t1WJ 6= t2WJ , that is, distinct

reflections in T \WJ lie in distinct left cosets of WJ .

According to [29], to prove Theorem 12.3.1, we need to prove the following

proposition.

Proposition 12.3.3. Let (W,S) be an irreducible, infinite, non-affine Coxeter

system and let J ( S. Then there exists M ∈ N such that for all k ≥ 0 there are

at least 2k elements t ∈ T \WJ such that l(t) ≤M(2k + 1).

If this proposition holds, Theorem 12.3.1 holds by the same argument given in

[29].

12.4 Proof of Proposition 12.3.3

Proof. For all x ∈ W , any reduced expression for x uses the same simple reflec-

tions. We write supp(x) = r ∈ S| r appears in all reduced expression for x. It

is also well known that x ∈ WJ if and only if supp(x) ⊆ J .

Now, let (W,S) be an irreducible, infinite, non-affine Coxeter system and let

s′ 6∈ J . According to Theorem 11.8.2, (W,S) contains a reflection subgroup W ′

such that W ′ ∼= W(3). Suppose χ(W ′) = t1, t2, t3. Without loss of generality,

we may assume that s′ ∈ supp(t1). Indeed, if s′ 6∈ supp(ti) for all i, then let

Z ⊂ S be the smallest subset such that W ′ ⊂ WZ . Then, take a minimal path

s′ = sn − sn−1 − · · · − s1 − z in the Coxeter diagram with si 6∈ Z for all i and

z ∈ Z; let the word corresponding to the path be x = s1 · · · sn ∈ W , and by

reordering we assume z ∈ supp(t1). Now, we define a sequence of reflections in

the following way. Let r0 = t1 and ri = si · · · s1t1s1 · · · si for all i ≥ 1. Let αri ∈ Φ+

132

be the root corresponding to ri. Now, since (s1, ..., si−1 ∪ Z) ∩ (si) = ∅ for

all i ≥ 1 and si is connected to si−1, we have (αri−1, αsi) < 0 for all i and so

l(ri) = l(ri−1) + 2 for all i ≥ 1. This implies that l(x−1t1x) = 2l(x) + l(t1);

hence, s′ = sn ∈ supp(x−1tx). Then, xW ′x−1 is a Coxeter system, isomorphic to

W(3) with χ(xW ′x−1) = xtx−1, xt2x−1, xt3x

−1 = t′1, t′2, t′3 with s′ ∈ supp(t′1)

as required.

Next, by the properties listed in Section 12.2, we know that there are at least

2k elements t ∈ W ′ ∩ T such that l(W ′,χ(W ′))(t) ≤ 2k + 1 such that t1 ∈ supp(t).

Define M := maxl(ti), l(t2), l(t3). Then, we have l(w) ≤ M(l(W ′,χ(W ′))(w)) for

all w ∈ W ′. This implies that for each t ∈ W ′∩T above, we have l(t) ≤M(2k+1).

For each t ∈ W ′ ∩ T described above, t1 appears in a reduced expression for t;

in terms of roots this means αt =∑3

i=1 ciαti with c1 > 0. Since s′ ∈ supp(t1), we

have αt1 =∑

s∈S csαs with cs′ > 0. Therefore, it follows that αt =∑

s∈S dsαs with

ds′ > 0 so that s′ ∈ supp(t). Thus, we have t 6∈ WJ . This implies that t ∈ T \WJ .

Therefore, by Theorem 12.3.2, we get that there are at least 2k elements t ∈ T \WJ

such that l(t) ≤M(2k + 1). The proposition follows.

133

REFERENCES

1. Alexander Barvinok. A course in convexity, volume 54 of Graduate Studiesin Mathematics. American Mathematical Society, Providence, RI, 2002.

2. Anders Bjorner and Francesco Brenti. Combinatorics of Coxeter groups, vol-ume 231 of Graduate Texts in Mathematics. Springer, New York, 2005.

3. Nicolas Bourbaki. Lie groups and Lie algebras. Chapters 4–6. Elements ofMathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968French original by Andrew Pressley.

4. Brigitte Brink. The set of dominance-minimal roots. J. Algebra, 206(2):371–412, 1998.

5. Brigitte Brink and Robert B. Howlett. A finiteness property and an automaticstructure for Coxeter groups. Math. Ann., 296(1):179–190, 1993.

6. Pierre de la Harpe. Groupes de Coxeter infinis non affines. Exposition. Math.,5(1):91–96, 1987.

7. Pierre de la Harpe. Topics in geometric group theory. Chicago Lectures inMathematics. University of Chicago Press, Chicago, IL, 2000.

8. Ronald de Man. The generating function for the number of roots of a Coxetergroup. J. Symbolic Comput., 27(6):535–541, 1999.

9. Vinay V. Deodhar. On the root system of a Coxeter group. Comm. Algebra,10(6):611–630, 1982.

10. Vinay V. Deodhar. A note on subgroups generated by reflections in Coxetergroups. Arch. Math. (Basel), 53(6):543–546, 1989.

11. M. J. Dyer. Hecke algebras and shellings of Bruhat intervals. II. TwistedBruhat orders. In Kazhdan-Lusztig theory and related topics (Chicago, IL,1989), volume 139 of Contemp. Math., pages 141–165. Amer. Math. Soc.,Providence, RI, 1992.

12. M. J. Dyer. Hecke algebras and shellings of Bruhat intervals. CompositioMath., 89(1):91–115, 1993.

134

13. M. J. Dyer. Quotients of twisted Bruhat orders. J. Algebra, 163(3):861–879,1994.

14. Matthew Dyer. Imaginary cone and dominance order in Coxeter groups. inpreparation.

15. Matthew Dyer. Imaginary cone and reflection subgroups of Coxeter groups.in preparation.

16. Matthew Dyer. On rigidity of abstract root systems of Coxeter groups.preprint.

17. Matthew Dyer. Questions on initial sections of reflection orders. in prepara-tion.

18. Matthew Dyer. Regular subsets of Coxeter groups. in preparation.

19. Matthew Dyer. Reflection subgroups of Coxeter systems. J. Algebra,135(1):57–73, 1990.

20. Tom Edgar. Sets of reflections defining twisted Bruhat orders. J. AlgebraicCombin., 26(3):357–362, 2007.

21. David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy,Michael S. Paterson, and William P. Thurston. Word processing in groups.Jones and Bartlett Publishers, Boston, MA, 1992.

22. Leslie Hogben, editor. Handbook of linear algebra. Discrete Mathematics andits Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007.Associate editors: Richard Brualdi, Anne Greenbaum and Roy Mathias.

23. James E. Humphreys. Reflection groups and Coxeter groups, volume 29 ofCambridge Studies in Advanced Mathematics. Cambridge University Press,Cambridge, 1990.

24. Victor G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press,Cambridge, third edition, 1990.

25. K. Kuratowski. Topology. Vol. I. New edition, revised and augmented. Trans-lated from the French by J. Jaworowski. Academic Press, New York, 1966.

26. G. Lusztig. Hecke algebras with unequal parameters, volume 18 of CRM Mono-graph Series. American Mathematical Society, Providence, RI, 2003.

27. G. A. Margulis and E. B. Vinberg. Some linear groups virtually having a freequotient. J. Lie Theory, 10(1):171–180, 2000.

135

28. Richard P. Stanley. Enumerative combinatorics. Vol. 1, volume 49 of Cam-bridge Studies in Advanced Mathematics. Cambridge University Press, Cam-bridge, 1997. With a foreword by Gian-Carlo Rota, Corrected reprint of the1986 original.

29. Sankaran Viswanath. On growth types of quotients of Coxeter groups byparabolic subgroups. Comm. Algebra, 36(2):796–805, 2008.

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