DOMINANCE AND REGULARITY IN COXETER GROUPS A ...
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<strong>DOM<strong>IN</strong>ANCE</strong> <strong>AND</strong> <strong>REGULARITY</strong> <strong>IN</strong> <strong>COXETER</strong> <strong>GROUPS</strong><br />
A Dissertation<br />
Submitted to the Graduate School<br />
of the University of Notre Dame<br />
in Partial Fulfillment of the Requirements<br />
for the Degree of<br />
Doctor of Philosophy<br />
by<br />
Tom Edgar<br />
Graduate Program in Mathematics<br />
Notre Dame, Indiana<br />
July 2009<br />
Matthew Dyer, Director
<strong>DOM<strong>IN</strong>ANCE</strong> <strong>AND</strong> <strong>REGULARITY</strong> <strong>IN</strong> <strong>COXETER</strong> <strong>GROUPS</strong><br />
Abstract<br />
by<br />
Tom Edgar<br />
The main purpose of this thesis is to generalize many of the results of Dyer,<br />
[18]. Dyer introduces a large family of finite state automata that are demonstrated<br />
to recognize many natural subsets of Coxeter groups. In the current work, we<br />
consider generalized length functions in Coxeter systems and show that these<br />
length functions lead to a larger family of finite state automata, which in turn<br />
recognize a larger class of natural subsets of Coxeter groups. By a well-known<br />
result, these subsets (called regular subsets) will then have a rational multivariate<br />
Poincaré series.<br />
Dominance order on the set of reflections (or positive roots) of a Coxeter system<br />
plays a major role in the study of regular subsets. We draw the connection between<br />
dominance in root systems and a notion of closure in root systems. We describe<br />
some conjectures, due to Dyer, [17], and introduce a conjectural normal form for<br />
Coxeter group elements related to closure and dominance.<br />
In [14], Dyer introduced the notion of the imaginary cone to characterize domi-<br />
nance. We investigate the imaginary cone in the special case of hyperbolic Coxeter<br />
systems. We show that any infinite, irreducible, non-affine Coxeter system has<br />
universal reflection subgroups of arbitrarily large rank. This allows us to deduce<br />
some consequences about the growth type of Coxeter systems.
For Laura, Sue, Kristin, and Mark<br />
ii
CONTENTS<br />
FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi<br />
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . vii<br />
SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix<br />
CHAPTER 1: <strong>IN</strong>TRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 1<br />
CHAPTER 2: BACKGROUND MATERIAL . . . . . . . . . . . . . . . . 4<br />
2.1 Coxeter Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
2.2 Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5<br />
2.3 Reflection Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />
2.4 The Weak Order on T and the Set of Elementary Reflections . . . 7<br />
2.5 Dominance Order on T . . . . . . . . . . . . . . . . . . . . . . . . 9<br />
CHAPTER 3: GENERALIZED LENGTH FUNCTIONS . . . . . . . . . . 13<br />
3.1 Commutative Monoids . . . . . . . . . . . . . . . . . . . . . . . . 14<br />
3.2 A General Length for Coxeter Systems . . . . . . . . . . . . . . . 15<br />
3.3 Infinite Heights and Generalized Partitions for Reflections . . . . 18<br />
3.4 Infinite Height and the Weak Order . . . . . . . . . . . . . . . . . 19<br />
3.5 The Root Poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20<br />
3.6 Monotonicity Properties in Dominance Order on T . . . . . . . . 22<br />
3.7 A Partition of the Reflections . . . . . . . . . . . . . . . . . . . . 25<br />
3.8 Recurrence Formulae . . . . . . . . . . . . . . . . . . . . . . . . . 25<br />
3.9 Monoids with Poset Structure . . . . . . . . . . . . . . . . . . . . 26<br />
3.10 General Length and the Reflection Cocycle . . . . . . . . . . . . . 29<br />
CHAPTER 4: F<strong>IN</strong>ITE STATE AUTOMATA . . . . . . . . . . . . . . . . 32<br />
4.1 Regular Languages . . . . . . . . . . . . . . . . . . . . . . . . . . 32<br />
4.2 Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34<br />
4.3 Canonical Automata for Coxeter Systems . . . . . . . . . . . . . . 35<br />
iii
4.4 Some Subsets of S ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
4.5 The Boolean Algebra of Regular Sets of S ∗ . . . . . . . . . . . . . 39<br />
4.6 The Reflection Cocycle and Automata . . . . . . . . . . . . . . . 42<br />
CHAPTER 5: REGULAR SUBSETS OF <strong>COXETER</strong> <strong>GROUPS</strong> . . . . . . 44<br />
5.1 Boolean Algebra of Regular Sets in W . . . . . . . . . . . . . . . 44<br />
5.2 Coxeter Groups are Automatic . . . . . . . . . . . . . . . . . . . . 45<br />
5.3 Different Types of Regularity . . . . . . . . . . . . . . . . . . . . 45<br />
5.4 p-Regularity for Different p . . . . . . . . . . . . . . . . . . . . . . 47<br />
5.5 Some Properties of p-Completely Regular Sets . . . . . . . . . . . 48<br />
5.6 Poincaré Series and Examples of Regular Subsets of W . . . . . . 50<br />
5.7 More Regular Subsets of W . . . . . . . . . . . . . . . . . . . . . 52<br />
5.8 Regularity in Direct Products . . . . . . . . . . . . . . . . . . . . 56<br />
5.9 Product Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
5.10 Multivariate Product Regularity . . . . . . . . . . . . . . . . . . . 58<br />
CHAPTER 6: <strong>REGULARITY</strong> <strong>IN</strong> T <strong>AND</strong> MIXED PRODUCTS . . . . . 63<br />
6.1 Regularity of sets of Reflections . . . . . . . . . . . . . . . . . . . 63<br />
6.2 Types of Regularity in T . . . . . . . . . . . . . . . . . . . . . . . 65<br />
6.3 p-Complete Regularity in T . . . . . . . . . . . . . . . . . . . . . 66<br />
6.4 Maps of Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . 72<br />
6.5 Regularity in Mixed Products . . . . . . . . . . . . . . . . . . . . 79<br />
6.6 General Regularity in Mixed Products . . . . . . . . . . . . . . . 80<br />
6.7 Multivariate Poincaré Series . . . . . . . . . . . . . . . . . . . . . 82<br />
CHAPTER 7: HECKE ALGEBRA MODULES . . . . . . . . . . . . . . . 84<br />
7.1 The Generic Iwahori-Hecke Algebra . . . . . . . . . . . . . . . . . 84<br />
7.2 A Module for H . . . . . . . . . . . . . . . . . . . . . . . . . . . 85<br />
7.3 The Hecke Algebra Modules from hW,p,M∞ . . . . . . . . . . . . . 86<br />
7.4 The 0-Hecke Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 89<br />
CHAPTER 8: <strong>IN</strong>ITIAL SECTIONS OF REFLECTION ORDERS . . . . 90<br />
8.1 Classification of Twisted Bruhat Orders . . . . . . . . . . . . . . . 90<br />
8.2 Proof of the Classification . . . . . . . . . . . . . . . . . . . . . . 92<br />
CHAPTER 9: CLOSURE <strong>IN</strong> THE ROOT SYSTEM . . . . . . . . . . . . 101<br />
9.1 2-closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101<br />
9.2 More types of subsets of Φ + . . . . . . . . . . . . . . . . . . . . . 102<br />
9.3 A Conjectural Normal Form for Elements of W . . . . . . . . . . 103<br />
iv
CHAPTER 10: HYPERBOLIC <strong>COXETER</strong> <strong>GROUPS</strong> . . . . . . . . . . . 108<br />
10.1 Coxeter Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108<br />
10.2 Symmetric Matrices and Sylvester’s Law . . . . . . . . . . . . . . 109<br />
10.3 Hyperbolic Coxeter Systems . . . . . . . . . . . . . . . . . . . . . 109<br />
10.4 Non-affine Coxeter Systems . . . . . . . . . . . . . . . . . . . . . 110<br />
10.5 A Class of Coxeter Systems . . . . . . . . . . . . . . . . . . . . . 112<br />
10.6 A Smaller Class of Coxeter Systems . . . . . . . . . . . . . . . . . 112<br />
10.7 Parabolic Subsystems of Non-affine Coxeter Systems . . . . . . . 114<br />
10.8 Characterizations of Hyperbolic Coxeter Systems . . . . . . . . . 114<br />
CHAPTER 11: LARGE REFLECTION SUB<strong>GROUPS</strong> . . . . . . . . . . . 116<br />
11.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116<br />
11.2 A Metric on Convex Sets . . . . . . . . . . . . . . . . . . . . . . . 117<br />
11.3 Approximation of a Sphere by Polytopes . . . . . . . . . . . . . . 118<br />
11.4 Cones in Coxeter Systems . . . . . . . . . . . . . . . . . . . . . . 120<br />
11.5 Topology on Rays in Cones . . . . . . . . . . . . . . . . . . . . . 121<br />
11.6 Cones in Hyperbolic Coxeter Systems . . . . . . . . . . . . . . . . 122<br />
11.7 Universal Coxeter Systems . . . . . . . . . . . . . . . . . . . . . . 123<br />
11.8 Large Reflection Subgroups . . . . . . . . . . . . . . . . . . . . . 125<br />
CHAPTER 12: EXPONENTIAL GROWTH <strong>IN</strong> <strong>COXETER</strong> <strong>GROUPS</strong> . . 129<br />
12.1 Growth Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130<br />
12.2 Properties of W(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 130<br />
12.3 Parabolic Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . 131<br />
12.4 Proof of Proposition 12.3.3 . . . . . . . . . . . . . . . . . . . . . . 132<br />
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134<br />
v
FIGURES<br />
3.1 This diagram shows the idea of generalized length in � B2 = 〈r, s, t |<br />
r 2 = s 2 = t 2 = (rt) 2 = 1, (rs) 4 = (st) 4 = 1〉. Each group element<br />
corresponds to one of the alcoves, with the yellow triangle<br />
representing the identity. Each reflection corresponds to one of the<br />
affine hyperplanes, and the color of the hyperplane distinguishes the<br />
conjugacy class. There are three conjugacy classes. The standard<br />
length counts the number of affine hyperplanes separating w from<br />
the yellow alcove. The generalized length counts this same number<br />
but retains additional information about the conjugacy class<br />
of each affine hyperplane separating w from the identity. For the<br />
w specified above we have l(w) = 12 while lp(w) = (3, 6, 3) where<br />
p represents the function taking distinct values on each conjugacy<br />
class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17<br />
5.1 This diagram shows the tiling of the plane corresponding to � B2 as<br />
in Figure 3.1. Each of the gray regions is an intersection of two half<br />
spaces, and thus each is p-completely regular for any p. Then, the<br />
total gray region is a union of two p-completely regular sets and so<br />
is p-completely regular itself. Therefore, this set also has a rational<br />
multivariate Poincaré series. . . . . . . . . . . . . . . . . . . . . . 53<br />
8.1 This is a schematic diagram of the root system for (W, S) infinite.<br />
Φ + consists of the roots which are non-negative linear combinations<br />
of αr and αs. The dotted ray through the origin represents a limit<br />
line of roots. If C1 and C2 are not both satisfied by Φ + then Ψ is<br />
pictured above with the roots in Ψ labeled by •, and those not in<br />
Ψ labeled by ◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97<br />
vi
ACKNOWLEDGMENTS<br />
First and foremost, I would like to thank my advisor, Matthew Dyer. I am<br />
always amazed by how much he knows and is willing to teach me, whether it be<br />
for an hour in his office or over the computer thousands of miles away. Without<br />
his endless patience, encouragement, and ideas, I would not have finished this<br />
work. I would also like to thank his family, Annette and Katy, for their wonderful<br />
hospitality and amusing conversations.<br />
I would like to thank the members of my thesis committee, Mark Colarusso,<br />
Sam Evens, and Michael Gekhtman, who took much of their valuable time to read<br />
the thesis and provide thoughtful comments. I would further like to thank Sam<br />
Evens for his conversations about research throughout my fifth year.<br />
Additionally, I would like to thank many of the mathematicians I have had the<br />
pleasure of working with and learning from during my years as a student. I would<br />
like to thank Lorelei Koss, Dave Richeson, and Barry Tesman for their excellent<br />
teaching and for inspiring me to choose a career in mathematics. I thank Anton<br />
Betten for his guidance in my master’s program and for teaching me a great deal<br />
of interesting combinatorics. I would like to acknowledge and thank Brian Hall<br />
for the many enlightening conversations about root systems and mathematics in<br />
general. I would like to thank fellow graduate students Benjamin Jones, Don<br />
Brower, and John Wallbaum for many interesting and helpful conversations; I<br />
would moreover like to thank all the Notre Dame graduate students for the good<br />
vii
times spent here.<br />
Without my family, I would not be where I am today. I would like to express<br />
deep gratitude to Margot and Dick for treating us like their own children during<br />
our time in Indiana. I thank my sister, Kristin, for her loving support and will-<br />
ingness to just call and talk. I thank my mom, Sue, for everything she has done<br />
for me in my life. She is an amazing woman and someone I will always look up<br />
to. I would also like to thank my dad, Mark. He taught me how to work hard,<br />
and he always believed in me. I have never met a better man, and I will forever<br />
miss him.<br />
Finally, and most of all, I thank my beautiful wife, Laura. Her love and support<br />
have been an integral part of my success as a graduate student. She makes life<br />
fun, and I couldn’t imagine what it would be like if she were not around. I owe<br />
her immensely for her willingness to help me achieve this goal.<br />
viii
SYMBOLS<br />
W Coxeter group<br />
S Coxeter group generators<br />
l Standard length function on W<br />
lp<br />
General length function on W<br />
T Reflections<br />
N Reflection cocycle<br />
Φ Root system<br />
Φ + Positive roots<br />
Π Simple roots<br />
( , ) Bilinear pairing on root system<br />
≤A<br />
Twisted Bruhat order on group<br />
W ′ Reflection subgroup<br />
M Monomials in Z[X]<br />
� Dominance order on reflections<br />
≤ Weak order on T ; Order on M<br />
ℓp<br />
General length function on S ∗<br />
ix
CHAPTER 1<br />
<strong>IN</strong>TRODUCTION<br />
In this dissertation, we investigate some of the applications of dominance order<br />
on the root system (or reflections) of a Coxeter group. Dominance was first defined<br />
by Brink and Howlett in [5], where it is used to show that Coxeter groups are<br />
automatic. Brink further investigates the dominance minimal roots in [4].<br />
Despite this work, the dominance order is still poorly understood in general.<br />
More recently, in [15] Dyer has employed the idea of using the imaginary cone<br />
as a combinatorial tool for studying dominance in the root system. Our general<br />
goal will be to further the understanding of the role dominance plays in different<br />
aspects of study in Coxeter systems. The dissertation is organized as follows.<br />
Chapter 2 introduces the basic background information necessary to develop<br />
the results in the later chapters. These ideas will be used throughout the thesis.<br />
The thesis is broken up into three independent parts, which are loosely related<br />
through their connections to dominance order.<br />
In Chapter 3, we examine general length functions on Coxeter systems. These<br />
give rise to a class of height functions, which we demonstrate can be understood<br />
by studying the weak order and give a nice characterization of dominance order.<br />
These height functions allow us to define certain finite sets of reflections that parti-<br />
tion the entire set of reflections. We begin Chapter 4 by defining regular languages<br />
and finite state automata. We are then able to use the partition of the reflections<br />
1
described in Chapter 3 to define a class of canonical finite automata for the Cox-<br />
eter system. These automata generalize the automata defined by Dyer in [18]. In<br />
Chapter 5, we use the automata defined in Chapter 4 to describe large classes of<br />
regular subsets of Coxeter groups, i.e. sets that can be recognized by a finite state<br />
automaton. We introduce different types of regularity; in particular, our notion<br />
of p-complete regularity allows us to easily describe certain regular subsets of a<br />
Coxeter group. Regularity has some interesting and important implications for<br />
(multivariate) Poincaré series of subsets of a Coxeter system, and we will discuss<br />
these briefly. Due to the fact that many natural subsets of Coxeter systems cannot<br />
be described as completely regular in the sense of Chapter 5, we use Chapter 6<br />
to define an alternate type of regularity, which will allow us to demonstrate the<br />
regularity of many subsets of the set of reflections in a Coxeter system. We end<br />
with a characterization of regular sets inside of products W n × T m where W is a<br />
Coxeter system with T the set of reflections. Finally, in Chapter 7, we describe<br />
how the finite sets that give regularity also lead to a large family of modules for<br />
the generic Iwahori-Hecke algebra.<br />
Next, in Chapters 8 and 9, we discuss a notion of closure in the root system<br />
(or reflections). Chapter 8 classifies possible partial orders on a Coxeter system,<br />
generalizing Bruhat order, using certain closed subsets of the roots. These results<br />
appear in [20] as well. Chapter 9 focuses on discussing closure in general and other<br />
types of subsets of roots. We outline some conjectures, posed by Dyer (see [17]),<br />
involving closure in the root system. We also describe an application of these<br />
conjectures related to closure of the sets defined in Chapter 4, which would lead<br />
to a normal form for elements of the Coxeter group.<br />
Finally, in Chapters 10-12, we investigate some properties of the imaginary<br />
2
cone, especially in hyperbolic Coxeter systems. In Chapter 10, we provide some<br />
characterizations of hyperbolic Coxeter systems. As a byproduct of our charac-<br />
terizations, we obtain a simple, case-free proof of the well known fact that any<br />
Coxeter system contains a hyperbolic subsystem as a standard parabolic subsys-<br />
tem. Chapter 11 focuses on studying the imaginary cone in the case of hyperbolic<br />
Coxeter systems. We use the imaginary cone to show that any Coxeter system con-<br />
tains universal reflection subgroups of arbitrarily large rank. Furthermore, in the<br />
hyperbolic case, the positive spans of the simple roots of the universal reflection<br />
subgroups are shown to approximate the imaginary cone (using an appropriate<br />
topology on the set of roots). This answers in the affirmative a question due to<br />
Dyer [15] in the special case of hyperbolic Coxeter systems. Lastly, Chapter 12<br />
uses the results from Chapter 11 describing large universal reflection subgroups<br />
in any Coxeter system to extend the results of [29] regarding exponential growth<br />
in parabolic quotients in Coxeter groups.<br />
3
CHAPTER 2<br />
BACKGROUND MATERIAL<br />
We begin with a chapter introducing the basic objects and terminology used<br />
throughout the thesis.<br />
2.1 Coxeter Groups<br />
Definition 2.1.1. We define a Coxeter system to be a pair (W, S) consisting of<br />
a group W and a set of generators S ⊂ W subject only to relations of the form<br />
(ss ′ ) m(s,s′ ) = 1 where m(s, s) = 1 and m(s, s ′ ) = m(s ′ , s) ≥ 2 for s �= s ′ . If s and<br />
s ′ have no such relation, we say that m(s, s ′ ) = ∞.<br />
We may sometimes abuse terminology and call (W, S), or W , a Coxeter group.<br />
For general results about Coxeter groups, we refer the reader to [23] or [2]. In<br />
general, S does not need to be finite. For many of the results we will state we<br />
need S to be finite. We shall specify when this is the case.<br />
Every element w ∈ W can be written as a product of the generators w =<br />
s1s2 · · · sm, each si ∈ S. If m is minimal among all such expressions for w we will<br />
say that s1s2 · · · sn is a reduced expression for w and we say that w has length n,<br />
i.e. l(w) = n. So l : W → N denotes the standard length function on (W, S).<br />
We let T denote the set of W -conjugates of S, T := �<br />
w∈W wSw−1 , and call<br />
T the set of reflections. Then, following [19] we define N : W → P(T ) to be<br />
4
the “reflection cocycle” given by N(w) = {t ∈ T | l(tw) < l(w)}. We call N(w) a<br />
cocycle because it satisfies the cocycle condition:<br />
N(xy) = N(x) + xN(y)x −1<br />
where + represents symmetric difference on the power set of T , A + B = (A ∪<br />
B) \ (A ∩ B). The existence of this cocycle implies that W acts on P(T ) by<br />
w · A = N(w) + wAw −1 for A ⊆ T .<br />
2.2 Root Systems<br />
Without loss of generality, we will assume that (W, S) is realized in its standard<br />
reflection representation on a real vector space V with Π denoting the set of simple<br />
roots and (−, −) : V × V → R representing the associated bilinear form given<br />
by (α, β) = − cos π<br />
m(sα,sβ) if m(sα, sβ) �= ∞ and (α, β) ≤ −1 otherwise. Then<br />
we denote the roots and positive roots by Φ and Φ + respectively. Recall that<br />
Φ = Φ + ˙∪ − Φ + , and we may sometimes write Φ − := −Φ + .<br />
For α ∈ Φ we let sα ∈ T represent the corresponding reflection. Recall that<br />
the set of reflections are in bijective correspondence with the set of positive roots<br />
under the bijection α ↦→ sα : Φ + → T . For t ∈ T , we will let αt ∈ Φ + denote the<br />
unique positive root α with sα = t. We recall the following two facts<br />
l(wt) < l(w) ⇔ w(αt) ∈ Φ − for t ∈ T and w ∈ W (2.2.1)<br />
l(tw) < l(w) ⇔ w −1 (αt) ∈ Φ − for t ∈ T and w ∈ W (2.2.2)<br />
5
We can then describe the W -action on Φ in terms of the length function l by<br />
the following formula<br />
for any w ∈ W and t ∈ T .<br />
2.3 Reflection Subgroups<br />
⎧<br />
⎪⎨ 1, if l(wt) < l(w)<br />
w(±αt) = ±ɛαwtw−1, ɛ =<br />
⎪⎩ −1, if l(wt) > l(w)<br />
(2.2.3)<br />
We call a subgroup W ′ of W a reflection subgroup of W if it is generated by<br />
the reflections it contains, W ′ = 〈W ′ ∩ T 〉. It was shown by Dyer ([19]) and<br />
Deodhar ([10]) independently that any reflection subgroup is also a Coxeter sys-<br />
tem; moreover, any reflection subgroup has a canonical set of Coxeter generators<br />
χ(W ′ ) = {t ∈ T | N(t) ∩ W ′ = {t}}. We say a reflection subgroup is dihedral<br />
if it is generated by two distinct reflections or equivalently |χ(W ′ )| = 2. We let<br />
l(W ′ ,χ(W ′ )) : W ′ → N be the length function for (W ′ , χ(W ′ )) and T ′ = W ′ ∩ T<br />
be the set of reflections of W ′ . Due to [19], N(W ′ ,χ(W ′ ))(x) = N(x) ∩ W ′ for all<br />
reflection subgroups W ′ where N(W ′ ,χ(W ′ )) is the reflection cocycle for (W ′ , χ(W ′ )).<br />
Any dihedral reflection subgroup, W ′ = 〈sα, sβ〉, is contained in a unique<br />
maximal dihedral reflection subgroup, namely 〈sγ| γ ∈ Φ ∩ (Rα + Rβ)〉. We let<br />
M = {W ′ < W | W ′ is a maximal dihedral reflection subgroup}<br />
M∞ = {W ′ ∈ M | |W ′ | = ∞}.<br />
6
For any reflection t ∈ T , we get the following set:<br />
The previous remarks imply that<br />
Mt = {W ′ ∈ M | t ∈ W ′ }.<br />
T \ {t} = ˙�<br />
where ˙∪ represents a disjoint union.<br />
W ′ ∈Mt<br />
((W ′ ∩ T ) \ {t}) (2.3.1)<br />
Since any reflection subgroup is also a Coxeter system (W ′ , χ(W ′ )), we let ΦW ′,<br />
Φ +<br />
W ′, ΠW ′ ⊆ Φ be the set of roots, positive roots, and simple roots for (W ′ , χ(W ′ ))<br />
sitting inside the root system for (W, S). We have that ΦW ′ = {α ∈ Φ| sα ∈ W ′ }.<br />
For any w ∈ W , there exists a unique x ∈ W ′ w with N(x)∩W ′ = ∅ (likewise a<br />
unique x ∈ wW ′ with N(x −1 )∩W ′ = ∅). According to [19], we have χ(w −1 W ′ w) =<br />
x −1 χ(W ′ )x. Let w = yx with y ∈ W ′ . Then α ↦→ w −1 (α) = x −1 y −1 (α) induces a<br />
bijection ΦW ′<br />
α → y −1 (α) : ΦW ′<br />
∼=<br />
→ Φw−1W ′ w. This bijection is obtained by composing the bijection<br />
∼=<br />
→ ΦW ′ given by the action of y−1 ∈ W ′ on the root system of<br />
∼=<br />
→ Φx−1W ′ x = Φw−1W ′ w give by the action<br />
W ′ and the bijection α → x −1 (α) : ΦW ′<br />
of x. Since N(x) ∩ W ′ = ∅, the latter bijection preserves positive roots as well.<br />
2.4 The Weak Order on T and the Set of Elementary Reflections<br />
Much of the terminology of this section is described in terms of the root system<br />
in [2]. We will instead use terminology associated to the set of reflections when<br />
possible. For any reflection t ∈ T , we define the standard height to be h(t) :=<br />
h(W,S)(t) = l(t)−1<br />
2 .<br />
At this point, we define a partial order on the set of reflections. For any<br />
7
t, t ′ ∈ T , we set t → t ′ if there is some s ∈ S (it is unique if it exists) such that<br />
t ′ = sts and l(t ′ ) = l(t) + 2 (or equivalently h(t ′ ) = h(t) + 1). Let ≤ be the<br />
partial order on T given as the reflexive transitive closure of →, and call (T, ≤)<br />
the reflection poset or the weak order on T . It is clear that h is the rank function<br />
for (T, ≤).<br />
From the formula given in (2.3.1), we see that for t ∈ T we have<br />
h(t) = �<br />
W ′ ∈Mt<br />
h(W ′ ,χ(W ′ ))(t)<br />
where h(W ′ ,χ(W ′ )) is the height function for (W ′ , χ(W ′ )). Now, we use this charac-<br />
terization of height to introduce the “∞-height” of a reflection as follows.<br />
h ∞ (t) := �<br />
W ′ ∈Mt<br />
|W ′ |=∞<br />
h(W ′ ,χ(W ′ ))(t). (2.4.1)<br />
We now introduce a special subset of the reflections. Let T0 := {t ∈ T | h ∞ (t) =<br />
0}. We call this set the set of elementary reflections. Using the bijection between<br />
T and Φ + , this set is in bijective correspondence with the elementary roots defined<br />
in [5]. In that paper, Brink and Howlett also demonstrate that this set is finite.<br />
For another description of this fact, see [2].<br />
More recently, Dyer has shown similar sets to be finite. More precisely, we<br />
define, for any n ∈ N, Tn := {t ∈ T | h ∞ (t) = n}. For these sets, we have the<br />
following theorem, due to Dyer, found in [14].<br />
Theorem 2.4.1. Let (W, S) be a Coxeter system with reflections T . If S is a<br />
finite set, then Tn is a finite set for all n ∈ N.<br />
These sets of reflections, as well as others like it, will be defined in more<br />
8
generality in Chapter 3, and we will make extensive use of these types of sets in<br />
Chapter 5 in order to demonstrate nice regularity properties of Coxeter systems.<br />
We will also use these sets to define certain modules for the generic Iwahori-Hecke<br />
algebra associated to (W, S) in Chapter 7.<br />
2.5 Dominance Order on T<br />
The set of elementary reflections also has a nice characterization in terms of<br />
another partial order on the set of reflections; it is the set of minimal elements of<br />
this partial order. Dominance order was initially defined in [5]. The elementary<br />
reflections and the dominance order are studied in more detail in [4]. The basic<br />
properties of dominance can be found in [2],[4], or [5], but we will include a few<br />
results here.<br />
We will first define dominance on the root system associated to (W, S) and use<br />
the canonical bijection of positive roots with reflections to transport the definition<br />
to the reflections.<br />
We say that α ∈ Φ dominates β ∈ Φ written β � α if, for all w ∈ W ,<br />
w(α) ∈ Φ − implies that w(β) ∈ Φ − .<br />
Lemma 2.5.1. 1. If α, β ∈ Φ, then α dominates β in Φ if and only if for<br />
some (respectively every) reflection subgroup W ′ of W with sα, sβ ∈ W ′ , α<br />
dominates β in ΦW ′ (with respect to W ′ ).<br />
2. � is a partial order on Φ.<br />
3. Multiplication by −1 is an order-reversing bijection of (Φ, �) with itself.<br />
4. The W -action on Φ is as a group of poset automorphisms of (Φ, �).<br />
9
Proof. First, suppose that β, α ∈ Φ and that sα and sβ are contained in a reflection<br />
subgroup W ′ . According to section 2.3, we have that for any w ∈ W , there is<br />
x ∈ wW ′ with N(x −1 ) ∩ W ′ = ∅, and α ↦→ x(α) is a bijection, which preserves<br />
positive roots, between ΦW ′ and Φ wW ′ w −1. Let w = xy with y ∈ W ′ . Suppose<br />
that β � α in ΦW ′ with respect to W ′ . Then if w(α) ∈ Φ − , we have xy(α) ∈ Φ −<br />
and so y(α) ∈ Φ −<br />
W ′. Thus, w(β) = xy(β) ∈ x(Φ −<br />
W ′) = Φ −<br />
W ′ ⊂ Φ − and so β � α<br />
in Φ. If β � α in Φ, then by definition, w(α) ∈ Φ − implies w(β) ∈ Φ − for all<br />
w ∈ W and so this is true for all w ′ ∈ W as required for 1. We now describe<br />
the dominance order on dihedral Coxeter systems. Let (W, S) be dihedral, with<br />
S = {r, s}. Then if W is finite, all the (finite) elements of Φ are incomparable<br />
in �. If W is infinite, then the dominance order is given by the coarsest partial<br />
order satisfying the relations<br />
and<br />
· · · − αsrsrs ≺ −αsrs ≺ −αs ≺ αr ≺ αrsr ≺ αrsrsr ≺ · · ·<br />
· · · − αrsrsr ≺ −αrsr ≺ −αr ≺ αs ≺ αsrs ≺ αsrsrs ≺ · · · .<br />
In either of these cases, it is clear that (Φ, �) is a partial order. For general<br />
(not necessarily dihedral) (W, S), it is clear that � is reflexive and transitive by<br />
definition. Then, antisymmetry in general follows by using antisymmetry in a<br />
reduction to a dihedral reflection subgroup. Namely, suppose β � α � β with<br />
α �= β in Φ, and then consider W ′ = 〈sα, sβ〉. Due to 1, we see that β � α � β<br />
in ΦW ′, and by antisymmetry in the dihedral case, we get that α = β, which<br />
is a contradiction. Finally, 3 and 4 are simple consequences of the definition of<br />
dominance. In particular if β � α then w(β) � w(α) since for any u ∈ W we have<br />
10
uw(α) ∈ Φ − implies that uw(β) ∈ Φ − .<br />
Then, as stated before, we use the canonical bijection between Φ + and T to<br />
define the notion of dominance on the set of reflections. We have that sβ � sα if<br />
and only if β � α in Φ + . This is equivalent to t ′ � t if for all w ∈ W we have<br />
l(wt) < l(w) implies that l(wt ′ ) < l(w); in this case, we say that t dominates t ′ .<br />
We will write (T, �) when referring to the set of reflections partially ordered by<br />
dominance.<br />
We recall that in a poset, we say an element a covers an element b if b < a and<br />
there is no z ∈ P with b < z < a.<br />
Lemma 2.5.2. Let (W, S) be a Coxeter system.<br />
1. If s, t ∈ T0 generate an infinite subgroup, then {s, t} is the set of canonical<br />
Coxeter generators for the maximal dihedral reflection subgroup of (W, S)<br />
containing {s, t}.<br />
2. If α, β ∈ Φ + are distinct elementary roots (i.e. sα and sβ are in T0) and<br />
〈sα, sβ〉 is an infinite dihedral reflection subgroup then β covers −α in dom-<br />
inance order.<br />
Proof. Due to the definition of h ∞ in equation (2.4.1), we know that t ∈ T0 if and<br />
only if<br />
0 = h ∞ (t) = �<br />
W ′ ∈Mt<br />
|W ′ |=∞<br />
h(W ′ ,χ(W ′ ))(t)<br />
which implies that h(W ′ ,χ(W ′ ))(t) = 0 for all infinite dihedral reflection subgroups<br />
W ′ . Therefore, by the definition of the height, we see that t ∈ χ(W ′ ) for all<br />
infinite, maximal dihedral reflection subgroups of W with t ∈ W ′ ∩ T . We see<br />
that 1 follows from this fact. To prove 2, since 〈sα, sβ〉 is infinite then −α ≺ β<br />
11
y the description of dominance in infinite dihedral reflection subgroups given in<br />
2.5.1. Suppose that γ ∈ Φ and −α ≺ γ ≺ β. Since β is elementary, we cannot have<br />
γ ∈ Φ + ; similarly, since α is elementary and −γ ≺ α we cannot have −γ ∈ Φ + so<br />
that γ �∈ Φ − . This contradicts the fact that Φ = Φ + ˙∪Φ − .<br />
Remark 2.5.3. In [14], Dyer uses the notion of the imaginary cone in a Coxeter<br />
system (see Chapter 11) to characterize dominance. He demonstrates that every<br />
covering α ′ ≺ β ′ in dominance order is in the W -orbit of a covering α ≺ β as in<br />
part 2 of 2.5.2.<br />
12
CHAPTER 3<br />
GENERALIZED LENGTH FUNCTIONS<br />
In this chapter, we begin by introducing a family of generalized length functions<br />
on a Coxeter system. These length functions replace the standard length function<br />
on (W, S) by taking values in N k for some k ≥ 1 instead of simply taking values in<br />
N. In a similar manner to section 2.4, these length functions will allow us to define<br />
corresponding height functions and, more importantly, a class of “infinite-height”<br />
functions on the set of reflections. These height functions lead to a partition of the<br />
set of reflections with interesting applications. Due to Theorem 2.4.1, these sets<br />
of reflections turn out to be finite. We will use the finiteness property to develop<br />
many canonical automata in Chapter 4.<br />
We proceed by interpreting these height functions in terms of the poset (T, ≤)<br />
(weak order) as defined in section 2.4. In turn, we further obtain monotonicity<br />
properties of these length and height functions with respect to the dominance<br />
order.<br />
Finally, we demonstrate a basic recurrence formula for the reflection cocycle<br />
on (W, S) in terms of this new partition of T . We end with a few results that will<br />
be important in the following chapters. Many of the results generalize some of the<br />
results in [18].<br />
13
3.1 Commutative Monoids<br />
At this point we want to define a commutative monoid M. We have two<br />
equivalent definitions for M which we introduce below. Both definitions will be<br />
useful for different applications, but it will be clear from context how we want to<br />
view M since one definition will have multiplicative notation and the other will<br />
have additive notation.<br />
Let X be a finite set of indeterminates and Z[X] be the corresponding integral<br />
polynomial ring. We say m ∈ Z]X] is a monomial if m = �<br />
x∈X x ax where ax ∈ N<br />
for all x ∈ X. Let M := MX be the set of all monomials of Z[X], that is<br />
M := {m| m ∈ Z[X]; m is a monomial}. For any m ∈ M, m = �<br />
x∈X x ax ,<br />
define the degree of m to be deg(m) := �<br />
x∈X ax.<br />
If we fix an order for X = {x1, ..., xk}, we can also consider the commutative<br />
monoid N k defined by the map ζ : M → N k given by m = x a1<br />
1 x a2<br />
2 · · · x ak<br />
k ↦→<br />
(a1, ..., ak). Since ζ is an isomorphism, we will abuse notation by setting N k = M.<br />
The manner in which we view M will be clear depending on whether we use<br />
additive or multiplicative notation. We note that for m ∈ M, deg(m) = �<br />
x∈X ax<br />
is independent of how we view M. For any monomial m = (a1, ..., ak) we let<br />
m(i) := ai.<br />
For X as above, let X −1 = {x −1 | x ∈ X}. We can define M ′ := {m| m ∈<br />
Z[X, X −1 ]; m is a monomial} ∼ = Z k . This set has a natural group structure. We<br />
will also view M as a subset of M ′ ; in the case when we view M additively,<br />
this will explain using the notation m − n (and will explain using division when<br />
viewing M multiplicatively).<br />
We place a partial order ≤M on M as follows. For any m ∈ M and n ∈ M,<br />
we say m ≤M n if and only if n<br />
m ∈ M ⊂ M′ (additively this means n − m ∈<br />
14
M ⊂ M ′ ) . For x ∈ X, we say that m ✁x n if n<br />
m<br />
= x and call this a covering<br />
relation of type x. It is clear that if m ✁x n and m ≤M m ′ ≤M n then m ′ = m or<br />
m ′ = n. From this point on, we will write ≤:=≤M; this will not cause confusion<br />
with ≤ on T as the context will make clear which ≤ is intended.<br />
Now, by an ordered generalized partition of a monomial m we will mean a<br />
tuple M = (m1, m2, ...), mn = 0 for n ≫ 0, with deg(mi) ≥deg(mj) for all i < j<br />
along with �<br />
i mi = m. Next, we define an equivalence relation, ∼, on the set of<br />
ordered generalized partitions in the following way. Let (m1, m2, ...) ∼ (n1, n2, ...)<br />
if there exists a permutation of the natural numbers, σ, such that mi = nσ(i)<br />
for all i. For (m1, m2, ...) an ordered partition, we let M = [(m1, m2, ...)] be<br />
the equivalence class of the ordered partition. We call these equivalence classes<br />
generated by this relation unordered generalized partitions. For any unordered<br />
partition M = [(m1, m2, ...)], we define |M| := �<br />
i mi. All the terminology of the<br />
partial order on M can be phrased additively if necessary.<br />
At this point, we can define a partial order on the set of unordered generalized<br />
partitions. Let M and N be any two generalized partitions. Then we say that<br />
M ≤ N if there exists (m1, m2, ...) ∈ M and (n1, n2, ...) ∈ N such that mi ≤ ni in<br />
M for all i. For any x ∈ X, we define the covering relation of type x by M ✁x N<br />
if there exists (m1, m2, ...) ∈ M and (n1, n2, ...) ∈ N and j ∈ N such that mi = ni<br />
for all i �= j and mj ✁x nj in M.<br />
3.2 A General Length for Coxeter Systems<br />
Suppose we have a fixed Coxeter system (W, S). Let M be the additive,<br />
commutative monoid described in section 3.1 where M is generated by a finite<br />
set X = {x1, ..., xk} of indeterminates. Suppose that we are given a surjective<br />
15
function p : T → X satisfying p(t) = p(t ′ ) if t is conjugate to t ′ , i.e. if there exists<br />
v ∈ W with t = vt ′ v −1 . For any reflection t ∈ T , we say that t is of type xi if<br />
p(t) = xi where xi ∈ X (for simplicity we will usually say that t is of type p(t)).<br />
Until further notice, we will view M with additive notation and we will, for ease,<br />
let p(t) represent ζ(p(t)).<br />
We define a family of “length” functions in the following way. Let W ′ be any<br />
reflection subgroup of W and w ∈ W ′ where w = r1...rn is a reduced expression<br />
with ri ∈ χ(W ′ ). Then we define lp,W ′ : W ′ → M by lp,W ′(w) = lp,W ′(r1...rn) =<br />
p(r1) + p(r2) + · · · + p(rn). See figure 3.1 for an example.<br />
Similarly, for any reflection r ∈ W ′ ∩ T with r = r1...rmrm+1...r2m+1, we define<br />
a corresponding “height function” hp,W ′ : T → M by hp,W ′(r) = p(r1) + p(r2) +<br />
· · · + p(rm). Finally, for any r ∈ W ′ ∩ T where r = r1...rmrm+1...r2m+1, we define<br />
c(r) := p(r) = p(rm+1), which is called the center of r. Note that for t ∈ T ,<br />
lp,W ′(t) = c(t) + 2hp,W ′(t).<br />
From these height functions, we get the following functions on reflections as<br />
well. For any union O ⊂ M of W -orbits of maximal dihedral reflection subgroups<br />
on M , we define<br />
and<br />
hW,p,O : t ↦→ �<br />
W ′ ∈Mt<br />
W ′ ∈O<br />
hp,W ′(t) (3.2.1)<br />
lW,p,O : t ↦→ c(t) + 2hW,p,O(t). (3.2.2)<br />
In this thesis, we will only consider the case where O = M∞.<br />
Remark 3.2.1. We note that, considering terminology from section 2.4, we have<br />
h ∞ (t) = deg(hW,p,M∞(t)). Also, if M = N and q(t) = 1 for all t ∈ T , then<br />
16
Figure 3.1. This diagram shows the idea of generalized length in � B2 =<br />
〈r, s, t | r 2 = s 2 = t 2 = (rt) 2 = 1, (rs) 4 = (st) 4 = 1〉. Each group element<br />
corresponds to one of the alcoves, with the yellow triangle representing the<br />
identity. Each reflection corresponds to one of the affine hyperplanes, and<br />
the color of the hyperplane distinguishes the conjugacy class. There are<br />
three conjugacy classes. The standard length counts the number of affine<br />
hyperplanes separating w from the yellow alcove. The generalized length<br />
counts this same number but retains additional information about the<br />
conjugacy class of each affine hyperplane separating w from the identity.<br />
For the w specified above we have l(w) = 12 while lp(w) = (3, 6, 3) where<br />
p represents the function taking distinct values on each conjugacy class.<br />
17<br />
w
h ∞ (t) = hW,q,M∞(t).<br />
Lemma 3.2.2. Let (W, S) be a Coxeter system, and let ≤∅ be ordinary Bruhat<br />
order on (W, S) (see Chapter 8). Let x, y ∈ W with x ≤∅ y. Then for all reflection<br />
subgroups W ′ ⊆ W with x, y ∈ W ′ , the following hold.<br />
1. lp,W ′(x) ≤ lp,W ′(y), and<br />
2. hp,W ′(x) �> hp,W ′(y) if x, y ∈ T ∩ W ′ .<br />
3. hp,W ′(x) ≤ hp,W ′(y) if x, y ∈ T ∩ W ′ and c(x) = c(y).<br />
Proof. We know that x ≤∅ y in W if and only if x ≤∅ y in W ′ for all x, y ∈ W ′ ⊆<br />
W . The subword characterization of Bruhat order and the definition of lp,W in<br />
terms of reduced expressions imply 1. Finally, 2 and 3 follow from 1.<br />
3.3 Infinite Heights and Generalized Partitions for Reflections<br />
According to equation (3.2.1) we can attach an unordered generalized partition<br />
to each reflection t, MW,p,M∞(t), where each part, mi, is given by some distinct<br />
hp,W ′(t) with W ′ ∈ M∞.<br />
Lemma 3.3.1. Let t ∈ T and s ∈ S with lp(sts) = lp(t) + 2p(s). Then<br />
1. hW,p,M∞(sts) = hW,p,M∞(t) if 〈s, t〉 is finite and hW,p,M∞(sts) = hW,p,M∞(t)+<br />
p(s) if 〈s, t〉 is infinite.<br />
2. MW,p,M∞(sts) = MW,p,M∞(t) if 〈s, t〉 is finite. If 〈s, t〉 is infinite then<br />
MW,p,M∞(t) ✁p(s) MW,p,M∞(sts).<br />
Proof. Let W ′ be a maximal dihedral reflection subgroup. According to 2.3 if<br />
s �∈ W ′ , then N(s) ∩ W ′ = ∅ so (W ′ , χ(W ′ )) ∼ = (sW ′ s, sχ(W ′ )s) and thus<br />
18
hp,sW ′ s(sts) = hp,W ′(t). If s ∈ W ′ , then s ∈ χ(W ′ ) and sW ′ s = W , and it<br />
follows that hp,W ′(sts) = hp,W ′(t) + p(s). Now, since Msts = {sW ′ s| W ′ ∈ Mt}, if<br />
〈s, t〉 ∈ Mt ∩ M∞ then we have<br />
Otherwise we have<br />
hW,p,M∞(sts) = �<br />
W ′ ∈Msts<br />
W ′ ∈M∞<br />
= p(s) + �<br />
hW,p,M∞(sts) = �<br />
W ′ ∈Msts<br />
W ′ ∈M∞<br />
= �<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
�<br />
hp,W ′(sts) =<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
hp,sW ′ s(sts)<br />
hp,W ′(t) = p(s) + hW,p,M∞(t).<br />
�<br />
hp,W ′(sts) =<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
hp,W ′(t) = hW,p,M∞(t).<br />
Based on these results, the lemma follows clearly.<br />
3.4 Infinite Height and the Weak Order<br />
hp,sW ′ s(sts)<br />
Consider the Hasse diagram of the weak order (T, ≤) as a directed graph with<br />
vertex set T and directed edges (t, t ′ ) from t to t ′ for all t, t ′ ∈ T with t → t ′ .<br />
Furthermore, we can regard this graph as a labeled directed graph by labeling<br />
the edges as follows. For any edge (t, t ′ ), we know there is a unique s ∈ S with<br />
t ′ = sts. From this, we can label the edge (t, t ′ ) with label p(s), and we call this<br />
edge an edge of type p(s). We say that an edge (t, t ′ ) is p(s)-short if it is labeled<br />
with p(s) and if hW,p,M∞(t ′ ) = hW,p,M∞(t), i.e. if 〈t, s〉 is finite. We say that the<br />
edge is p(s)-long if it is labeled with p(s) and hW,p,M∞(t ′ ) = hW,p,M∞(t) + p(s),<br />
i.e. if 〈t, s〉 is infinite. Based on this, we see that for any t ≤ t ′ in (T, ≤), the<br />
19
number of p(s)-long edges (xi−1, xi) in a chain t = x0 → x1 → ... → xn = t ′ is<br />
the m(j) = hW,p,M∞(t ′ )(j) − hW,p,M∞(t)(j) where p(s) = xj. Furthermore, for any<br />
fixed t, then maximal number of p(s)-long edges in chains x0 → x1 → ... → xn = t<br />
is hW,p,M∞(t)(j) and this maximum is realized if x0 ∈ S.<br />
Corollary 3.4.1. The map t ↦→ MW,p,M∞(t) is an order preserving map from the<br />
reflection poset of (W, S) to the poset of generalized partitions.<br />
Proof. This follows directly from part 2 of Lemma 3.3.1.<br />
Remark 3.4.2. Notice that for t ≤ t ′ , we have that c(t) = c(t ′ ) by definition of the<br />
weak order.<br />
3.5 The Root Poset<br />
The reflections endowed with weak order as a poset is isomorphic to a subposet<br />
of another poset which we briefly introduce. We call this poset the root poset<br />
denoted (Φ, ≤). Define → on Φ by α → β if and only if there is some γ ∈ Π such<br />
that β = sγ(α) and (γ|α) < 0, i.e. β − α ∈ R>0γ. Then, we let ≤ be the reflexive<br />
transitive closure of →.<br />
We can consider the Hasse diagram of (Φ, ≤) as a directed graph with vertex set<br />
Φ and edges (α, β) whenever α → β. We label all edges by their type p(sγ) where<br />
sγ is uniquely determined by β = sγ(α) where γ ∈ Π. We say an edge is p(sγ)-long<br />
if it is labeled with p(sγ) and if (α|γ) ≤ −1, otherwise we call it p(sγ)-short. It<br />
is clear that the relations (Φ + , →) and (Φ + , ≤) are obtained by transporting the<br />
relations from (T, →) and (T, ≤) by means of the bijection sα ↦→ α from T to Φ + .<br />
We note that this correspondence also respects edge lengths and types, i.e. the<br />
edge (α, β) is p(s)-long if and only if the corresponding edge (sα, sβ) is p(s)-long.<br />
20
Using this correspondence, we translate the basic properties of the reflection<br />
poset into corresponding properties of the root poset. In particular, we introduce<br />
the following length function on the roots.<br />
⎧<br />
⎪⎨ lW,p,M∞(sα) if α ∈ Φ<br />
LW,p,M∞(α) =<br />
⎪⎩<br />
+<br />
−lW,p,M∞(sα) if α ∈ Φ− Lemma 3.5.1. 1. For α ≤ β in Φ, the interval [α, β] is finite.<br />
2. The map α ↦→ −α is an order-reversing bijection of (Φ, ≤) with itself.<br />
(3.5.1)<br />
3. Let α = α0 → α1 → ... → αn = β be a maximal chain in (Φ, ≤). Then for<br />
any xi ∈ X, the number of xi-long edges (αi−1, αi) in the chain (regarded<br />
as a path from α to β in the Hasse diagram) is equal to m(i) where m =<br />
LW,p,M∞<br />
(β)−LW,p,M∞ (α)<br />
.<br />
2<br />
Proof. First, 1 follows from the corresponding fact that intervals are finite in<br />
(T, ≤), and we remark that 2 is clearly true since −α → α for all α ∈ Π.<br />
Now, suppose that α, β ∈ Φ + , then sα ≤ sβ in (T, ≤). Let d := sα ≤ sα1 ≤<br />
· · · ≤ sαn−1 ≤ sβ be the maximal chain from sα to sβ. Then according to 3.4,<br />
hW,p,M∞(sβ)(i) − hW,p,M∞(sα)(i) is the number of xi-long edges in the chain. By<br />
definition of lW,p,M∞ we have that<br />
hW,p,M∞(sβ) − hW,p,M∞(sα) = lW,p,M∞(sβ) − c(sβ)<br />
2<br />
� �<br />
lW,p,M∞(sα) − c(sα)<br />
−<br />
2<br />
and by Remark 3.4.2 we know that since sα ≤ sβ then c(sα) = c(sβ) and the result<br />
follows in this case. Similarly, if α, β ∈ Φ − then −α, −β ∈ Φ + and −β ≤ −α and<br />
the proof follows from above.<br />
This leaves us with the case that α ∈ Φ − and β ∈ Φ + (notice we cannot have<br />
21
α ∈ Φ + and β ∈ Φ − ). Let αi ∈ Φ − and αi+1 ∈ Φ + , then since αi → αi+1 we<br />
must have −αi = αi+1 (since the only positive root made negative by sγ is γ for<br />
γ ∈ Π). In this case, we have that the edge (αi, αi+1) is p(sαi+1 )-long. Note, that<br />
since α ≤ αi+1 ≤ β we have c(sα) = c(sβ) = c(sαi+1 ) = p(sαi+1 ) by Remark 3.4.2<br />
and the definition of c. Then, using the previous argument, we know that the<br />
number of xi-long edges in α → α1 → · · · → αi is equal to the number of xi-long<br />
edges in −αi → −αi−1 → · · · → −α which is hW,pM∞(sα)(i). Also, the number of<br />
xi-long edges in αi+1 → αi+2 → · · · → β is hW,p,M∞(sβ)(i). Now we have the total<br />
number of xi long edges is given by m(i) where<br />
m : = hW,pM∞(sβ) + hW,pM∞(sα) + c(sαi+1 )<br />
= lW,p,M∞(sβ) − c(sβ)<br />
2<br />
= lW,p,M∞(sα) + lW,p,M∞(sβ)<br />
.<br />
2<br />
+ lW,p,M∞(sα) − c(sα)<br />
2<br />
3.6 Monotonicity Properties in Dominance Order on T<br />
+ c(sαi+1 )<br />
Recall that using the canonical bijection between Φ + and T , given by α ↦→ sα,<br />
we transport the dominance order on Φ + to dominance order on T . We have that<br />
sβ � sα if and only if β � α. This is equivalent to t ′ � t if for all w ∈ W we have<br />
l(wt) < l(w) implies that l(wt ′ ) < l(w); in this case, we say that t dominates t ′ .<br />
Lemma 3.6.1. Let r, t ∈ T .<br />
(a) t dominates r in T if and only if for some (respectively every) reflection<br />
group W ′ of W with t, r ∈ W ′ , t dominates r in dominance order on the set<br />
W ′ ∩ T of reflections of (W ′ , χ(W ′ )).<br />
22
(b) t dominates r if and only if either (1) t = r, or (2) 〈t, r〉 is infinite and<br />
hp,W (rtr) < hp,W (t).<br />
(c) t dominates r if and only if either (1) t = r, or (2) 〈t, r〉 is infinite and<br />
lp,W (rt) < lp,W (t) and hp,W (r) �> hp,W (t).<br />
(d) If r � t then t = r if and only if hp,W (r) = hp,W (t).<br />
(e) Let t ∈ T , xi ∈ X, and m = hW,p,M∞(t). Then, m(i) = |{t ′ ∈ T |t ′ ≺<br />
t; p(t ′ ) = xi}|.<br />
(f) If r � t, then hW,p,M∞(r) ≤ hW,pM∞(t).<br />
Proof. First of all, (a) follows from part 1 of Lemma 2.5.1 via the bijection between<br />
positive roots and T . We simply transfer the structure of dominance by this<br />
bijection and the result holds.<br />
Next, (e) is clear for finite dihedral groups. Now, suppose we have an infinite<br />
dihedral group (W, S) with S = {r, s}. Then, as in 2.5.1, we know that the<br />
dominance order on T in this case is the coarsest partial order satisfying<br />
and<br />
s ≺ srs ≺ srsrs ≺ srsrsrs ≺ · · ·<br />
r ≺ rsr ≺ rsrsr ≺ rsrsrsr ≺ · · · .<br />
Using this description, (e) is readily checked. Then, for general (W, S), for any<br />
23
xi ∈ X we have<br />
hW,p,M ∞(t)(i) = �<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
= �<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
= �<br />
W ′ ∈Mt<br />
W ′ ∈M∞<br />
hp,W ′(t)(i)<br />
|{t ′ ∈ W ′ ∩ T | t ′ ≺W ′ ,χ(W ′ ) t; p(t ′ ) = xi}|<br />
| t ′ ∈ W ′ ∩ T | t ′ ≺ t; p(t ′ ) = xi}|<br />
= |{t ′ ∈ T | t ′ ≺ t ′ ; p(t ′ ) = xi}|<br />
where the first equality is the definition, the second equality follows from the<br />
dihedral case, the third equality follows from (a), and the final equality follows<br />
from equation (2.3.1).<br />
Now, we can also prove (b) and (c) using this reduction to rank 2 as well. We<br />
note that by the description for dihedral reflection groups above, (b) and (c) hold<br />
by inspection. For the general case of (b) with arbitrary (W, S), suppose that<br />
t, r ∈ T with r � t and t �= r. Let W ′ = 〈r, t〉. Since r � t in W ′ , according to [4]<br />
we have that W ′ is infinite. The dihedral case shows that we have hp,W ′(rtr) <<br />
hp,W ′(t). This implies that rtr
≤∅ t or t ≤∅ r (by inspection of infinite dihedral case). Due to the fact that<br />
hp,W (r) �> hp,W (t), we must have r ≤∅ t and so hp,W ′(r) �> hp,W ′(t) by Lemma<br />
3.2.2; consequently, r � t in W ′ and thus in W by (a). Finally, (d) follows from<br />
(b) and (f) follows from (e) directly.<br />
3.7 A Partition of the Reflections<br />
At this point, we define certain subsets of the set of reflections, T , of a given<br />
Coxeter group (W, S). These sets are analogs of Tn from section 2.4. For any<br />
monomial m ∈ M, define Tm := {t ∈ T | hW,p,M∞(t) = m}.<br />
Proposition 3.7.1. Suppose S is finite. Then Tm is finite for any m ∈ M.<br />
Proof. This follows from Theorem 2.4.1 since Tm ⊂ T deg(m) and deg(m) ∈ N.<br />
Recall (M, ≤) from section 3.1. For any m ∈ M, let T≤m := �<br />
n≤m Tn.<br />
3.8 Recurrence Formulae<br />
We now further extend the results of [18]. Recall the reflection cocycle N :<br />
W → P(T ). For any m ∈ M and w ∈ W , we define Nm(w) := N(w) ∩ T≤m.<br />
Proposition 3.8.1. Let w ∈ W , s ∈ S, and m ∈ M.<br />
(a) If s �∈ Nm(w) then Nm(sw) = ({s} ∪ sNm(w)s) ∩ T≤m.<br />
(b) If s ∈ Nm(w) and m ≥ p(s), then Nm−p(s)(sw) = (sNm(w)s \ {s}) ∩<br />
T≤m−p(s)<br />
Proof. Recall that N(sw) = {s} + sN(w)s so the containment of the right hand<br />
side into the left hand side is clear for both parts. Now assume that s �∈ Nm(w).<br />
25
Then s �∈ N(w). Let t ∈ Nm(sw) = N(sw) ∩ T≤m = ({s} ∪ sN(w)s) ∩ T≤m, and<br />
then let t ′ = sts. If t = s then we clearly have t = s ∈ ({s} ∪ sNm(s)s) ∩ T≤m. If<br />
t �= s, then t ∈ sN(w)s and so t ′ ∈ N(w). If t = t ′ then we already have assumed<br />
t ′ = t ∈ T≤m and so this proves that t ′ ∈ Nm(w) so that t ∈ sNm(w)s and thus t<br />
is in the right hand side. If 〈s, t〉 is finite or if lp(t) = lp(t ′ )+2p(s) then by Lemma<br />
3.3.1 we know that hW,p,M∞(t ′ ) ≤ hW,p,M∞(t) ≤ m by assumption that t ∈ T≤m;<br />
thus, t ′ ∈ Nm(w) and t ∈ sNm(w)s as well as the right hand side. This only<br />
leaves the case where 〈s, t〉 is infinite and lp(t ′ ) = lp(t) + 2p(s). Let W ′ = 〈t, s〉.<br />
Since s ∈ S, then we know that χ(W ′ ) = {r, s} for some r ∈ T . By assumption,<br />
s �∈ N(t) and t �= 1 ∈ W ′ so we must have r ∈ N(t). Therefore, we know by the<br />
proof of Lemma 2.5 that t dominates r. We have assumed that t ∈ N(sw) and the<br />
dominance then implies that r ∈ N(sw) as well. Also, by our first assumption, we<br />
know that s ∈ N(sw) as well, but this forms a contradiction since we now have<br />
〈t, s〉 = 〈r, s〉 ⊂ N(sw) (see [17]). However N(sw) is finite, and we know that<br />
〈t, s〉 is infinite. Thus, we have a contradiction proving (a).<br />
To prove part (b), assume that s ∈ Nm(w). Thus, we have that s ∈ N(w)<br />
and so by symmetric difference we must have N(sw) = sN(w)s \ {s}. Let t ∈<br />
Nm−p(s)(sw) = (sN(w)s \ {s}) ∩ T≤m−p(s). Set t ′ := sts. Clearly we have that<br />
hW,p,M∞(t ′ ) ≤ m − p(s) + p(s) = m so t ′ ∈ T≤m. Also, t ′ ∈ N(w) by assumption<br />
that t ∈ sN(w)s \ {s}. Thus, t ′ ∈ Nm(w) and so t = st ′ s ∈ sNm(w)s \ {s}, and<br />
we already have assumed that t ∈ T≤m−p(s).<br />
3.9 Monoids with Poset Structure<br />
Let (P, ≤) be any poset and let M be an additively written commutative<br />
monoid. Suppose we have a partial order ≤M on M such that 0 ≤M m for all<br />
26
m ∈ M and m ≤ n if and only if m + k ≤ n + k for all m, n, k ∈ M. In this<br />
case, we say ≤M is compatible with the monoid structure on M. Furthermore,<br />
let q : P → M \ {0} be a map. Define an operator | − | : P fin (P ) → M in the<br />
following way, where P fin (P ) represents all of the finite subsets of P . For finite<br />
A ⊂ P , we have<br />
|A| := �<br />
q(a) (3.9.1)<br />
a∈A<br />
Remark 3.9.1. Notice that if M = N with the usual total order and q(x) = 1 for<br />
all x ∈ P then this is the usual cardinality of a finite set.<br />
An ideal I of P is a subset of P satisfying the following property: if x ∈ I<br />
and y ≤ x in P then y ∈ I as well. Any subset X ⊂ P is contained in a<br />
smallest (under inclusion) ideal of P called the ideal generated by X. We say<br />
an ideal, I, is principal if I is generated by a single element {x} with x ∈ P .<br />
We say that P is lower finite if every principal ideal is finite. If we have a lower<br />
finite poset, we define h ′ : P → M by h ′ (x) = |{z ∈ P | z < x}|. For any<br />
Γ ⊂ M define PΓ := {z ∈ P | h ′ (z) ∈ Γ}. We will consider certain cases like<br />
Γ =≤M m := {n ∈ M| n ≤M m} and Γ =
The fact that the right hand side implies the left is clear. So assume that |I| �≤M<br />
m. Now, we define B := {I ′ ⊆ I| I ′ is an ideal; |I ′ | �≤M m}. Since I is finite, B<br />
is clearly finite and nonempty since I ⊂ B. Let I0 ∈ B be a minimal element of B<br />
(with respect to inclusion). We note that |∅| = 0 and due to the fact that m ≥ 0<br />
for all m ∈ M, we must have |I0| �= ∅. Then, for any maximal element t ∈ I0, we<br />
know that J := I0 \ {t} is an ideal satisfying J � I0 ⊂ I. So by the minimality of<br />
I0, we must have |J| ≤M m. Similarly, let K = {x ∈ I0| {y ∈ I0| y > x} = ∅} be<br />
the set of maximal elements in I0, then J ′ = I0 \ K must also have |J ′ | ≤M m.<br />
So, for any t ∈ I0 we must have that h ′ (t) = |{s ∈ P | s < t}| ≤M |I0 \ K| =<br />
|J ′ | ≤M m. Therefore, t ∈ P≤Mm. So we have shown that I0 ⊂ I ∩ P≤Mm and<br />
thus |I ∩ P≤Mm| ≥M |I0|. If |I ∩ P≤Mm| ≤M m, then this would contradict the<br />
fact that |I0| �≤M m; thus, we have shown (a).<br />
To show (b), we will use (a). First, suppose that |I| = m. This implies<br />
that |I| �≤M n for all n ✁M m (where ✁M represents the covering relation for<br />
≤M). Using part (a), we have that |I ∩ P≤Mn| �≤M n for all n ✁M m. Putting<br />
these all together we get that |I ∩ ∪k✁MmP≤Mk| �≤M n for all n ✁M m. Thus,<br />
|I ∩ P
Suppose, for the sake of contradiction, that |I| >M m. This implies that |I| �≤M m<br />
and so by (a) we conclude that |I ∩ P≤Mm| �≤M m. However, this contradicts the<br />
assumption that I ∩ P m.<br />
5. lp,W (xy) = lp,W (x) + lp,W (y) − 2m iff |Nm(x −1 ) ∩ Nm(y)| = m and<br />
∪n✁m (Nn(x −1 ) ∩ Nn(y)) = Nm(x −1 ) ∩ Nm(y).<br />
Proof. First, we note that if w = r1...rn is a reduced expression, then N(w) =<br />
{ti| i ∈ {1, ..., n}} where ti := r1...ri−1riri−1...r1 (see [19]). From this we see clearly<br />
that p(ri) = p(ti) so that we have lp,W = |N(w)|. Also, we remark that N(x) is an<br />
ideal of T under dominance order by definition, and so for any finite family (xi)i∈I<br />
of elements of W , the intersection ∩iN(xi) is a finite ideal. From these remarks,<br />
(1) and (2) follow directly from the previous lemma.<br />
29
The points (3) and (5) also follow using the cocycle condition N(xy) = N(x)+<br />
xN(y)x −1 where + denotes symmetric difference. This condition along with the<br />
interpretation of N(x) in terms of lp,W (x) imply that<br />
lp,W (xy) = lp,W (x) + lp,W (y) − 2|N(x −1 ) ∩ N(y)|.<br />
Then let I = N(x −1 ) ∩ N(y) and the Lemma implies (3) and (5).<br />
Finally, if |Nm(x −1 ) ∩ Nm(y)| > m then by (3) we know lp,W (xy) �≥ lp,W (x) +<br />
lp,W (y) − 2m. However, we also know that |N(x −1 ) ∩ N(y)| = k ≥ |Nm(x −1 ) ∩<br />
Nm(y)| > m due to the containment of sets. So since<br />
we have that<br />
as required for (4).<br />
lp,W (x) + lp,W (y) + 2m < lp,W (x) + lp,W (y) + 2k<br />
lp,W (xy) = lp,W (x) + lp,W (y) − 2k < lp,W (x) + lp,W (y) − 2m<br />
We finish this section with a lemma that has the same ideas as the previous<br />
lemma.<br />
Lemma 3.10.2. Suppose that r, s ∈ S and x, y ∈ W with lp,W (xr) > lp,W (x) and<br />
lp,W (ys) > lp,W (y). If ysy −1 � xrx −1 , then lp,W (y −1 x) ≤ lp,W (x) + lp,W (y) − 2n<br />
where n = hW,p,M∞(ysy −1 ).<br />
Proof. We have s = y −1 ysy −1 y � y −1 xrx −1 y since lp,W (ys) > lp,W (y). Therefore,<br />
we know that lp,W (y −1 xr) = lp,W (y −1 x) + lp(r). Thus, by the exchange condition<br />
we know that lp,W (sy −1 xr) ≥ lp,W (y −1 x) + lp(r) − lp(s).<br />
30
Now, ysy −1 � xrx −1 and xrx −1 ∈ N(xr) so we must have ysy −1 ∈ N(xr).<br />
Also, ysy −1 ∈ N(ys) by definition. So<br />
ysy −1 ∈ N(xr) ∩ N(ys) = N(xr) ∩ N((sy −1 ) −1 ).<br />
Therefore, we see that if n = hW,p,M∞(ysy −1 ), we have |N((sy −1 ) −1 ) ∩ N(xr)| ≥<br />
n + p(s) using Lemma 3.6.1. As in the proof of (4) from the previous lemma, this<br />
implies that lp,W (sy −1 xr) ≤ lp,W (sy −1 ) + lp,W (xr) − 2(n + p(s)). Putting together<br />
the information we have, we get<br />
lp,W (y −1 x) ≤ lp,W (sy −1 xr) − lp(r) + lp(s)<br />
≤ lp,W (xr) + lp,W (sy −1 ) − 2(n + p(s)) − lp(r) + lp(s)<br />
≤ lp,W (x) + lp(r) + lp,W (y) + lp(s) − 2n − 2p(s) − lp(r) + lp(s)<br />
= lp,W (x) + lp,W (y) − 2n.<br />
31
CHAPTER 4<br />
F<strong>IN</strong>ITE STATE AUTOMATA<br />
In this chapter, we begin by introducing some standard terminology involving<br />
finite state automata. For the basic theory of using automata in groups, we use<br />
[21]. In [5], Brink and Howlett described a method of creating a finite state<br />
automata that would recognize any finitely generated Coxeter system. For any<br />
finite Coxeter system, the corresponding automata can be described by using the<br />
Cayley graph of the group.<br />
More recently, Dyer ([18]) has demonstrated a much larger class of finite state<br />
automata that can recognize many different subsets of Coxeter systems as well.<br />
We generalize the results of Dyer using generalized length functions from Chapter<br />
3. These length functions give rise to large class of finite state automata, which<br />
are more powerful than the family of automata introduced by Dyer. We then<br />
describe the corresponding languages (subsets of the free monoid S ∗ ) recognized<br />
by our automata. These automata will be used in Chapter 5 to show that many<br />
subsets of Coxeter systems are regular.<br />
4.1 Regular Languages<br />
We use this section to introduce the idea of a regular language of a monoid.<br />
Let M be a monoid. By a (left) M-set, we mean a set A along with a function<br />
32
M × A → A given by (m, a) ↦→ ma such that 1Ma = a for all a ∈ A if 1M is the<br />
identity of M and m1(m2a) = (m1m2)a for all a ∈ A and mi ∈ M. In this case,<br />
we say that M acts on A.<br />
A subset M ′ of M is called (monoid) regular if there is a left M-set A with<br />
A finite, an element a0 ∈ A and a subset Af ⊂ A with the property that M ′ =<br />
{m ∈ M| ma0 ∈ Af}. The following lemma is well known (see [21]). We include<br />
a proof here for completeness.<br />
Lemma 4.1.1. The family of regular subsets of M forms a Boolean subalgebra<br />
R(M) of P(M). If M is finitely (or countably) generated, then R(M) is count-<br />
able.<br />
Proof. Suppose that M1 and M2 are both monoid regular subsets of M. We will<br />
demonstrate that M c 1 = M \ M1, M1 ∪ M2 and M1 ∩ M2 are all monoid regular as<br />
well.<br />
Since Mi is monoid regular, there exists M-sets A1 and A2 with ai ∈ Ai and<br />
(Ai)f ⊂ Ai such that Mi = {m ∈ M| ma0 ∈ (Ai)f}. Now, A1 is finite and so<br />
A ′ f = A1 \ (A1)f is also finite. Then M c 1 = {m ∈ M| ma1 ∈ A ′ f } so M c 1 is regular.<br />
Next, let B = A1×A2 and let m(a, a ′ ) = (ma, ma ′ ) be the action of M on B. Note<br />
that B is a finite set. Then M1 ∩ M2 = {m ∈ M| ma1 ∈ (A1)f; ma2 ∈ (A2)f} =<br />
{m ∈ M| m(a1, a2) ∈ (A1)f × (A2)f}. Let Bf = (A1)f × (A2)f} and then M1 ∩ M2<br />
is regular. Finally, M1 ∪ M2 = {m ∈ M| ma1 ∈ (A1)f; or ma2 ∈ (A2)f}. Let<br />
B ′ f = (A1)f ×A2 ∪A1 ×(A2)f ⊂ B and then M1 ∪M2 = {m ∈ M| m(a1, a2) ∈ B ′ f }<br />
so M1 ∪ M2 is regular.<br />
Now, let S be a set, called an alphabet. Elements of the free monoid S ∗ on<br />
S are called words. We call subsets of S ∗ languages. Let N be an additively<br />
written commutative monoid and let map L ′ : S → N be a map. Then, for<br />
33
any word sn...s1 ∈ S ∗ , we can define the length ℓ : S ∗ → N by ℓ(sn...s1) :=<br />
L ′ (sn) + · · · + L ′ (s1). For example, if we let N = N and L ′ (s) = 1 for all s ∈ S,<br />
then ℓ(sn...s1) = n. Also, for any word w = sn...s1 ∈ S ∗ we define the transpose<br />
† : S ∗ → S ∗ by †(w) := w † := s1...sn.<br />
Now, if A is a finite S ∗ -set, then we can describe the action by restriction of<br />
the multiplication map S ∗ × A → A to a function µ : S × A → A. Typically, µ<br />
is called the transition function of the S ∗ -set A. Since the transition function µ<br />
can be an arbitrary function, we get a bijection between functions µ : S × A → A<br />
where A is a set and S ∗ -sets A. If S is a finite set, then we define a regular<br />
language on S to be a monoid regular subset of S ∗ . Therefore, a regular language<br />
is determined by a tuple (µ : S × A → A, a0, Af) where A is a finite set, µ is<br />
a function, a0 ∈ A and Af ⊆ A. The language determined by previous tuple is<br />
L = {m ∈ S ∗ | ma0 ∈ Af} where the S ∗ action is determined by µ.<br />
4.2 Automata<br />
We can equivalently describe the monoid regular subsets of S ∗ by the termi-<br />
nology of finite state automata. A finite state automaton for the alphabet S is a<br />
tuple (A, a0, µ : S × A → A, Af) such that A is a finite set with a distinguished<br />
element a0 called the initial state, a finite subset Af ⊆ A called the accepting<br />
states (or final states) and a transition function µ. The automaton reads a word<br />
w = sn...s1 from right to left, one letter at a time. The machine begins in the<br />
initial state a0; upon reading a letter si, the state moves from the current state,<br />
a, to the state µ(si, a). We say the automaton accepts a word w = sn...s1 if after<br />
reading w, the current state of the automaton is a member of Af. The set of words<br />
accepted by the automaton is called the language accepted by the automaton. It<br />
34
is clear that the language accepted by the automaton (A, a0, µ : S × A → A, Af)<br />
corresponds to the regular subset of S ∗ associated to A viewing A as an S ∗ -set.<br />
Example 4.2.1. Consider the affine Weyl group of type � A1 = 〈r, s | r 2 = s 2 = 1〉.<br />
We know that the set of reduced expressions in � A1 is given by<br />
{e, r, s, rs, sr, rsr, srs, rsrs, srsr, ...}.<br />
Below, we have a finite state automaton that recognizes the set of reduced expres-<br />
sions of words in � A1. A is the vertex set of the given graph, a0 is the blue vertex,<br />
and Af is the set of non-red vertices. µ : S × A → A is given by the labeled edges.<br />
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4.3 Canonical Automata for Coxeter Systems<br />
At this point, we want to describe some finite state automata for Coxeter<br />
systems.<br />
Almost all objects defined in the rest of this chapter depend on the choice of p<br />
as in 3.2. We indicate when defining anything if it is dependent on p by including<br />
an appropriate subscript or superscript. However, when using the terminology,<br />
if there is no confusion about which function p we are referring to, we will leave<br />
off the corresponding subscript or superscript for ease of notation. Also, since<br />
(W, S) will be fixed, we let lp := lp,W . Recall, that lp takes values in M (which<br />
35<br />
r,s
depends on p) and that M has a partial order, which is described in section 3.1.<br />
In this chapter, we view M additively; therefore, M ∼ = N k and M ⊂ M ′ ∼ = Z k<br />
(see section 3.1).<br />
Let (W, S) be a Coxeter system and S ∗ be the free monoid on S. We denote<br />
the natural monoid homomorphism from S ∗ to W extending the identity map on<br />
S by ν : S ∗ → W . Let ℓp : S ∗ → M be the length of w = s1 · · · sn ∈ S ∗ given by<br />
ℓp(s1 · · · sn) = p(s1) + · · · + p(sn). Note ℓp(w) ≥ lp,W (ν(w)).<br />
We associate to (W, S) an S ∗ -set A = A(W,S). As a set we have<br />
A(W,S,p) := A = {∞} ∪ {(m, A)| m ∈ M, A ⊆ T≤m}.<br />
The S ∗ -action can be described by its restriction to a function S × A → A<br />
given by (s, ∞) ↦→ ∞ and<br />
⎧<br />
⎪⎨<br />
(s, (m, A)) ↦→<br />
⎪⎩<br />
∞, if p(s) �≤ m and s ∈ A<br />
� �<br />
m − p(s), (sAs \ {s}) ∩ T≤(m−p(s)) , if p(s) ≤ m and s ∈ A<br />
(m, (sAs ∪ {s}) ∩ T≤m) , if s �∈ A<br />
We may define sub-S ∗ -sets An = A(W,S,p,n) of A for n ∈ M defined by<br />
An := {∞} ∪ {(m, A)| m ≤ n, A ⊆ T≤m}<br />
We know that if S is finite then An is a finite S ∗ -set.<br />
36
Lemma 4.3.1. Let x ∈ W , w ∈ S ∗ , and m ∈ M. Then w · ∞ = ∞ and<br />
⎧<br />
⎪⎨ ∞, if k �≤ m<br />
w · (m, Nm(x)) ↦→<br />
⎪⎩ (m − k, Nm−k(ν(w)x)) , if k ≤ m<br />
in A , where k := lp(x)+ℓp(w)−lp(ν(w)x)<br />
2<br />
Proof. This follows by induction on ℓp(w) since it is clearly true for each w =<br />
s ∈ S. Then suppose that w = sw ′ and the result is known for sw ′ . Then, by<br />
definition of the S-action on pairs, the result is clear.<br />
Finally, from the previous lemma, we can define<br />
M = M(W,S,p) := {∞} ∪ {(m, Nm(x))| m ∈ M, x ∈ W }<br />
and M(W,S,p,n) = Mn = M ∩ An for n ∈ M, which are all sub-S ∗ -sets of A . We<br />
call these the canonical and n-canonical S ∗ -sets for (W, S) respectively.<br />
4.4 Some Subsets of S ∗<br />
Let Aa,b := {w ∈ S ∗ | w · a = b} for any a, b ∈ M. By Lemma 4.3.1, we see<br />
that A∞,∞ = S ∗ and A∞,x = ∅ for x �= ∞. We now intend to describe the sets<br />
Aa,b. To do this, we introduce some subsets of S ∗ in the following way.<br />
To begin with, for any word w ∈ S ∗ , we define<br />
Lp(w) := ℓp(w) − lp(ν(w))<br />
2<br />
∈ M.<br />
For any m ∈ M, we can then define S ∗ m := S ∗ p,m = {w ∈ S ∗ | Lp(w) = m}. The<br />
elements of S ∗ m will be called the m-nonreduced words of (W, S). If S is finite,<br />
37
then for any w ∈ W , there are only finitely many m-nonreduced expressions of w,<br />
namely the elements of the set ν −1 (w) ∩ S ∗ m.<br />
Next, for x, y ∈ W and n ∈ M ′ , we define more general sets<br />
F p x,y,n := {w ∈ S ∗ | lp(yν(w)x) = lp(y) + ℓp(w) + lp(x) − 2n} (4.4.1)<br />
and also define, for m ∈ M ′<br />
With this notation, we see that S ∗ m = F1,1,m.<br />
F p x,y,n,m := S ∗ m ∩ F p<br />
x,y,n+m. (4.4.2)<br />
Remark 4.4.1. It is clear from the definition that F p x,y,n = ∅ if n �∈ M, i.e. if<br />
n < 0.<br />
Example 4.4.2. For s ∈ S, we have that the set F1,s,lp(s) is given by<br />
{w ∈ S ∗ |lp(sw) = lp(s) + ℓp(w) − 2lp(s) = ℓp(w) − lp(s)},<br />
which can also be described by<br />
[ν −1 ({w ∈ W | lp(sw) < lp(w)}) ∩ S ∗ 0] ˙∪[ν −1 ({w ∈ W | lp(sw) > lp(w)}) ∩ S ∗ lp(s)].<br />
Next for any x ∈ W , k ∈ M ′ and finite subsets A, B of T , we let<br />
H p<br />
x,k,A,B := {w ∈ S∗ | lp(ν(w)x) = ℓp(w)+lp(x)−2k; N(ν(w)x)∩A = B} (4.4.3)<br />
so it is clear the H p<br />
x,k,A,B<br />
such that N(y) ∩ A = B.<br />
= ∅ unless k ∈ M, B ⊆ A, and there exists a y ∈ W<br />
38
According to Lemma 4.3.1, we see that for m ∈ M and x ∈ W we get<br />
and then for m, n ∈ M,<br />
A(m,Nm(x)),∞ = �<br />
k�≤m<br />
Fx,1,k<br />
⎧<br />
⎪⎨ ∅ if n �≤ m<br />
A(m,Nm(x)),(n,B) =<br />
⎪⎩ Hx,m−n,T≤n,B if n ≤ m<br />
where both sides are empty unless B is of the form Nn(y) for some y ∈ W .<br />
4.5 The Boolean Algebra of Regular Sets of S ∗<br />
(4.4.4)<br />
At this point, we assume that S is finite. Then, for a, b ∈ M, we can view<br />
Aa,b in the following way. We choose m large enough so that a, b ∈ Mm. Then<br />
Aa,b is the regular language accepted by an automaton where Mm is the state set,<br />
the transition function described in 4.3.1, initial state a, and single accept state<br />
b. At this stage, we characterize the Boolean subalgebra of S ∗ generated by all<br />
Aa,b where a �= ∞, b �= ∞.<br />
Lemma 4.5.1. The following Boolean subalgebras of S ∗ are all equal:<br />
1. The Boolean subalgebra Dp of P(S ∗ ) generated by all sets Aa,b where a, b ∈<br />
M \ {∞}.<br />
2. The Boolean subalgebra D ′ p of P(S ∗ ) generated by all sets Hx,k,A,B.<br />
3. The Boolean subalgebra D ′′<br />
p of P(S ∗ ) generated by all sets Fx,y,k,m.<br />
4. The Boolean subalgebra D ′′′<br />
p of P(S ∗ ) generated by all sets Fx,y,k.<br />
39
Moreover, Dp is stable under the automorphism † : S ∗ → S ∗ given by †(sn....s1) =<br />
s1...sn.<br />
Proof. According to equation (4.4.4), we know that Dp is generated by all the sets<br />
Hx,k,T≤n,B where x, y ∈ W , n, k ∈ M and B ⊂ T is finite. Suppose that k ∈ M,<br />
x ∈ W , and A, B ⊆ T are finite sets. We choose an element n ∈ M such that<br />
A ⊆ T≤n. Then<br />
Hx,k,A,B = �<br />
B ′ ⊆T≤n<br />
B ′ ∩A=B<br />
Hx,k,T≤n,B ′<br />
and so Hx,k,A,B is generated by a finite union of Aa,b for some a, b ∈ M \ {∞}.<br />
Thus Dp = D ′ p.<br />
Next, we note that for w ∈ Fx,y,k,m we get (since w ∈ S ∗ m)<br />
lp(yν(w)x) = lp(y) + ℓp(w) + lp(x) − 2(k + m)<br />
= lp(y) + lp(ν(w)) + 2m + lp(x) − 2(k + m)<br />
and so lp(yν(w)x) = lp(y) + lp(ν(w)) + lp(x) − 2k. This implies that Fx,y,k,m = ∅<br />
unless 0 ≤ k ≤ lp(y) + lp(x). This shows that S ∗ m = ∪0≤k≤lp(x)+lp(y)Fx,y,k,m and<br />
that we also have<br />
Fx,y,k =<br />
�<br />
k−lp(x)−lp(y)≤m≤k<br />
Fx,y,k−m,m<br />
thus showing that D ′′<br />
p = D ′′′<br />
p since it is clear that Fx,y,k,m = S ∗ m ∩ Fx,y,k+m =<br />
F1,1,m ∩ Fx,y,k+m.<br />
Now, we show that D ′′′<br />
p ⊆ D ′ p. For fixed x ∈ W and A ⊆ T finite, we have<br />
S ∗ = �<br />
�<br />
k∈M B⊆A<br />
Hx,k,A,B<br />
Recall from section 3.10 the terminology |A| ∈ M for A ⊆ T finite. Let y ∈ W .<br />
40
Then for w ∈ Hx,k,A,B we have<br />
lp(yν(w)x) = lp(y) + lp(ν(w)x) − 2|N(ν(w)x) ∩ N(y −1 )|<br />
From this we can see that<br />
= lp(x) + ℓp(w) + lp(y) − 2k − 2|N(ν(w)x) ∩ N(y −1 )|.<br />
Fx,y,m =<br />
�<br />
�<br />
k,n∈M:k+n=m B⊆N(y−1 ):|B|=n<br />
H x,k,N(y −1 ),B.<br />
This union is finite since there are only finitely many pairs k, n in M less than m<br />
and N(y −1 ) is a finite set, and so D ′′′<br />
p ⊆ D ′ p.<br />
have<br />
So we now must show that D ′ p ⊆ D ′′′ p. Let x, y ∈ W and w ∈ S ∗ , then we<br />
Fx,1,k ∩ Fx,y,m = {w ∈ W | lp(ν(w)x) = ℓp(w) + lp(x) − 2k and<br />
lp(yν(w)x) = lp(y) + ℓp(w) + lp(x) − 2m},<br />
(4.5.1)<br />
and by replacing ℓp(w) + lp(x) = lp(ν(w)x) + 2k in the second equation (using the<br />
first equation) we get<br />
Fx,1,k ∩ Fx,y,m = {w ∈ W | lp(ν(w)x) = ℓp(w) + lp(x) − 2k and<br />
lp(yν(w)x) = lp(ν(w)x) + lp(y) − 2m + 2k}.<br />
(4.5.2)<br />
Now, we know that lp(ν(w)x) ≤ lp(y) + lp(yν(w)x) ≤ lp(y) + lp(y) + lp(ν(w)x)<br />
consequently 0 ≤ lp(y)+lp(yν(w)x)−lp(ν(w)x) ≤ 2lp(y). Substituting appropriate<br />
equations from (4.5.1) and (4.5.2), we note that Fx,1,k ∩ Fx,y,m is empty unless<br />
0 ≤ 2lp(y) − 2m + 2k ≤ 2lp(y) and this implies that k ≤ m ≤ k + lp(y). We also<br />
have ∪m∈MFx,y,m = S ∗ . Now, let y from above be a given reflection y = t. Then<br />
41
we have that the following sets<br />
and<br />
Lx,1,k,t := {w ∈ Fx,1,k| t ∈ N(ν(w)x)} = Fx,1,k ∩<br />
L ′ x,1,k,t := {w ∈ Fx,1,k| t �∈ N(ν(w)x)} = Fx,1,k ∩<br />
�<br />
m: k≤m≤k+lp(t)<br />
0≤lp(t)≤2m−2k<br />
�<br />
m: k≤m≤k+lp(t)<br />
lp(t)≥2m−2k∈M<br />
Fx,t,m<br />
Fx,t,m<br />
are clearly in D ′′′<br />
p since these intersections are finite. If B ⊆ A, we thus get<br />
Hx,k,A,B =<br />
� �<br />
t∈B<br />
Lx,1,k,t<br />
� ⎛<br />
∩ ⎝ �<br />
t∈A\B<br />
L ′ x,1,k,t<br />
and as we have noted before, Hx,k,A,B = ∅ if B �⊆ A. Therefore, Hx,k,A,B is in D ′′′<br />
p ,<br />
which is what we were trying to show.<br />
Finally, suppose w ∈ Fx,y,m for some m. Now, we have that ν(†(w)) = ν(w) −1 .<br />
Since lp(yν(w)x) = lp(y) + ℓp(w) + lp(x) − 2m then we can reverse this to see that<br />
we must also have lp(x −1 ν(†(w))y −1 ) = lp(x −1 ) + ℓp(†(w)) + lp(y −1 ) − 2m and so<br />
†(w) ∈ F y −1 ,x −1 ,m, which shows that Dp is stable under †.<br />
4.6 The Reflection Cocycle and Automata<br />
Recall that for A ⊆ T a finite subset, we write |A| = �<br />
t∈A p(t). Thus, we<br />
note that lp(w) = |N(w)|. Now, we state a lemma that describes the essence<br />
of the previous proof. The problem of determining, for all w ∈ W , the value of<br />
lp(xw)−lp(w) for all x in some given finite subset of W is equivalent to finding, for<br />
42<br />
⎞<br />
⎠ ,
all w ∈ W , the intersection N(w) ∩ A for some given finite subset A of reflections.<br />
Recall that M ′ = Z k if M = N k .<br />
Lemma 4.6.1. (a) Let x1, ..., xn be elements of W , and l1, ..., ln ∈ M ′ . Let<br />
A := ∪ n i=1N(x −1<br />
i ). Then for w ∈ W , lp(xiw)−lp(w) = li for all i ∈ {1, ..., n}<br />
if and only if |(N(w) ∩ A) ∩ N(x −1 lp(xi)−li<br />
i )| = 2<br />
(b) Let A be a finite subset of T . Then for w ∈ W , we have<br />
for all i ∈ {1, ..., n}.<br />
N(w) ∩ A = {t ∈ A| lp(tw) = lp(w) − k for some k ∈ M≤lp(t)}<br />
Proof. Recall that N(xy) = N(x) + xN(y)x −1 = x(N(x −1 ) + N(y))x −1 . This<br />
implies that<br />
lp(xy) = lp(x) + lp(y) − 2|N(x −1 ) ∩ N(y)|<br />
where |−| is as above. Therefore, for w ∈ W , if xi ∈ W , we have lp(xiw)−lp(w) =<br />
lp(x) − 2|N(x −1<br />
i ) ∩ N(w)|. Since N(w) ∩ N(x−1 ) = (N(w) ∩ A) ∩ N(x −1<br />
i ), then<br />
lp(xiw) − lp(w) = li if and only if lp(xi) − 2|(N(w) ∩ A) ∩ N(x −1<br />
i )| = li, and (a)<br />
follows.<br />
Then, (b) follows since N(w) = {t ∈ T | lp(tw) < lp(w)}. We have lp(tw) =<br />
lp(t) + lp(w) − 2|N(t) ∩ N(w)|, we see that lp(t) − 2|N(t) ∩ N(w)| = −k for some<br />
k ∈ M.<br />
43
CHAPTER 5<br />
REGULAR SUBSETS OF <strong>COXETER</strong> <strong>GROUPS</strong><br />
This chapter is devoted to the definition of various notions of regularity in<br />
Coxeter systems. We introduce three different types of regularity. The strongest<br />
type of regularity, p-complete regularity, will be examined in detail, and we will<br />
describe many subsets of Coxeter systems that are p-completely regular. The set<br />
of m-nonreduced expressions of elements of any p-completely regular subset of W<br />
is a regular language in S ∗ for all m ∈ M. This implies that the multivariate<br />
Poincaré series of p-completely regular subsets are rational, providing a natural<br />
explanation (and strengthening) of known rationality results on Poincaré series of<br />
many natural subsets of W . We also use the results in this section to describe<br />
regularity of subsets in products of Coxeter systems.<br />
5.1 Boolean Algebra of Regular Sets in W<br />
For any x, y ∈ W and m ∈ M ′ , we can now define a subset<br />
Wx,y,m := W p x,y,m = {w ∈ W | lp(ywx) = lp(y) + lp(w) + lp(x) − 2m} ⊂ W.<br />
We have that W =<br />
�<br />
0≤k≤lp(x)+lp(y)<br />
Wx,y,k. Notice that Fx,y,k,m = ν −1 (Wx,y,k) ∩ S ∗ m<br />
so that Dp is the Boolean subalgebra of P(S ∗ ) generated by all intersections<br />
ν −1 (Wx,y,k) ∩ S ∗ m.<br />
44
Remark 5.1.1. Again, it is clear that Wx,y,m = ∅ if m �∈ M. This allows us to<br />
only consider sets Wx,y,m with m ≥ 0.<br />
Let Bp(W ) denote the Boolean subalgebra of P(W ) generated by all the sets<br />
Wx,y,k with x, y ∈ W and k ∈ M. Then Dp consists of all the sets ∪m∈Mν −1 (Bm)∩<br />
S ∗ m where Bm ∈ Bp(W ) for all m and Bm is either ∅ for all but finitely many<br />
m or Bm is W for all but finitely many m. Thus, we see that Dp is uniquely<br />
determined by the sets Bp(W ) together with the set of m-nonreduced expressions<br />
of w, for all m ∈ M and w ∈ W .<br />
Example 5.1.2. Let J ⊆ S be a spherical subset, i.e. WJ is a finite standard<br />
parabolic subgroup. Then, let wJ be the longest element of WJ and suppose<br />
lp(wJ) = k. Then D(J) := {w ∈ W | DL(w) = J} ∈ Bp(W ) since it is the set<br />
{w ∈ W | lp(wJw) = lp(wJ) + lp(w) − 2k = lp(w) − lp(wJ)} = W1,wJ ,k. This set is<br />
known as the descent class associated to J.<br />
5.2 Coxeter Groups are Automatic<br />
The following result is due to Brink and Howlett, [5], in their proof that Coxeter<br />
groups are automatic.<br />
Theorem 5.2.1. Fix a total order of S and give S ∗ the corresponding reverse<br />
lexicographic order �L. Then for w ∈ W define ˙w as the minimal (under �L)<br />
element of the non-empty and finite set ν −1 (w) ∩ S ∗ 0. Then ˙ W = { ˙w| w ∈ W } is<br />
a regular subset of S ∗ .<br />
5.3 Different Types of Regularity<br />
Fix any map µ : W → S ∗ with the following properties:<br />
45
(i) For all w ∈ W , µ(w) ∈ S ∗ 0<br />
(ii) ν ◦ µ = IdW .<br />
(iii) µ(W ) is a regular subset of S ∗ .<br />
Properties (i) and (ii) require that for w ∈ W , we have µ(w) is a reduced<br />
expression of W , and we know such a map exists due to Theorem 5.2.1.<br />
We say that a subset A of W is µ-regular if µ(A) is a regular subset of S ∗ .<br />
Definition 5.3.1. 1. A subset A of W is weakly regular if there is some map<br />
w ↦→ ˙w : A → S ∗ 0 such that ν( ˙w) = w for all w ∈ A and { ˙w| w ∈ A} is a<br />
regular subset of S ∗ .<br />
2. A subset of W is regular (resp. p-strongly regular) if ν −1 (A)∩S ∗ m is a regular<br />
subset of S ∗ for m = 0 (resp. for all m ∈ M).<br />
3. A subset of W is called p-completely regular if A ∈ Bp(W ).<br />
Example 5.3.2. Due to example 5.1.2, D(J) is p-completely regular.<br />
Lemma 5.3.3. (a) Any p-strongly regular subset of W is regular.<br />
(b) If A is a regular subset of W , then A is µ-regular for any map µ satisfying<br />
conditions (i)-(iii) above.<br />
(c) If A is µ-regular for some µ, then A is weakly regular.<br />
Proof. Parts (a) and (c) follow directly from the definitions. To show part (b),<br />
we assume that A is a regular subset of W . This means that ν −1 (A) ∩ S ∗ 0 is a<br />
regular subset of S ∗ . Additionally, since µ satisfies (i)-(iii), we know that µ(W ) is<br />
a regular subset of S ∗ as well. Now, (b) follows since µ(A) = µ(W )∩(ν −1 (A) ∩ S ∗ 0)<br />
which is a finite intersection of regular sets and is thus regular.<br />
46
5.4 p-Regularity for Different p<br />
Suppose that p : T → Xp and p ′ : T → Xp ′ are two different functions<br />
satisfying the properties of section 3.2 with corresponding sets of indeterminates<br />
Xp and Xp ′ respectively. Additionally, let Mp and Mp ′ be the corresponding<br />
sets of monomials for Xp and Xp ′. Since Mp and Mp ′ are the free commutative<br />
monoids on Xp and Xp ′ respectively, then for any map ϕ : Xp → Xp ′ we get<br />
a corresponding map, denoted ˜ϕ, with ˜ϕ : Mp → Mp ′ given by the universal<br />
property. We note that deg(m) = deg( ˜ϕ(m)) for any m ∈ Mp.<br />
Proposition 5.4.1. Let ϕ : Xp → Xp ′ be a surjective map such that ϕ(p(t)) =<br />
p ′ (t) for all t ∈ T . Then the following are true:<br />
1. Dp ′ ⊂ Dp, and<br />
2. Bp ′(W ) ⊂ Bp(W ).<br />
Proof. For q ∈ {p, p ′ }, Dq is generated by the sets F q x,y,m with x, y ∈ W and<br />
m ∈ Mq according to Lemma 4.5.1. Now, since ϕ is surjective then ˜ϕ is surjective<br />
as well. Let m ∈ Mp ′. Then define Q := ˜ϕ−1 (m). By surjectivity, Q �= ∅ and<br />
since ˜ϕ preserves degree, |Q| < ∞. Then, it follows that<br />
F p′<br />
x,y,m = �<br />
k∈Q<br />
F p<br />
x,y,k ∈ Dp<br />
and therefore Dp ′ ⊂ Dp, proving 1. Then, 2 is analogous replacing F q x,y,m by<br />
W q x,y,m.<br />
47
5.5 Some Properties of p-Completely Regular Sets<br />
sets.<br />
In this section we describe some of the basic properties of p-completely regular<br />
Proposition 5.5.1. (a) Any p-completely regular set is p-strongly regular (and<br />
thus also regular and µ-regular for any µ satisfying (i)-(iii) in section 5.3.<br />
(b) For any finite subsets B, C ⊆ A of T the subset<br />
GA,B,C := {w ∈ W | N(w) ∩ A = B, N(w −1 ) ∩ A = C}<br />
of W is p-completely regular.<br />
(c) If A ∈ Bp(W ) then A −1 := {w −1 | w ∈ A} ∈ Bp(W ).<br />
(d) Bp(W ) is closed under action by automorphisms of (W, S) that commute<br />
with p, i.e. automorphisms satisfying θ(p(t)) = p(θ(t)) for all t ∈ T .<br />
(e) Bp(W ) is closed under left and right multiplication by elements of W .<br />
(f) Bp(W ) contains all finite subsets of W .<br />
(g) Bp(W ) is countable.<br />
Proof. Suppose that A is p-completely regular. Then as we have seen ν −1 (A) ∩<br />
S ∗ m ∈ Dp for all m ∈ M. But Dp contains only regular languages by Lemma<br />
4.5.1 part (1), which proves (a). To prove (b), we let Wt := {w ∈ W | lp(tw) <<br />
lp(w)} = � lp(t)−c(t)<br />
lp(w)} =<br />
�<br />
W1,t,k ∈ Bp(W ) since both are finite unions. We also have sets<br />
0≤k< lp(t)+c(t)<br />
2<br />
Xt := {w ∈ W | lp(wt) < lp(w)} and X ′ t := {w ∈ W | lp(wt) > lp(w)}, and both<br />
48
are in Bp(W ) using finite unions of appropriate sets Wt,1,k. Now, since A is finite,<br />
we get<br />
YB := �<br />
Wt ∩ �<br />
t∈B<br />
t∈A\B<br />
YC := �<br />
Xt ∩ �<br />
t∈C<br />
t∈A\C<br />
and then GA,B,C := YB ∩ YC. In particular, this process describes an application,<br />
in detail, of Lemma 4.6.1.<br />
Next, (c) holds since (Wx,y,k) −1 = {w −1 | lp(ywx) = lp(y)+lp(w)+lp(x)−2k} =<br />
{w ∈ W | lp(yw −1 x) = lp(y) + lp(w) + lp(x) − 2k} = {w ∈ W | lp(x −1 wy −1 ) =<br />
lp(x −1 ) + lp(w) + lp(y −1 ) − 2k} = W y −1 ,x −1 ,k.<br />
Now, for any automorphism, θ, of (W, S) commuting with p, we have lp(w) =<br />
lp(θ(w)) for all w ∈ W . Then, we see that if w ∈ Wx,y,k then lp(θ(ywx)) =<br />
lp(ywx) = lp(y) + lp(x) + lp(w) − 2k = lp(θ(y)) + lp(θ(x)) + lp(θ(w)) − 2k so that<br />
θ(Wx,y,k) ⊂ Wθ(x),θ(y),k. Using the same argument with the automorphism θ −1<br />
W ′<br />
t<br />
demonstrates that in face θ(Wx,y,k) = Wθ(x),θ(y),k, proving (d).<br />
Next, (e) holds using the following observations. Let a ∈ W and b ∈ W<br />
and x, y ∈ W . Then lp(ya −1 ) = lp(y) + lp(a −1 ) − 2P for some P ≥ 0 and<br />
lp(b −1 x) = lp(x) + lp(b −1 ) − 2Q for some Q ≥ 0. If w ∈ Wx,y,k we have lp(ywx) =<br />
lp(y) + lp(w) + lp(x) − 2k. Then awb ∈ W satisfies lp(ya −1 awbb −1 x) = lp(y) +<br />
lp(w)+lp(x)−2k = lp(ya −1 )−lp(a −1 )+2P+lp(w)−lp(b −1 )+lp(b −1 x)+2Q−2k =<br />
lp(ya −1 ) + lp(awb) − 2n + lp(b −1 x) + 2P + 2Q − 2k where 0 ≤ n ≤ lp(a) + lp(b)<br />
and awb ∈ W b −1 ,a −1 ,n. Therefore we have<br />
aWx,y,kb =<br />
�<br />
0≤n≤lp(a)+lp(b)<br />
X ′ t<br />
(W b −1 x,ya −1 ,k−P−Q+n ∩ W b −1 ,a −1 ,n)<br />
49
where we recall that Wu,v,k = ∅ if k < 0.<br />
Finally, (f) holds since we have GS,∅,∅ = {1} ∈ Bp(W ) by (b). Then for any<br />
w ∈ W , w{1} = {w} ∈ Bp(W ) by (e), and so any finite union of singleton<br />
elements of W is in Bp(W ). Then, (g) follows from the nature of the fact that<br />
there are countably many sets Wx,y,k and we can only take finite unions and<br />
intersections.<br />
Example 5.5.2. For any J ⊆ S spherical, the subgroup WJ is p-completely<br />
regular by (f). Since W = W1,1,0, then W is p-completely regular. If (W, S) is<br />
finite, we can easily demonstrate that (W, S) is regular by letting the Cayley graph<br />
of (W, S) be the corresponding S ∗ -set.<br />
Remark 5.5.3. Given a completely regular set A ⊆ W , described explicitly as<br />
a finite union of finite intersections of the sets W p<br />
x,y,k , we could construct an<br />
automaton on S accepting ν −1 (A) ∩ S ∗ n for any n ∈ M using Lemma 4.5.1. In<br />
addition, following from Lemma 5.3.3, we could construct an automaton on S<br />
accepting µ(A) for any µ satisfying the (i)-(iii) from 5.3.3.<br />
5.6 Poincaré Series and Examples of Regular Subsets of W<br />
The previous result leads to some results about the rationality of Poincaré<br />
series of subsets of Coxeter groups. First we recall the definition of a Poincaré<br />
series.<br />
Definition 5.6.1. Let u be an indeterminate and consider the power series ring<br />
Z[[u]]. For a subset A ⊆ W we define the Poincaré series of A to be<br />
P (A; W ) = �<br />
u deg(lp(w)) .<br />
w∈A<br />
50
If we consider the multiplicative version of M generated by X, then we can con-<br />
sider the completion Z[[X]] of Z[X]. We define the multivariate Poincaré series of<br />
A to be<br />
Pp(A; W ) = �<br />
lp(w).<br />
w∈A<br />
The following examples are previously known in the case where lp = l is the<br />
standard length; we use our results to show the examples are p-completely regular<br />
for general p as well.<br />
Example 5.6.2. 1. For any finitely generated reflection subgroup W ′ and any<br />
standard parabolic subgroup WJ of (W, S), any double coset W ′ wWJ has<br />
a unique element of minimal length determined by the conditions N(w) ∩<br />
χ(W ′ ) = ∅ and N(w −1 ) ∩ J = ∅ where χ(W ′ ) is the set of canonical Coxeter<br />
generators of (W, S). In particular the set of these shortest coset represen-<br />
tatives, denoted W ′ \ W/WJ is thus given by the following:<br />
W ′ \ W/WJ = �<br />
C⊂χ(W )<br />
Gχ(W ′ ),∅,C ∩ �<br />
B⊂J<br />
GJ,B,∅<br />
and so is p-completely regular. This applies to shortest coset representatives<br />
W ′ \ W of W and to shortest double coset representatives WK \ W/WJ of<br />
standard parabolic subgroups.<br />
2. The weak right order ≤ on W is defined by x ≤ y if N(x) ⊆ N(y) or<br />
equivalently if lp(x −1 y) = lp(y)−lp(x). Thus, if we fix x ∈ W with lp(x) = m,<br />
the set<br />
W 1,x −1 ,m = {y ∈ W | lp(x −1 y) = lp(y) + lp(x) − 2lp(x)} = {y ∈ W | y ≥ x}<br />
51
is the upper rays in weak right order, and so this set is p-completely regular.<br />
A similar argument shoes that the upper rays in weak left order are also<br />
p-completely regular.<br />
3. As we have seen in the proof of Proposition 5.5.1 the sets {w ∈ W | lp(tw) <<br />
lp(w)} and {w ∈ W | lp(wt) < lp(w)} are p-completely regular for fixed<br />
t ∈ T . These spaces (and their complements) are known as left and right<br />
half spaces respectively. It follows that any finite intersection and union of<br />
left (or right) half spaces is p-completely regular. See Figure 5.1 for example.<br />
All of the sets in the previous examples thus have rational Poincaré series and<br />
multivariate Poincaré series due to a well-known result, called the transfer matrix<br />
method (see [2] or [28]), which states that regular languages have rational Poincaré<br />
series.<br />
5.7 More Regular Subsets of W<br />
The sets described in this section are shown to be p-completely regular in [18]<br />
for the case where p : T → N with p(t) = 1 for all t. We now prove that these<br />
sets are also p-completely regular for general p.<br />
Recall that W acts as a group of permutations on the set T × {±1} such that<br />
for s ∈ S, ɛ ∈ {±1}, and t ∈ T \{s} we have s(s, ɛ) = (s, −ɛ) and s(t, ɛ) = (sts, ɛ).<br />
We note that as a W × {±1}-set, T × {±1} is isomorphic to the standard root<br />
system Φ, with Φ + corresponding to T × {1} and simple roots Π corresponding to<br />
S × {1}. With this terminology, we have the following p-completely regular sets.<br />
Proposition 5.7.1. 1. For u, v ∈ T , W (u, v) = {w ∈ W | wuw −1 = v} is<br />
p-completely regular.<br />
52
Figure 5.1. This diagram shows the tiling of the plane corresponding to<br />
�B2 as in Figure 3.1. Each of the gray regions is an intersection of two<br />
half spaces, and thus each is p-completely regular for any p. Then, the<br />
total gray region is a union of two p-completely regular sets and so is<br />
p-completely regular itself. Therefore, this set also has a rational multivariate<br />
Poincaré series.<br />
53
2. For α, β ∈ T × {±1}, W (α, β) = {w ∈ W | w(α) = β} is p-completely<br />
regular.<br />
3. For finite sets Γ, ∆ ⊂ T ×{±1}, the set {w ∈ W | w(Γ) = ∆} is p-completely<br />
regular.<br />
4. For J, K ⊆ S, {w ∈ W | wJw −1 = K} is p-completely regular.<br />
5. For J, K ⊆ S, {w ∈ W | w((WJ ∩T )×{1}) = (WK ∩T )×{1}} is p-completely<br />
regular.<br />
Proof. We first prove 1 for u = r and v = s with r, s ∈ S. In this case, we have<br />
W (r, s) = {w ∈ W | wrw −1 = s} = {w ∈ W | wr = sw}. Now, W (r, s) will be<br />
empty (and thus p-completely regular) unless r is conjugate to s. So, if r and<br />
s are conjugate, then lp(r) = lp(s) and according to the exchange condition, we<br />
have that W (r, s) = Z1 ∪ Z2 where<br />
and<br />
Z1 = {w ∈ W | lp(wr) = lp(w) − lp(r) = lp(sw); lp(swr) = lp(w)}<br />
Z2 = {w ∈ W | lp(wr) = lp(w) + lp(r) = lp(sw); lp(swr) = lp(w)}.<br />
Then, each Zi is p-completely regular due to the following descriptions.<br />
and<br />
Z1 = Wr,1,lp(r) ∩ W1,s,lp(s) ∩ Wr,s,lp(r)<br />
Z2 = Wr,1,0 ∩ W1,s,0 ∩ Wr,s,lp(r).<br />
54
Now, in general, let u = xrx −1 and let v = ysy −1 , then we know that W (r, s)<br />
is p-completely regular by above and<br />
yW (r, s)x −1 = {ywx −1 ∈ W | wrw −1 = s}<br />
= {w ′ ∈ W | y −1 w ′ xrx −1 w ′−1 y = s}<br />
= {w ′ ∈ W | w ′ uw ′−1 = v} = W (u, v)<br />
is p-completely regular since p-completely regular sets are closed under left and<br />
right multiplication by elements of W by Proposition 5.5.1, proving 1.<br />
Now, according to equation (2.2.3), we know that the W -action on T × {±1}<br />
is given by w(t, ɛ) = (wtw −1 , τw,tɛ) where τw,t ∈ {±1} is given by τw,t = 1 if<br />
lp(wt) > lp(w) and τw,t = −1 if lp(wt) < lp(w). Then, let α = (u, ɛ) and β = (v, ɛ ′ )<br />
then<br />
W (α, β) = W (u, v) ∩ {w ∈ W | τw,uɛ = ɛ ′ }.<br />
Now, W (u, v) is p-completely regular by 1. If ɛ = ɛ ′ then τw,u must be 1 and<br />
so {w ∈ W | τw,u = 1} = {w ∈ W | lp(wu) > lp(w)} is p-completely regular by<br />
Example 3 in 5.6. Likewise if ɛ �= ɛ ′ then τw,u must be -1 and so {w ∈ W | τw,u =<br />
−2} = {w ∈ W | lp(wu) < lp(w)} is p-completely regular by the same example.<br />
Thus, {w ∈ W | τw,uɛ = ɛ ′ } is p-completely regular so that W (α, β) also is as<br />
required.<br />
Finally, to demonstrate 3, we note that the set described is empty unless<br />
|Γ| = |∆|. Let {α1, ..., αn} = Γ and {β1, ..., βn} = ∆. Then we have<br />
YΓ,∆ := {w ∈ W | w(Γ) = ∆} = ∪σ∈Sn ∩ n i=1 W (αi, βσ(i)),<br />
where Sn denotes the symmetric group. Then, YΓ,∆ is p-completely regular by 2.<br />
55
Similarly, we have 4 following from 1 in the exact same way replacing W (αi, βσ(i))<br />
with W (si, rσ(i)). Finally, according to section 2.3, we know that w((WJ ∩ T ) ×<br />
{1}) = (WK ∩ T ) × {1} if and only if w(J × {1}) = K × {1}, so 5 follows from 3<br />
directly using Γ = J × {1} and ∆ = K × {1} since both sets are finite.<br />
Example 5.7.2. The centralizer of a reflection, t, is CW (t) = {w ∈ W | w −1 tw =<br />
t}. Due to the previous Proposition, this set is p-completely regular. So the cen-<br />
tralizer of any finitely generated reflection subgroup is also p-completely regular.<br />
5.8 Regularity in Direct Products<br />
We now turn to looking at direct products of Coxeter systems. We have the<br />
following result describing the p-completely regular subsets of a Coxeter system<br />
in terms of the p-completely regular subsets of its irreducible components.<br />
Lemma 5.8.1. Suppose that (W, S) is the direct product of Coxeter systems<br />
(Wi, Si) for i = 1, ..., n. We identify W = W1 × · · · × Wn and S = ˙∪Si. Then<br />
Bp(W ) is equal to the Boolean subalgebra Bp(W1) ∗ · · · ∗ Bp(Wn) of P(W ) gen-<br />
erated by all subsets of W of the form A1 × · · · × An with Ai ∈ Bp(Wi).<br />
Proof. This is clearly true for n = 1. Now, suppose it is true for n = k, we will<br />
show that it is true for n = k + 1. We consider W ′ = W1 × · · · × Wk and Wk+1<br />
as subgroups of W . For any x, y ∈ W , we write x = x ′ xk+1 and y = y ′ yk+1 where<br />
56
x ′ , y ′ ∈ W ′ and xk+1, yk+1 ∈ Wk+1. Let m ∈ M. Then<br />
W p x,y,m = {w ∈ W | lp(ywx) = lp(y) + lp(w) + lp(x) − 2m}<br />
= {w ′ wk+1 ∈ W ′ × Wk+1 |<br />
lp(y ′ w ′ x ′ yk+1wk+1xk+1) =<br />
lp(y ′ ) + lp(yk+1) + lp(w ′ ) + lp(wk+1) + lp(x ′ ) + lp(xk+1) − 2m}<br />
= �<br />
W<br />
k1,k2∈M<br />
k1+k2=m<br />
′p<br />
x ′ ,y ′ p<br />
× (Wk+1)<br />
,k1 xk+1,yk+1,k2<br />
(5.8.1)<br />
so that Bp(W ) ⊆ Bp(W ′ ) ∗ Bp(Wk+1). Also, equation (5.8.1) implies that<br />
W p<br />
x ′ ,y ′ ′p<br />
= W ,k1 x ′ ,y ′ ,k1 × Wk+1 and W p<br />
xk+1,yk+1,k2 = W ′ × (Wk+1) p<br />
. Therefore,<br />
xk+1,yk+1,k2<br />
we have<br />
W ′p<br />
x ′ ,y ′ p<br />
p<br />
× (Wk+1) = W ,k1 xk+1,yk+1,k2 x ′ ,y ′ p<br />
∩ W ,k1 xk+1,yk+1,k2<br />
so that Bp(W ′ ) ∗ Bp(Wk+1) ⊆ Bp(W ). However, by induction, we know that<br />
Bp(W ′ ) = Bp(W1) ∗ · · · ∗ Bp(Wk) and the result follows.<br />
5.9 Product Regularity<br />
Complete regularity of a subset in a product W = W1 × · · · × Wn implies<br />
regularity of many subsets of S ∗ and so regularity of various subsets of S ∗ 1 ×· · ·×S ∗ n.<br />
Then we regard each S ∗ i as a submonoid of S ∗ and each Wi as a subgroup of W ;<br />
then we can define natural homomorphisms νi : S ∗ i → Wi. We say that a subset of<br />
S ∗ 1 × · · · × S ∗ n is product regular if it is in the Boolean algebra of subsets generated<br />
by the products A1 × · · · × An with each Ai regular in S ∗ i .<br />
Lemma 5.9.1. For any p-completely regular subset A of W1 × · · · × Wn, and any<br />
57
n-tuple (k1, ..., kn) ∈ M n , the subset<br />
{(x1, ..., xn) ∈ (S1) ∗ k1 × · · · × (Sn) ∗ kn | (ν1(x1), ..., νn(xn)) ∈ A}<br />
is product regular in (S1) ∗ × · · · × (Sn) ∗ .<br />
Proof. For any subset A of W , we define<br />
A ′ := {x1, ..., xn) ∈ (S1) ∗ k1 × · · · × (Sn) ∗ kn | (ν1(x1), ..., νn(xn)) ∈ A}.<br />
If A is a product then A = A1 × · · · × An, then we have<br />
A ′ = � (S1) ∗ k1<br />
∩ ν−1 1 (A1) � × · · · × � (Sn) ∗ kn ∩ ν−1 n (An) � .<br />
In this, if each Ai is p-completely regular (or just p-strongly regular), then each<br />
(Si) ∗ ki<br />
∩ ν−1<br />
i (Ai) is regular in (Si) ∗ by Lemma 5.3.3 and thus A ′ is product regular.<br />
Now if A = ∪mBm, where the union is finite, then A ′ = ∪mB ′ m. Now, this implies<br />
that A ′ is product regular for any p-completely regular set since any such set A is<br />
a finite union of products of p-completely regular sets.<br />
5.10 Multivariate Product Regularity<br />
For any Coxeter system (W, S), define a subset of W n to be p-completely<br />
regular if it is completely regular as a subset of the n-fold product Coxeter system<br />
(W, S) n . We have the following result.<br />
Lemma 5.10.1. Let n, m be natural numbers, m ∈ M, f1, ..., fm ∈ {1, ..., n},<br />
ɛ1, ..., ɛm ∈ {±1}, x0, x1, ..., xm ∈ W and A, B, C be finite subsets of W satisfying<br />
B, C ⊆ A. Then let L be the subset of W n consisting of n-tuples satisfying the<br />
58
following conditions:<br />
(i) If<br />
v := xm(wfm) ɛm xm−1(wfm−1) ɛm−1 xm−2 · · · x1(wf1) ɛ1 x0<br />
and k := � m<br />
i=0 lp(xi) + � m<br />
i=1 lp(wfi ), then lp(v) = k − 2m.<br />
(ii) N(v) ∩ A = B and N(v −1 ) ∩ A = C.<br />
Then L is a p-completely regular subset of W n .<br />
Proof. We will prove this by induction on m. To make things more tractable, we<br />
denote L by<br />
L = L(n, m, m, (fj), (ɛj), (xj), A, B, C)<br />
to show that L depends on its parameter set.<br />
First, suppose that m is fixed. If<br />
is p-completely regular, then<br />
L(n, m, m, (fj), (ɛj), (xj), ∅, ∅, ∅)<br />
L(n, m, m, (fj), (ɛj), (xj), A, B, C)<br />
is also p-completely regular using Lemma 4.6.1 (similar to the proof of Proposition<br />
5.5.1 part (b)).<br />
Now, suppose that m = 1. The set<br />
L(1, 1, m, (1), (1), (x0, x1), ∅, ∅, ∅) = W p x0,x1,m<br />
and thus is p-completely regular by definition. Also, as in the proof of Proposition<br />
59
5.5.1 part (c), we have<br />
L(1, 1, m, (1), (−1), (x0, x1), ∅, ∅, ∅) = L(1, 1, m, (1), (1), (x −1<br />
1 , x −1<br />
0 ), ∅, ∅, ∅).<br />
It follows that the sets<br />
for ɛ ∈ {±1} are p-completely regular.<br />
Let n ∈ N. Then any set<br />
with k ∈ {1, ..., n} is of the form<br />
L(1, 1, m, (1), (ɛ), (x0, x1), ∅, ∅, ∅)<br />
L ′ = L(n, 1, m, (k), (ɛ), (x0, x1), ∅, ∅, ∅)<br />
L ′ = W k−1 × L(1, 1, m, (1), (ɛ), (x0, x1), ∅, ∅, ∅) × W n−k<br />
and thus is p-completely regular. Therefore, we have shown that any set<br />
is p-completely regular.<br />
L(n, 1, m, (k), (ɛ), (x0, x1), A, B, C)<br />
At this point, we have proved that the set L is p-completely regular if m = 1,<br />
and if m = 0 it is clear that L is p-completely regular. So, we now move to the<br />
induction step. We will show that for fixed m, p-complete regularity of all sets of<br />
the form<br />
L(n, m, m, (fj), (ɛj), (xj), A, B, C)<br />
60
implies p-complete regularity of any sets of the form<br />
L ′′ := L(n, m + 1, m, (fj), (ɛj), (xj), ∅, ∅, ∅).<br />
Now, suppose that w = (w1, ..., wn) ∈ W n is an element of L ′′ above. Then we<br />
define the following two words:<br />
and<br />
vm := xm(wfm) ɛm xm−1(wfm−1) ɛm−1 xm−2 · · · x1(wf1) ɛ1 x0<br />
vm+1 := xm+1(wfm+1) ɛm+1 xm(wfm) ɛm xm−1(wfm−1) ɛm−1 xm−2 · · · x1(wf1) ɛ1 x0<br />
and two p-lengths km := � m<br />
i=0 lp(xi) + � m<br />
i=1 lp(wfi ) and km+1 := km + lp(xm+1) +<br />
lp(wfm+1). Let v := xm+1(wfm+1) ɛm+1 so that vm+1 = vvm, and let k := lp(xm+1) +<br />
lp(wfm+1) so that km+1 = km + k. Using these equalities, we have<br />
lp(vm+1) = lp(v) + lp(vm) − 2|N(v −1 ) ∩ N(vm)|<br />
= km+1 − k − km + lp(v) + lp(vm) − 2|N(v −1 ) ∩ N(vm)|<br />
�<br />
k − lp(v)<br />
= km+1 − 2<br />
+<br />
2<br />
km − lp(vm)<br />
+ |N(v<br />
2<br />
−1 �<br />
) ∩ N(vm)| .<br />
By definition, lp(v) = lp(xm+1) + lp(wfm+1) − 2|N(x −1<br />
m+1) ∩ N((wfm+1) ɛm+1 )| ≤ k,<br />
and similarly we have lp(vm) ≤ km. We know that w ∈ L ′′ so we must have<br />
lp(vm+1) = km+1 − 2m.<br />
According to Corollary 3.10.1 part (4) we know that lp(vm+1) < km+1 − 2m if<br />
lp(v) < k − 2m, lp(vm) < km − 2m or |Nm(v −1 ) ∩ Nm(vm)| > m. Otherwise, we<br />
61
have<br />
�<br />
k − lp(v)<br />
lp(vm+1) = km+1 − 2<br />
+<br />
2<br />
km − lp(vm)<br />
+ |Nm(v<br />
2<br />
−1 �<br />
) ∩ Nm(vm)| .<br />
(5.10.1)<br />
Suppose lp(v) = k − 2m ′ and lp(vm) = km − 2m ′′ . Then the equation (5.10.1) is<br />
lp(vm+1) = km+1 − 2 � m ′ + m ′′ + |Nm(v −1 ) ∩ Nm(vm)| � .<br />
Since lp(vm+1) = km+1 − 2m, then we have the following:<br />
where<br />
L ′′ =<br />
�<br />
m ′ ,m ′′ ,B,C<br />
B,C⊂T≤m<br />
m ′ +m ′′ +|B∩C|=m<br />
Lm ′ ,m ′′ ,B,C<br />
Lm ′ ,m ′′ ,B,C = L(n, m, m ′ , (fj) m j=1, (ɛj) m j=1, (xj) m j=0, T≤m, B, ∅)<br />
∩ L(n, 1, m ′′ , (fm+1), (ɛm+1), (1, xm+1), T≤m, ∅, C)<br />
both of which are p-completely regular by induction. Thus L ′′ is a finite union of<br />
p-completely regular sets and so L ′′ is p-completely regular as needed.<br />
62
CHAPTER 6<br />
<strong>REGULARITY</strong> <strong>IN</strong> T <strong>AND</strong> MIXED PRODUCTS<br />
The p-complete regularity in W as we have described has been useful for show-<br />
ing that many natural subsets of W are regular, and therefore have rational<br />
Poincaré series. However, there are many natural subsets of W , in particular<br />
subsets of T , that can be shown to not be p-complete regular in W (or even µ-<br />
regular in W ). Nonetheless, we would like to be able to show that certain subsets<br />
of T are regular in some sense. We will use the fact that every reflection has a<br />
palindromic reduced expression to introduce a notion of p-complete regularity in<br />
T . We can then extend our notion of regularity to regularity in products of T<br />
or even mixed products of W and T . We include some examples of p-completely<br />
regular subsets of T , and the regularity implies that these subsets have rational<br />
(multivariate) Poincaré series.<br />
6.1 Regularity of sets of Reflections<br />
Let X ⊂ S ∗ be the set of all non-empty words in S ∗ . We see that X is<br />
clearly a regular subset of S ∗ being the complement of the set consisting of the<br />
empty word. We say that a subset of X is regular if it is regular as a subset of<br />
S ∗ . Define η : X → T by η(rn+1rn · · · r1) = r1 · · · rnrn+1rn · · · r1. Let ℓ ′ p(x) =<br />
2ℓp(x) − lp(rn+1) = 2ℓp(x) − p(η(x)). Let � ′ L<br />
lexicographic order on S ∗ to X.<br />
63<br />
be the restriction of some reverse
Proposition 6.1.1. 1. For any m ∈ M let Xm := {x ∈ X| lp(η(x)) = ℓ ′ p(x)−<br />
2m}. Then Xm is a regular subset of X.<br />
2. For t ∈ T , define ˙xt as the minimal (under the order � ′ L ) element of the<br />
non-empty finite set η −1 (t) ∩ X0. Then ˙<br />
T = { ˙xt| t ∈ T } is a regular subset<br />
of X.<br />
Proof. We prove 1 first. Let m ∈ M, and let x ∈ Xm. Then, we can write<br />
x = rx ′ with r ∈ S and x ′ ∈ S ∗ k<br />
for some k ∈ M. By definition, we know that<br />
lp(ν(x ′ )) = ℓp(x ′ ) − 2k. Since x ∈ Xm, and substituting the previous formula we<br />
get<br />
lp(ν(x ′ ) −1 rν(x ′ )) = lp(η(x)) = ℓ ′ p(x) − 2m = 2ℓp(x ′ ) + lp(r) − 2m<br />
= 2(2lp(ν(x ′ )) + 2k) + lp(r) − 2m<br />
= 2lp(ν(x ′ )) + lp(r) − 2(m − 2k)<br />
. (6.1.1)<br />
Equation (6.1.1) implies that m − 2k ≥ 0 so 2k ≤ m; furthermore, (6.1.1) also<br />
implies that ν(x ′ ) ∈ Ar,m−2k where<br />
Ar,m−2k := L(1, 2, m − 2k, (1, 1), (1, −1), (e, r, e), ∅, ∅, ∅)<br />
where e ∈ W is the identity, and L is as in 5.10. According to Lemma 5.10.1<br />
Ar,m−2k is p-completely regular. Therefore, we have x ′ ∈ S ∗ k ∩ ν−1 (Ar,m−2k) which<br />
is regular by Lemma 5.5.1. So, for any x ∈ Xm, there is an r ∈ S and k ∈ M<br />
with 2k ≤ m such that x ∈ r(S ∗ k ∩ ν−1 (Ar,m−2k)). Thus<br />
Xm ⊆ �<br />
�<br />
�<br />
r<br />
r∈S<br />
0≤2k≤m<br />
64<br />
S ∗ k ∩ ν −1 (Ar,m−2k)<br />
�
Now, suppose that x ∈ r(S ∗ k ∩ ν−1 (Ar,m−2k)) for some r ∈ S and k ∈ M with<br />
2k ≤ m. We have from above that<br />
Ar,m−2k = {w ∈ W | lp(w −1 rw) = 2lp(w) + lp(r) − 2(m − 2k)}.<br />
Thus x = rx ′ with x ′ ∈ ν −1 (Ar,m−2k) and lp(ν(x ′ )) = 2ℓp(x ′ ) − 2k. This gives that<br />
lp(η(x)) = lp(ν(x ′ ) −1 rν(x ′ )) = 2lp(ν(x ′ )) + lp(r) − 2(m − 2k)<br />
so that x ∈ Xm. Thus we have shown<br />
Xm = �<br />
�<br />
�<br />
r<br />
r∈S<br />
= 2(ℓp(x ′ ) − 2k) + lp(r) − 2(m − 2k)<br />
= 2ℓp(x ′ ) + lp(r) − 2m = 2ℓ ′ p(x) − 2m,<br />
0≤2k≤m<br />
S ∗ k ∩ ν −1 (Ar,m−2k)<br />
and the right hand side is a finite union of regular sets as already shown hence<br />
Xm is regular as well.<br />
Part (2) follows from and unpublished result due to Bob Howlett, which is the<br />
analog of 5.2.1 for T instead of W . The result is mentioned in [18].<br />
6.2 Types of Regularity in T<br />
Fix any map τ : T → X, denoted t → xt for t ∈ T , with the following<br />
properties:<br />
(i) For all t ∈ T , τ(t) ∈ X0.<br />
(ii) η ◦ τ = IdT .<br />
(iii) ˙<br />
T := τ(T ) is a regular subset of X.<br />
65<br />
�<br />
,
We know that maps, τ, satisfying (i)-(iii) above exist due to the previous propo-<br />
sition. We say a subset A of T is τ-regular if τ(A) is regular in X.<br />
Definition 6.2.1. A subset A of T is regular (respectively p-strongly regular) in<br />
T if η −1 (A) ∩ Xm is a regular subset of X for m = 0 (respectively all m ∈ M).<br />
Lemma 6.2.2. (a) Any p-strongly regular subset of T is regular.<br />
(b) If A is a regular subset of T then A is τ-regular for any map satisfying<br />
(i)-(iii) above.<br />
Proof. Part (a) is a restatement of the definition since 0 ∈ M. For part (b), we<br />
have that for regular subset A ⊆ T and τ satisfying (i)-(iii) above, we get that<br />
τ(A) = (η −1 (A) ∩ X0) ∩ τ(T ) which is an intersection of regular sets and thus<br />
regular.<br />
6.3 p-Complete Regularity in T<br />
We now want to describe the analog of Bp(W ) when dealing with regular<br />
subsets of T . To do this, we first introduce a few subsets of P(T ).<br />
Definition 6.3.1. 1. For x ∈ W and m ∈ M, we let<br />
R p x,m := {t ∈ T | lp(xtx −1 ) = 2lp(x) + lp(t) − 2m}.<br />
Then, let B ′′<br />
p(T ) be the Boolean subalgebra of P(T ) generated by all sets<br />
R p x,m for x ∈ W and m ∈ M.<br />
2. For t ∈ T and A ⊆ T , we define At := {w ∈ W | w −1 tw ∈ A}. Then,<br />
let B ′ p(T ) be the family of all subsets A of T such that for all r ∈ S,<br />
Ar ∈ Bp(W ).<br />
66
3. For any t ∈ T and A ⊂ T and m ∈ M, we define<br />
A p<br />
t,m := {w ∈ W | w −1 tw ∈ A; lp(w −1 tw) = 2lp(w) + lp(t) − 2m}.<br />
We let Bp(T ) be the family of all subsets A of T such that for all r ∈ S<br />
and m ∈ M we have Ar,m ∈ Bp(W ). We call the elements of Bp(T ) the<br />
p-completely regular subsets of T .<br />
Properties of these families of subsets are described in the following proposi-<br />
tion. Again, for ease of notation, we omit the superscript p since p is understood<br />
to be fixed.<br />
Proposition 6.3.2. (a) If A ∈ B ′ p(T ) then At ∈ Bp(W ) for all t ∈ T . Like-<br />
wise, if A ∈ Bp(T ) then At,m ∈ Bp(W ) for all t ∈ T and m ∈ M.<br />
(b) B ′′<br />
p(T ), B ′ p(T ) ⊂ Bp(T ) are Boolean subalgebras of P(T ) closed under con-<br />
jugation by elements of W and closed under permutations of T induced by<br />
automorphisms, θ, of (W, S) that commute with p.<br />
(c) B ′ p(T ) contains all finite subsets of T and all conjugacy classes of simple<br />
reflections.<br />
(d) Any set A which is p-completely regular in T is p-strongly regular in T .<br />
Proof. Suppose A ∈ B ′ p(T ). Then Ar ∈ Bp(W ) for all r ∈ S. Now, let t ∈ T be<br />
given. Then, t = u −1 su for some u ∈ W and s ∈ S. Now we have<br />
At = {w ∈ W | w −1 tw ∈ A} = {w ∈ W | w −1 u −1 suw ∈ A}<br />
= {u −1 v ∈ W | v −1 sv ∈ A} = u −1 · {v ∈ W | v −1 sv ∈ A} = u −1 As,<br />
67
and u −1 As ∈ Bp(W ) since As ∈ Bp(W ) and by Proposition 5.5.1 Bp(W ) is<br />
closed under left multiplication by elements of W , thus the first part of (a) is true.<br />
For the second part of (a), we fix u ∈ W and s ∈ S such that t = u −1 su and<br />
lp(t) = 2lp(u) + lp(s). Now, we consider the sets<br />
Bk := u −1 As,m−2k ∩ W1,u,k ∈ Bp(W ).<br />
Notice, if w ∈ Bk then w = u −1 v with v ∈ As,m−2k, and so we have w −1 tw =<br />
v −1 uu −1 suu −1 v = v −1 sv ∈ A. Also, we have<br />
lp(w −1 tw) = lp(v −1 sv) = 2lp(v) + lp(s) − 2(m − 2k) = 2lp(uw) + lp(s) − 2(m − 2k),<br />
but w ∈ W1,u,k so lp(uw) = lp(u) + lp(w) − 2k so that we get<br />
lp(w −1 tw) = 2(lp(u) + lp(w) − 2k) + lp(s) − 2(m − 2k)<br />
= 2lp(w) + 2lp(u) + lp(s) − 2m<br />
= 2lp(w) + lp(t) − 2m.<br />
Therefore Bk ⊆ At,m. Similarly, At,m ∩ W1,u,k ⊆ Bk. We note that Bk = ∅ unless<br />
2k ≤ m, and thus<br />
At,m =<br />
�<br />
k:0≤2k≤m<br />
So At,m ∈ Bp(W ) since it is a finite union of sets in Bp(W ), finishing the proof<br />
of (a).<br />
For the first part of (b), we notice that for A, B ∈ B ′ p(T ) or A, B ∈ Bp(T ) and<br />
r ∈ S, it is clear that (A ∪ B)r = Ar ∪ Br, (A ∩ B)r = Ar ∩ Br and (A c )r = (Ar) c<br />
so that B ′ p(T ) and Bp(T ) are clearly Boolean subalgebras of P(T ) and B ′′<br />
p(T )<br />
68<br />
Bk.
is a Boolean subalgebra by definition. Now, suppose that A ∈ B ′ p(T ). Then<br />
Ar ∈ Bp(W ) by definition. Also, for any r ∈ S and m ∈ M, we have the set<br />
Br,m = {w ∈ W | lp(w −1 rw) = 2lp(w) + lp(r) − 2m}<br />
= L(1, 2, m, (1, 1), (1, −1), (e, r, e), ∅, ∅, ∅)<br />
following notation from section 5.10. Thus, by Lemma 5.10.1 we have Br,m ∈<br />
Bp(W ) for all r ∈ S and m ∈ M. Finally, we see that Ar,m = Ar ∩ Br,m, and<br />
thus Ar,m ∈ Bp(W ) for all r ∈ S and m ∈ M so that A ∈ Bp(T ).<br />
Now, we show that R p x,m ∈ Bp(T ) for all x ∈ W and m ∈ M. For any r ∈ S<br />
and n ∈ M we have<br />
(R p x,m)r,n = {w ∈ W | w −1 rw ∈ Tx,m; lp(w −1 rw) = 2lp(w) + lp(r) − 2n}<br />
= {w ∈ W | lp(xw −1 rwx −1 ) = 2lp(x) + lp(w −1 rw) − 2m;<br />
lp(w −1 rw) = 2lp(w) + lp(r) − 2n}<br />
= {w ∈ W | lp(xw −1 rwx −1 ) = 2lp(x) + 2lp(w) + lp(r) − 2(m + n);<br />
lp(w −1 rw) = 2lp(w) + lp(r) − 2n}<br />
= L(1, 2, m + n, (1, 1), (−1, 1), (x, r, x −1 ), ∅, ∅, ∅) ∩ Br,n<br />
using notation from the proof of Lemma 5.10.1, and by the result of that lemma,<br />
we get that (R p x,m)r,n ∈ Bp(W ). Thus, R p x,m ∈ Bp(T ) so that B ′′<br />
p(T ) ⊂ Bp(T ) as<br />
required.<br />
Next, suppose that A ∈ B ′ p(T ). Then Ar ∈ Bp(W ) for all r ∈ S. Let w ∈ W<br />
69
e fixed. Then wAw −1 ∈ B ′ p(T ) since for r ∈ S we have<br />
(wAw −1 )r = {u ∈ W | u −1 ru ∈ wAw −1 } = {u ∈ W | w −1 u −1 ruw ∈ A}<br />
= {vw −1 ∈ W | v −1 rv ∈ A} = {v ∈ W | v −1 rv ∈ A}w −1 = Arw −1<br />
is in Bp(W ) because Bp(W ) is closed under multiplication by elements of W .<br />
For A ∈ Bp(T ) and w ∈ W , we see that for r ∈ S and m ∈ M we have<br />
(wAw −1 )r,m = (wAw −1 )r ∩ �<br />
0≤k≤lp(w)<br />
(B(w, r, 2k) ∩ W w −1 ,1,k)w −1<br />
(6.3.1)<br />
where B(w, r, 2k) = {v ∈ W | lp(wv −1 rvw) = 2lp(w) + 2lp(v) + lp(r) − 2(m + 2k)}.<br />
Indeed, if u ∈ (wAw −1 )r,m, then u ∈ (wAw −1 )r and lp(u −1 ru) = 2lp(u) + lp(r) −<br />
2m. By the previous argument, u = vw −1 , and then we get lp(wv −1 rvw −1 ) =<br />
2lp(vw −1 ) + lp(r) − 2m = 2(lp(v) + lp(w) − 2k) + lp(r) − 2m where k ≤ lp(w) (and<br />
we note that this means v ∈ W w −1 ,1,k). Hence, we have inclusion of the left hand<br />
side in the right hand side of (6.3.1). However, working the equation backwards,<br />
since k ≤ lp(w) we get inclusion the other way proving that Bp(T ) is closed under<br />
conjugation.<br />
Suppose that θ is an automorphism of (W, S). Then we see if A ∈ B ′ p(T ) we<br />
have for any r ∈ S<br />
θ(A)r = {u ∈ W | u −1 ru ∈ θ(A)} = {u ∈ W | θ −1 (u −1 ru) ∈ A}<br />
= {u ∈ W | θ −1 (u −1 )θ −1 (r)θ −1 (u) ∈ A} = {θ(v) ∈ W | v −1 θ −1 (r)v ∈ A}<br />
= θ(A θ −1 (r))<br />
and so θ(A)r ∈ Bp(W ) since Bp(W ) is closed under automorphisms. Notice<br />
this holds for any automorphism. For A ∈ Bp(T ), and θ an automorphism of<br />
70
(W, S) commuting with p, recall that from the proof of Proposition 5.5.1 that<br />
θ(lp(w)) = lp(w). Then for any r ∈ S and m ∈ M we have<br />
θ(A)r,m = {u ∈ W | u −1 ru ∈ θ(A); lp(u −1 ru) = 2lp(u) + lp(r) − 2m<br />
= θ(A)r ∩ {u ∈ W | lp(u −1 ru) = 2lp(u) + lp(r) − 2m}.<br />
We know that θ(A)r ∈ Bp(W ) from above, and {u ∈ W | lp(u −1 ru) = 2lp(u) +<br />
lp(r) − 2m} ∈ Bp(W ) by Lemma 5.10.1. Thus, we have proved (b).<br />
To show (c), we first note that {w ∈ W | w −1 rw = t} ∈ Bp(W ) for all r ∈ S by<br />
Proposition 5.7.1. Therefore {t} ∈ B ′ p(T ) for all t ∈ T , and hence any finite subset<br />
of T is p-completely regular as well. Also, if A = {wrw −1 | w ∈ W } is the conjugacy<br />
class of r ∈ S, then for any s ∈ A we have As = {w ∈ W | w −1 sw ∈ A} = W if<br />
s is conjugate to r and As = ∅ if s is not conjugate to r. Since W ∈ Bp(W ) and<br />
∅ ∈ Bp(W ), then As ∈ Bp(W ) for all s ∈ S and so A ∈ B ′ p(T ).<br />
Finally, we show that (d) is true. We have to show that Bm := η −1 (A) ∩ Xm<br />
is regular for all m ∈ M if A is p-completely regular. If x ∈ η −1 (A) then x = rx ′<br />
for some r ∈ S and η(x) = ν(x ′ ) −1 rν(x ′ ). Thus, η −1 (A) = ∪r∈S{r}B ′ r where<br />
B ′ r = {x ∈ S ∗ | ν(x) −1 rν(x) ∈ A}. Since x ∈ Xm, we also know that lp(η(x)) =<br />
lp(ν(x ′ ) −1 rν(x ′ )) = 2ℓp(x ′ ) + lp(r) − 2m. Therefore, we see that Bm = ∪r∈SrBr,m<br />
where<br />
Br,m := {x ′ ∈ S ∗ | ν(x ′ ) −1 rν(x ′ ) ∈ A; lp(ν(x ′ ) −1 rν(x ′ )) = 2ℓp(x ′ ) + lp(r) − 2m}.<br />
Hence, to show that Bm is regular, we need to show that each Br,m is regular (as<br />
Bm is then a finite union of regular sets).<br />
Now, if x ′ ∈ S ∗ k we know that ℓp(x ′ ) = lp(ν(x ′ )) + 2k. Therefore, we know that<br />
71
Br,m ∩ S∗ k is given by the set<br />
{x ′ ∈ S ∗ |ν(x ′ ) −1 rν(x ′ ) ∈ A; lp(ν(x ′ ) −1 rν(x ′ )) = 2lp(x ′ ) + lp(r) − 2(m − 2k)},<br />
which is equal to ν−1 (Ar,m−2k) ∩ S∗ k . Now, we know that since A is p-completely<br />
regular in T then Ar,n ∈ Bp(W ) for all n ∈ M so that Ar,m−2k is p-completely<br />
regular in W for all k. Thus, by Proposition 5.5.1 we know that ν −1 (Am−2k) ∩ S ∗ k<br />
is regular. Also, we see that Ar,m−2k = ∅ unless 0 ≤ 2k ≤ m. Therefore we have<br />
Br,m = ∪k∈M(Br,m ∩ S ∗ k) = ∪0≤k≤2m(Br,m ∩ S ∗ k)<br />
which is a finite union and thus regular, completing the proof of (d).<br />
6.4 Maps of Boolean Algebras<br />
In this section, we prove some technical facts about Boolean subalgebras that<br />
will be useful for describing regularity in mixed products of T and W .<br />
Suppose that X1, X2, ..., Xn are sets with each Xi �= ∅. Let X := X1×· · ·×Xn.<br />
For each i let<br />
be the natural projection and let<br />
πi : X → Xi<br />
π c i : X → X1 × · · · × ˆ Xi × · · · × Xn<br />
be the natural projection where ˆ Xi means we omit Xi. For each i let Bi be a<br />
Boolean subalgebra of P(Xi). Then let B := B1 ∗ · · · ∗ Bn represent the Boolean<br />
subalgebra of P(X) generated by all A1 × · · · × An with Ai ∈ Bi.<br />
72
For any Y ∈ P(X1) ∗ · · · ∗ P(Xn) = P(X), we define an equivalence relation<br />
on Xi, ≡Y,i, given by a ≡Y,i b if π c i<br />
� π −1<br />
i (a) ∩ Y � = π c i<br />
� π −1<br />
i (b) ∩ Y � .<br />
Lemma 6.4.1. Suppose X = X1 × · · · × Xn, B1 ∗ · · · ∗ Bn are as above, and<br />
Y ∈ P(X). Then the following hold.<br />
(a) For each i, there exist finitely many equivalence classes of ≡Y,i, denoted<br />
Y1,i, Y2,i, ..., Yni,i.<br />
(b) Define a rectangle to be a set of the form Yj1,1 ×Yj2,2 ×· · ·×Yjn,n (where each<br />
set in the product is one of the finitely many equivalence classes in (a)). If<br />
a point of Y is in a rectangle, then that rectangle is a subset of Y , and thus,<br />
Y is a union of (finitely many) rectangles.<br />
(c) Y ∈ B1 ∗ · · · ∗ Bn if and only if Yj,i ∈ Bi for all i ∈ {1, .., n} and all<br />
j ∈ {1, ..., ni}.<br />
(d) Let Fi be a subset of Xi for each i such that Fi∩Yj,i �= ∅ for all j ∈ {1, ..., ni}.<br />
Consider F ′ := Y ∩F of F := F1×· · ·×Fn, and let ≡F ′ ,i be the corresponding<br />
equivalence relation on each Fi (in the same manner as ≡Y,i on Xi). Then<br />
the equivalence classes for ≡F ′ ,i are precisely the intersections Yj,i ∩ Fi. In<br />
particular, the rectangles for Y ∩ F that are contained in Y ∩ F are the<br />
intersections with F of the rectangles for Y which are contained in Y .<br />
Proof. Since Y ∈ P(X), we write<br />
Y = �<br />
Ak,1 × · · · × Ak,n<br />
k∈K<br />
where K is a finite index set. Then define Ci to be the Boolean subalgebra of<br />
P(Xi) generated by the finite set of elements {Ak,i| k ∈ K}. For each k ∈ K,<br />
73
let A r k,i := Ak,i and let A c k,i = Xi \ Ak,i (the complement). Then for a sequence<br />
ɛ := (ɛ1, ..., ɛ|K|) with each ɛi ∈ {r, c} we define Ai,ɛ := ∩k∈KA ɛk<br />
k,i . There are<br />
only finitely many such sequences, and so we call the finite set {Ai,ɛ| Ai,ɛ �= ∅}<br />
the set of atoms of Ci. We necessarily have Ai,ɛ ∩ Ai,ɛ ′ = ∅ for ɛ �= ɛ′ , and<br />
each set J ∈ Ci is a finite union of atoms. Suppose now that a, b ∈ Ai,ɛ for<br />
some ɛ. Let J ⊆ K such that ɛj = r for j ∈ J and ɛj = c for j �∈ J. Then<br />
{Aj,i| a ∈ Aj,i} = {Aj,i| j ∈ J} = {Aj,i| b ∈ Aj,i}. Therefore,<br />
π c i<br />
� π −1<br />
i (a) ∩ Y � = �<br />
j∈J<br />
Aj,1 × · · · × ˆ Aj,i × · · · × Aj,n = π c � −1<br />
i πi (b) ∩ Y � ;<br />
and thus a ≡Y,i b so that each atom Ai,ɛ is contained in an equivalence class of<br />
≡Y,i. Therefore, the atoms give a finer partition of Xi than ≡Y,i. Since there are<br />
only finitely many atoms, there are only finitely many equivalence classes of ≡Y,i<br />
proving (a).<br />
Now, suppose that Yj1,1 ×Yj2,2 ×· · ·×Yjn,n is a rectangle and that (a1, ..., an) ∈<br />
Y ∩ (Yj1,1 × Yj2,2 × · · · × Yjn,n). Let (b1, ..., bn) ∈ Yj1,1 × Yj2,2 × · · · × Yjn,n, as well.<br />
Then, since a1 ≡Y,1 b1, we know that (b1, a2, ..., an) ∈ Y . Working inductively,<br />
since ai ≡Y,i bi for all i, we have (b1, ..., bi, ai+1, ..., an) ∈ Y for all i. In particular,<br />
(b1, ..., bn) ∈ Y . Therefore, Yj1,1 × Yj2,2 × · · · × Yjn,n ⊆ Y , and so Y is a finite union<br />
of rectangles proving (b).<br />
Next, if Yji,i ∈ Bi for all i ∈ {1, ..., n} and ji ∈ {1, ..., ni} then Yj1,1 × Yj2,2 ×<br />
· · · × Yjn,n ∈ B1 ∗ · · · ∗ Bn for all rectangles and since Y is a finite union of<br />
rectangles by (b), we have Y ∈ B1 ∗ · · · ∗ Bn. Conversely, if Y ∈ B1 ∗ · · · ∗ Bn,<br />
then from the proof of (a), each Ak,i ∈ Bi, and so each Ai,ɛ ∈ Bi and so Ci ⊆ Bi.<br />
However, each Yji,i ∈ Ci ⊆ Bi as required for (c).<br />
Finally, let Fi, F , and F ′ be as in the statement of (d). Let a ≡Y,i b and let<br />
74
a, b ∈ Fi. Then we know that<br />
Since both a, b ∈ Fi, this implies that<br />
π c i<br />
π c i<br />
� −1<br />
πi (a) ∩ Y )� = π c � −1<br />
i πi (b) ∩ Y )� .<br />
� −1<br />
πi (a) ∩ (Y ∩ F ))� = π c � −1<br />
i πi (b) ∩ (Y ∩ F ))� ,<br />
and thus a ≡F ′ ,i b. Therefore Yj,i ∩ Fi ⊆ F ′ k,i for some k.<br />
Now, suppose that a ≡F ′ ,i b. Then a, b ∈ Fi and we have<br />
π c i<br />
� −1<br />
πi (a) ∩ (Y ∩ F ))� = π c � −1<br />
i πi (b) ∩ (Y ∩ F ))� .<br />
Let (x1, ..., a, ..., xn) ∈ π −1<br />
i (a)∩Y . Then, since Yj,m ∩Fm �= ∅, we have xm ≡Y,m x ′ m<br />
with x ′ m ∈ Fm for all m �= i. Therefore, we see that (x ′ 1, ..., ai, ..., x ′ n) ∈ π −1<br />
i (a) ∩<br />
(Y ∩ F ). Since a ≡F ′ ,i b, then (x ′ 1, ..., b, ..., x ′ n) ∈ π −1<br />
i (b) ∩ (Y ∩ F ), and using the<br />
equivalence xm ≡Y,m x ′ m in reverse, this implies that (x1, ..., b, ..., xn) ∈ π −1<br />
i (b)∩Y .<br />
The exact same argument starting with b instead of a shows<br />
π c i<br />
� π −1<br />
i (a) ∩ Y )� = π c i<br />
� −1<br />
πi (b) ∩ Y )�<br />
so that a ≡Y,i b, and thus Fk,i ⊆ Yj,i ∩ F for some j. Therefore, we have equality<br />
of equivalence classes. Equality of rectangles follows directly.<br />
Next, we assume we additionally have another family of sets X ′ 1, ..., X ′ n. Let<br />
X ′ := X ′ 1 × · · · × X ′ n, and suppose that we are given a subset Hi ⊆ Hom(Xi, X ′ i) ×<br />
Bi. For each i, we define a Boolean algebra B ′ i (which depends on Hi) of P(X ′ i)<br />
consisting of sets A ⊆ X ′ i such that for all (hi, Bi) ∈ Hi we have h −1<br />
i (A)∩Bi ∈ Bi.<br />
75
Then, let B ′ := B ′ 1 ∗ · · · ∗ B ′ n be the corresponding Boolean subalgebra of P(X ′ ).<br />
Define H := {((h1 × · · · × hn), (B1 × · · · × Bn))| (hi, Bi) ∈ Hi; ∀i}. Additionally,<br />
we set<br />
B ′′ := {Z ⊆ X ′ | h −1 (Z) ∩ B ∈ B, for all (h, B) ∈ H}.<br />
Lemma 6.4.2. Suppose that for all i ∈ {1, ..., n} there exists a finite subset H ′ i ⊂<br />
Hi such that ∪(hi,Bi)∈H ′ i hi(Bi) = Xi. Then B ′′ ⊆ P(X ′ 1) ∗ · · · ∗ P(X ′ n).<br />
Proof. Define H ′ := {((h1 × · · · × hn), (B1 × · · · × Bn))| (hi, Bi) ∈ H ′ i; ∀i} ⊆ H.<br />
Let Z ∈ B ′′ . Clearly we have<br />
�<br />
(h,B)∈H ′<br />
h � h −1 (Z) ∩ B � ⊆ Z, (6.4.1)<br />
and since each h −1 (Z) ∩ B ∈ P(X) and h : P(X) → P(X ′ ), we have that the<br />
left hand side of (6.4.1) is in P(X ′ ). Now, suppose that (z1, ..., zn) ∈ Z. By<br />
assumption, we can write zi = hi(bi) for some (hi, Bi) ∈ H ′ i with bi ∈ Bi. Then,<br />
let h := h1 × · · · × hn and B := B1 × · · · × Bn so that (h, B) ∈ H ′ by construction.<br />
Then, b := (b1, ..., bn) ∈ h −1 (Z) ∩ B and so (z1, ..., zn) ∈ h (h −1 (Z) ∩ B) for some<br />
(h, B) ∈ H ′ . Therefore, we have<br />
Z ⊆ �<br />
(h,B)∈H ′<br />
h � h −1 (Z) ∩ B � ,<br />
and thus Z ∈ P(X ′ ) since we have equality in (6.4.1).<br />
Lemma 6.4.3. With notation as above, the following hold.<br />
1. B ′ ⊆ B ′′ .<br />
2. Suppose that for all i ∈ {1, ..., n} and for all h ∈ Hi we have that h is<br />
76
surjective; furthermore, assume that for each hi, we have<br />
�<br />
Bi: (hi,Bi)∈Hi<br />
Then B ′′ ⊆ B ′ (and thus B ′ = B ′′ ).<br />
Bi = Xi. (6.4.2)<br />
Proof. We begin by showing 1. Let C ′ i ∈ B ′ i. Then if (hi, Bi) ∈ Hi we have<br />
h −1<br />
i (C′ i) ∩ Bi ∈ Bi. So we have that<br />
(h1 × · · · × hn) −1 (C ′ i × · · · × C ′ n) ∩ (B1 × · · · × Bn)<br />
= (h −1<br />
1 (C ′ 1) ∩ B1) × · · · × (h −1<br />
n (C ′ n) ∩ Bn) ∈ B1 ∗ · · · ∗ Bn = B.<br />
Therefore, C ′ 1 × · · · × C ′ n ∈ B ′′ . Since B ′′ is closed under finite unions and<br />
intersections, 1 follows.<br />
Now, suppose that for all i and for all h ∈ Hi, we have that h is surjective.<br />
Let Z ∈ B ′′ . According to Lemma 6.4.2, Z ∈ P(X ′ ), and thus, by Lemma 6.4.1,<br />
we have<br />
Z = �<br />
Zk,1 × · · · × Zk,n<br />
k∈K<br />
where K is a finite set, and each Zk,1 ×· · ·×Zk,n is a rectangle (in the terminology<br />
of 6.4.1). Now, let (h, B) ∈ H with h = (h1 × · · · × hn). We have,<br />
V := h −1 (Z) = �<br />
k∈K<br />
A quick calculation shows that if a ∈ Xi then<br />
π c i<br />
h −1<br />
1 (Zk,1) × · · · × h −1<br />
n (Zk,n).<br />
� −1<br />
πi (a) ∩ h−1 (Z) � = h −1 � π c � −1<br />
i πi (hi(a)) ∩ Z �� . (6.4.3)<br />
77
Using equation (6.4.3) and the assumption that all hi are surjective, then if a ≡V,i<br />
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 −1<br />
i (a) and<br />
b ′ ∈ h −1<br />
i (b). Therefore, each h−1 1 (Zk,1) × · · · × h−1 n (Zk,n) is a rectangle for h−1 (Z)<br />
(recalling the definition of a rectangle from 6.4.1).<br />
Now, fix h = (h1 × · · · × hn). Consider the sets Qi := {D (j)<br />
i | (hi, D (j)<br />
i ) ∈ Hi}.<br />
By assumption<br />
if D (ji)<br />
i<br />
h −1 (Z) ∩ (D (j1)<br />
1<br />
× · · · × D (jn)<br />
n ) ∈ B<br />
∈ Qi for all i. Then, we let ˜ Bi = ∪j∈JD (j)<br />
i<br />
Since these unions are finite, we have<br />
Z ′ := h −1 (Z) ∩ ( ˜ B1 × · · · × ˜ Bn) ∈ B.<br />
with J finite and all D(j) i ∈ Qi.<br />
Choose each ˜ Bi large enough so that ˜ Bi ∩ Vj,i �= ∅ for all j ∈ {1, ..., ni} (all<br />
equivalence classes of ≡V,i); we note that this is possible by our assumption (6.4.2).<br />
Then, we have<br />
Z ′ = �<br />
k∈K<br />
(h −1<br />
1 (Zk,1) ∩ ˜ B1) × · · · × (h −1<br />
n (Zk,n) × ˜ Bn) ∈ B.<br />
By Lemma 6.4.1 (d), we know that (h −1<br />
1 (Zk,1) ∩ ˜ B1) × · · · × (h −1<br />
n (Zk,n) × ˜ Bn) is<br />
a rectangle. Hence, by Lemma 6.4.1 (c), we have that each h −1<br />
i (Zk,i) ∩ ˜ Bi ∈ Bi.<br />
Finally, since Bi ∈ Bi for all i, we know that<br />
h −1<br />
i (Zk,i) ∩ Bi = h −1<br />
i (Zk,i) ∩ ˜ Bi ∩ Bi ∈ Bi.<br />
Therefore, for all i ∈ {1, ..., n} and k ∈ K, we have Zi,k ∈ B ′ i and thus Z ∈ B ′ as<br />
required.<br />
78
6.5 Regularity in Mixed Products<br />
We now extend our notions of p-complete regularity on T and W to the notion<br />
of a p-complete regularity of products W n × T m with n, m ∈ N.<br />
As done previously when dealing with Cartesian products, we say that a subset<br />
of a product B1 × · · · × Bn with each Bi equal to T or W is p-completely regular<br />
if it is in the Boolean subalgebra of P(B1 × · · · × Bn) generated by products<br />
A1 × · · · × An where each Ai is p-completely regular in W if Bi = W and each Aj<br />
is p-completely regular in T for Bj = T . To avoid trivial situations, if n = 0, we<br />
let B1 × · · · × Bn be the singleton set and declare all subsets to be p-completely<br />
regular. For ease of notation in the next following lemma, we assume that B =<br />
B1 × · · · × Bn = T k × W n−k with Bi = T for i ∈ {1, ..., k} and Bi = W otherwise.<br />
Lemma 6.5.1. Let B = B1 ×· · ·×Bn be as above, with each Bi = T for 1 ≤ i ≤ k<br />
and Bi = W for k < i ≤ n. For any family ¯t = (t1, ..., tk) in T k , we define<br />
f¯t : W n → B1 × · · · × Bn by f¯t(w1, ..., wn) = (u1, ..., un) where<br />
⎧<br />
⎪⎨ w<br />
ui :=<br />
⎪⎩<br />
−1<br />
i tiwi; if 1 ≤ i ≤ k<br />
wi; otherwise.<br />
Then a subset X ⊆ B1 × · · · × Bn is p-completely regular if and only if for all ¯s =<br />
(s1, ..., sk) ∈ Sk and (m1, ..., mk) ∈ Mk , f −1<br />
¯s (X)∩ � Ts1,m1 × · · · × Tsk,mk × W n−k�<br />
is p-completely regular in W n .<br />
Proof. This lemma follows from the lemmas in Section 6.4 because the map and<br />
sets satisfy the following. Using terminology from that section, we have H ′ i :=<br />
∪ ¯s∈S k(f¯s, T¯s,0), which is finite. Since each conjugacy class is p-completely regular,<br />
and T is a finite union of conjugacy classes, we reduce to the case where we replace<br />
79
Bi = T by a conjugacy class in T so that fsi is surjective. Also, for any ¯s ∈ Sk , we<br />
have ∪m∈MTsi,m equal to the conjugacy class of si, so we satisfy (6.4.2). Finally,<br />
by definition A ⊆ T is p-completely regular in T if and only if f −1<br />
s (A)∩Ts,m = As,m<br />
is p-completely regular in W for all s ∈ S and m ∈ M.<br />
Remark 6.5.2. We see that the above proof would work for mixed products B =<br />
B1 × · · · × Bn. We simply used the appropriate ordering for ease of notation in<br />
the statement.<br />
Remark 6.5.3. Let B ∼ = W n−k ×T k and B ′ ∼ n<br />
= W ′ −k ′<br />
×T k′ be products as above, so<br />
that B×B ′ ∼ n+n<br />
= W ′ −k−k ′<br />
×T k+k′ is another such product. Let R be a p-completely<br />
regular subset of B × B ′ . Then for b ′ ∈ B ′ the set Bb ′ ,R := {b ∈ B| (b, b ′ ) ∈ R}<br />
is p-completely regular in B. Also {b ∈ B| (b, b ′ ) ∈ R for some b ′ ∈ B ′ } and<br />
{b ∈ B| (b, b ′ ) ∈ R for all b ′ ∈ B ′ } are both p-completely regular.<br />
6.6 General Regularity in Mixed Products<br />
Now, we generalize Lemma 5.10.1 to the case of mixed products of copies of<br />
W and T .<br />
Consider B = B1 ×· · ·×Bn ∼ = W n−k ×T k as above. Then, we let F be the free<br />
group on n generators X1, ..., Xn. For any element b = (b1, ..., bn) ∈ B, we define<br />
a group homomorphism αb : F → W determined by αb(Xi) = bi. Now, fix an<br />
element g ∈ F , and suppose g = X n1<br />
i1<br />
for all i. Then αb(g) = b n1<br />
i1<br />
· · · Xnm<br />
im is a reduced word for g with ni ∈ Z<br />
· · · bnm.<br />
Then, we can define a length function on F ,<br />
im<br />
dependent on b, by lb p : F → M by lb p(g) = �m |nj|lp(bij j=1 ) where g is as above and<br />
|nj| represents the absolute value. Now, we can see that l b p(g) − lp(αb(g)) ∈ 2M.<br />
So for any g ∈ F and b ∈ B, we call the quantity l b p(g) − lp(αb(g)) the g-deficiency<br />
of b.<br />
80
Theorem 6.6.1. For any m ∈ M and g ∈ F , the set<br />
is a p-completely regular subset of B.<br />
Bg,m := {b ∈ B| lp(αb(g)) = l b p(g) − 2m}<br />
Proof. Fix m ∈ M and g ∈ F with g = X n1<br />
i1<br />
· · · Xnm<br />
im . Let D := {i1, ..., im}.<br />
Then let C := {j1, ..., jq} ⊂ D be such that Bj = T for all j ∈ C and Bj = W<br />
for all j ∈ D \ C. Then for any ¯r ∈ S k , we consider W ¯r g,m := f −1<br />
¯r (Bg,m). Let<br />
Zi := W if Bi = W and Zi = Tri,ni otherwise, and let (n1, ..., nk) ∈ M k . For<br />
w := (w1, ..., wn) ∈ W ¯r g,m ∩ (Z1 × · · · × Zn), we have<br />
where xik = wik if ik �∈ D and xik<br />
Kvw := �<br />
ik:ik∈D<br />
vw := x n1<br />
i1<br />
· · · xnm<br />
im<br />
= w−1<br />
ik rik wik if ik ∈ D. Thus, we have<br />
�<br />
(2|nik |lp(wik ) + 1) +<br />
Then, since ¯r is fixed, Lemma 5.10.1 proves that<br />
ik:ik�∈D<br />
(|nik |lp(wik ))<br />
W ¯r g,m ∩(Z1 ×· · ·×Zn) = {(w1, ..., wn) ∈ W n | lp(vm) = Kvm −2m}∩(Z1 ×· · ·×Zn)<br />
is p-completely regular. Therefore, by Lemma 6.5.1, since W ¯r g,m ∩ (Z1 × · · · × Zn)<br />
is p-completely regular for all ¯r ∈ S k and (n1, ..., nk) ∈ M k , then we have Bg,m is<br />
p-completely regular.<br />
81
6.7 Multivariate Poincaré Series<br />
Recall the notation from 5.6. In [8], it is shown that the Poincaré series for<br />
the set of reflections, P (T ; W ), is rational. Due to the results of this chapter,<br />
we obtain the stronger result that the multivariate Poincaré series of the set of<br />
reflections as well as any p-completely regular subset of T will be rational.<br />
Furthermore, if we have a mixed product B := B1 × · · · Bm+n ∼ = W m × T n ,<br />
let M1, ..., Mm+n be the corresponding multiplicative monomials for each Bi gen-<br />
erated by X1, ..., Xm+n with corresponding length functions lpi : Bi → Mi. Let<br />
Z[[X1, ..., Xm+n]] be the completion of the polynomial ring Z[X1, ..., Xm+n]. Then<br />
according to the transfer matrix method ([2] or [28]), for any p-completely regular<br />
subset of A ⊆ B, we have that the multivariate Poincaré series<br />
P (A; B) =<br />
�<br />
(x1,...,xm+n)∈A<br />
lp1(x1) · · · lpm+n(xm+n)<br />
is rational. In fact, P (A; B) is a finite sum of products, each of the form f1 · · · fm+n<br />
with each fi a rational expression in the variables Xi, i.e. fi ∈ Q(Xi).<br />
Example 6.7.1. Let lp : W → M, lp1 : W → M1, lp2 : W → M2, and<br />
lp3 : T → M3 be generalized length functions with M considered additively and<br />
M1, M2, and M3 considered multiplicatively.<br />
1. According to Theorem 6.6.1, for n ∈ M,<br />
Add(n) := {(x, y) ∈ W × W | lp(xy) = lp(x) + lp(y) − 2n}<br />
82
is p-completely regular in W × W . Therefore,<br />
P (Add(n); W × W ) =<br />
�<br />
(x,y)∈Add(n)<br />
lp1(x)lp2(y) =<br />
�<br />
x∈M1;x ′ ∈M2<br />
ax,x ′xx′<br />
has a rational expression, where ax,x ′ ∈ N is the number of pairs (x, y) ∈<br />
W × W with lp(xy) = lp(x) + lp(y) − 2n and lp1(x) = x and lp2(y) = x ′ .<br />
2. Also by Theorem 6.6.1, for n ∈ M,<br />
Un := {(x, t) ∈ W × T | lp(xtx −1 ) = 2lp(x) + lp(t) − 2n}<br />
is p-completely regular in W × T . It follows that<br />
P (Un; W × T ) = �<br />
(x,t)∈Un<br />
lp1(x)lp3(t)<br />
has a rational expression. Moreover, according to Remark 6.5.3, for any<br />
x ∈ W , the set<br />
Un(x) := {t ∈ T | lp(xtx −1 ) = 2lp(x) + lp(t) − 2n}<br />
is p-completely regular in T so<br />
has a rational expression.<br />
P (Un(x); T ) = �<br />
83<br />
t∈Un(x)<br />
lp3(t)
CHAPTER 7<br />
HECKE ALGEBRA MODULES<br />
The recurrence formulae given in section 3.7 look very similar to recurrence<br />
formulae for the generic Iwahori-Hecke Algebra. In this chapter, we will further<br />
discuss this algebra, and we introduce a large family of modules for the Iwahori-<br />
Hecke algebra. These modules arise from the sets T≤m. In, [18], Dyer uses modules<br />
like these to prove a weak form of Lusztig’s conjecture about the boundedness<br />
of the a-function. There is some hope that deeper regularity properties of the<br />
modules described in this section may lead to more results about the a-function.<br />
Throughout this chapter, we fix an arbitrary Coxeter system, (W, S), with<br />
reflections T , reflection cocycle N : W → P(T ), and with generalized length<br />
function lp,W where p is a surjective function from T to the set of indeterminates<br />
X as in section 3.2. In addition to X, let V be another set of indeterminates<br />
equipped with a surjective function q : T → V satisfying q(t) = q(t ′ ) if t is<br />
conjugate to t ′ . For ease of notation, we denote p(r) = pr and q(r) = qr for any<br />
r ∈ S.<br />
7.1 The Generic Iwahori-Hecke Algebra<br />
Let R denote the integral polynomial ring R = Z[X, V ] in indeterminates<br />
X ∪ V . We consider the generic Iwahori-Hecke algebra H of (W, S) over R with<br />
84
generators (as a unital R-algebra) tr for r ∈ S subject to the braid relations of<br />
(W, S)<br />
trts · · ·<br />
� �� �<br />
= tstr · · ·<br />
� �� �<br />
m(r,s) factors m(r,s) factors<br />
and the quadratic relations t 2 r = pr +qrtr. As an R-module, H is free with R-basis<br />
{tw}w∈W . The multiplication is determined by the following formulae: t1 = IdH<br />
and<br />
⎧<br />
⎪⎨ trw<br />
if r �∈ N(w)<br />
trtw =<br />
.<br />
⎪⎩ prtrw + qrtw if r ∈ N(w)<br />
By these formulae, it is clear that the set {tw| w ∈ W } is a spanning set for<br />
H as a Z[X, V ]-module. In fact, this set is also a Z[X, V ]-basis for H (see [26]).<br />
7.2 A Module for H<br />
We regard R as a free Z[V ] module with basis corresponding to M := MX,<br />
i.e. monomials in Z[X]. We will denote monomials in MX by boldface letters, e.g.<br />
m. For any m ∈ M, we denote the dual basis element of R † := HomZ[V ](R, Z[V ])<br />
by m ∗ so that m ∗ (n) = 0 if n �= m and m ∗ (m) = 1.<br />
R † is an R-module with the standard R-structure given by (rf)(r ′ ) = f(rr ′ )<br />
for f ∈ R † and r, r ′ ∈ R. Then, we note that<br />
prm ∗ ⎧ � �<br />
⎪⎨<br />
∗<br />
m if pr ≤ m<br />
pr<br />
=<br />
⎪⎩ 0 otherwise<br />
Let R ′ be the R-submodule of R † spanned by the dual basis elements of MX.<br />
Then R ′ is free as a Z[V ]-module with basis MX.<br />
Next, we can form the H -module H ′ := H ⊗R R ′ . For ease, we denote the<br />
85
element tw ⊗ m ∗ of H ′ by tw,m for m ∈ M and w ∈ W . These elements form a<br />
Z[V ]-basis of H ′ . We thus have<br />
for all r ∈ S and<br />
for all r ∈ S.<br />
⎧<br />
⎪⎨<br />
trtw,m =<br />
⎪⎩<br />
⎧<br />
⎪⎨ tw,<br />
prtw,m =<br />
⎪⎩<br />
m if pr ≤ m<br />
pr<br />
0 otherwise<br />
trw,m<br />
if r �∈ N(w)<br />
trw, m<br />
pr + qrtw,m if r ∈ N(w) and pr ≤ m<br />
qrtw,m<br />
if r ∈ N(w) and pr �≤ m<br />
For each n ∈ M, the Z[V ]-span of the elements tw,m with w ∈ W and m ≤ n<br />
is a H -submodule, called H ′<br />
n of H ′ . Provided S is finite, then H ′<br />
n has finite<br />
rank as a Z[V ]-module. For n ≥ m > 0, multiplication by m induces a short<br />
exact sequence of H -modules:<br />
0 −→ �<br />
k
tN(w)∩T≤m,m is an H -module epimorphism.<br />
2. For m ∈ M, H ′′<br />
m = ρ(H ′ m) is a submodule of H ′′ . So H ′′<br />
m is a submodule<br />
of H ′′<br />
n if m ≤ n.<br />
Proof. We define Z[V ]-linear endomorphisms p ′ r and θr on H ′′ for r ∈ S by<br />
and<br />
where<br />
⎪⎨<br />
θr(tA,m) =<br />
p ′ r(tA,m) =<br />
⎧<br />
⎪⎩<br />
tA ′ ,m<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
tA∩T ≤ m pr , m<br />
pr<br />
if pr ≤ m<br />
0 otherwise<br />
if r �∈ N(w)<br />
tA ′′<br />
r , m<br />
pr + qrtA,m if r ∈ N(w) and pr ≤ m<br />
qrtA,m<br />
if r ∈ N(w) and pr �≤ m<br />
A ′ := (rAr ∪ {r}) ∩ T≤m and A ′′<br />
r := (rAr \ {r}) ∩ T≤ m<br />
pr<br />
Now suppose that tw,m ∈ H ′ . Then on one hand we have<br />
⎧<br />
⎪⎨ ρ(tw,<br />
ρ(pr(tw,m)) =<br />
⎪⎩<br />
m<br />
pr ) if pr<br />
⎫<br />
⎪⎬ ≤ m<br />
ρ(0) otherwise<br />
⎪⎭ =<br />
⎧<br />
⎪⎨ tN(w)∩T≤ m ,<br />
pr<br />
⎪⎩<br />
m<br />
⎫<br />
⎪⎬<br />
if pr ≤ m<br />
pr<br />
⎪⎭<br />
0 otherwise<br />
and on the other hand we have that<br />
p ′ r(ρ(tw,m)) = p ′ ⎧<br />
⎪⎨ tN(w)∩T≤ m ,<br />
pr<br />
r(tN(w)∩T≤m,m) =<br />
⎪⎩<br />
m if pr ≤ m<br />
pr<br />
0 otherwise<br />
Thus, we see that p ′ r(ρ(tw,n)) = ρ(pr(tw,n)) and extending by linearity we get<br />
p ′ r(ρ(h)) = ρ(pr(h)) for all h ∈ H ′ .<br />
Next, let tw,m ∈ H ′ and tr ∈ H . Again we compute the following:<br />
87<br />
⎫<br />
⎪⎬<br />
⎪⎭ .
⎧<br />
⎪⎨<br />
ρ(trw,m) if r �∈ N(w)<br />
ρ(tr(tw,m)) =<br />
⎪⎩<br />
⎧<br />
⎪⎨<br />
=<br />
⎪⎩<br />
ρ(trw, m<br />
pr + qrtw,m) if r ∈ N(w) and pr ≤ m<br />
ρ(qrtw,m) if r ∈ N(w) and pr �≤ m<br />
tN(rw)∩T≤m,m<br />
tN(rw)∩T ≤ m pr , m<br />
pr<br />
qrtN(w)∩T≤m,m<br />
by linearity of ρ on Z[V ]. Also,<br />
⎪⎨<br />
θr(ρ(tw,m)) = θr(tN(w)∩T≤m,m) =<br />
if r �∈ N(w)<br />
+ qrtN(w)∩T≤m,m if r ∈ N(w) and pr ≤ m<br />
⎧<br />
⎪⎩<br />
tB ′ ,m<br />
if r ∈ N(w) and pr �≤ m<br />
if r �∈ N(w)<br />
tB ′′ m<br />
r , pr + qrtB,m if r ∈ N(w) and pr ≤ m<br />
qrtB,m<br />
if r ∈ N(w) and pr �≤ m<br />
where B = N(w) ∩ T≤m. Then B ′ = (rBr ∪ {r}) ∩ T≤m = N(rw) ∩ T≤m since<br />
r �∈ N(w) then N(rw) = {r} ∪ rN(w)r. Finally we have B ′′<br />
r = rBr \ {r} ∩ T≤ m<br />
pr =<br />
N(rw) ∩ T≤ m<br />
pr<br />
since r ∈ N(w) then N(rw) = rN(w)r \ {r}. These all follow from<br />
Proposition 3.8.1. Therefore we have shown that θr(ρ(tw,n)) = ρ(tr(tw,n)) for all<br />
tw,n and so extending by linearity we get θr(ρ(h)) = ρ(tr(h)) for all h ∈ H ′ .<br />
Now, ρ is clearly an epimorphism,, and so we thus have an H -module structure<br />
on H ′′ in which pr acts by p ′ r for all r ∈ S, qr acts via the natural Z[V ]-module<br />
structure on H ′′ for all r ∈ S, and tr acts by θr for all r ∈ S. Hence, 1 is true.<br />
Then 2 follows from the definition of ρ.<br />
88
7.4 The 0-Hecke Algebra<br />
Now, we regard Z[V ] as the R-module R/XR. Then for any m ∈ M we<br />
define � H ′′<br />
m := H ′′<br />
m/( � ′′<br />
n
CHAPTER 8<br />
<strong>IN</strong>ITIAL SECTIONS OF REFLECTION ORDERS<br />
In this chapter, we recall the definition due to Dyer (c.f. [11], [12], or [13])<br />
of reflection orders, which are certain total orders of the set of reflections T ,<br />
and of their initial sections, which are subsets of T that are known to lead to<br />
partial orders (twisted Bruhat orders) on W that are similar to Bruhat order.<br />
In particular, using the initial section ∅ ⊆ T we get the Bruhat order on W , and<br />
using T ⊆ T , we get the reverse Bruhat order on W . In this chapter, we determine<br />
all subsets of T that give rise to partial orders on W in the same manner. The<br />
subsets of T that have this property are those which are “locally” (at each dihedral<br />
reflection subgroup) initial sections; it has been conjectured by Dyer ([12]) that<br />
these sets coincide with the initial sections. Reflection orders are closely related to<br />
a notion of closure in the positive root system. In Chapter 9, we will examine the<br />
relationship between closure in root systems and dominance order on the positive<br />
roots. For general references about the properties of reflection orders and initial<br />
sections, see [2], [11],[12], and [13].<br />
8.1 Classification of Twisted Bruhat Orders<br />
Let (W, S) be a Coxeter system. Recall that W acts on P(T ) by w · A =<br />
N(w) + wAw −1 for any A ⊆ T and w ∈ W , where + represents symmetric<br />
90
difference. Now, for any A ⊆ T , we follow [11] by defining a directed graph Ω(W,A)<br />
with the vertex set of Ω(W,A) equal to W and edge set E(W,A) = {(tw, w)| t ∈ w·A}.<br />
We also have the following equivalent definition of E(W,A).<br />
E(W,A) = {(tw, w)| t ∈ N(w) + wAw −1 } = {(tw, w)| w −1 tw ∈ N(w −1 ) + A}<br />
= {(wt ′ , w)| t ′ ∈ N(w −1 ) + A},<br />
and we will find it convenient to use this description, E(W,A) = {(wt, w)| t ∈<br />
N(w −1 ) + A}. For x ∈ W , we have that t ∈ w · A if and only if t ∈ (wx −1 ) · x · A,<br />
and thus the map w ↦→ wx −1 defines an isomorphism Ω(W,A) ∼ = Ω(W,x·A). In<br />
addition, for A ⊆ T we can define a length function lA : W → Z in the following<br />
way:<br />
lA(v, w) = l(wv −1 �<br />
�<br />
) − 2�[N(vw<br />
−1 �<br />
�<br />
) ∩ v · A] � ∈ Z<br />
and then set lA(w) = lA(1, w). We can define a pre-order ≤A for any A ⊆ T given<br />
by the following: v ≤A w if and only if there exist t1, ..., tn ∈ T with w = vt1...tn<br />
such that ti �∈ [N((vt1...ti−1) −1 ) + A] for all i = 1, ..., n.<br />
Remark 8.1.1. The multivariate analog of such single variable length functions,<br />
lA, similar to that of Chapter 3, has not been studied but should be of interest.<br />
Definition 8.1.2. Following [12], a total order on the set T ′ in a dihedral Coxeter<br />
system, (W ′ , S ′ ) with S = {r, s}, is called a dihedral reflection order if it is either<br />
≺R or ≺ ′ R where ≺R and ≺ ′ R are defined by r ≺R rsr ≺R · · · ≺R srs ≺R s and<br />
s ≺ ′ R srs ≺′ R · · · ≺′ R rsr ≺′ R<br />
r. Then, for any Coxeter system (W, S) we call<br />
a total order ≺R on T a reflection order if the restriction ≺R � � W ′ ∩T is a dihedral<br />
reflection order for all dihedral reflection subgroups W ′ of (W, S) with respect to<br />
the canonical generators χ(W ′ ).<br />
91
Remark 8.1.3. We can also define a reflection order in terms of the root system Φ<br />
associated to the Coxeter system. This definition can be found in [2], and we will<br />
discuss this definition in more detail in Chapter 9.<br />
Recall from [12] that an initial section of a reflection order is a subset A ⊆ T<br />
such that there is a reflection order ≺R with the property that a ≺R b for all<br />
a ∈ A and b ∈ T \ A. It is shown in [11] that ≤A is a partial order of W if A is<br />
an initial section of a reflection order. Our main result in this chapter, Theorem<br />
8.1.4 below, describes all subsets A of T for which ≤A is a partial order.<br />
Let A(W,S) be the set of initial sections of reflection orders of T . Now, we<br />
define Â(W,S) = {A ⊆ T | A ∩ W ′ ∈ A(W ′ ,χ(W ′ )) ∀W ′ ⊆ W dihedral}. It has been<br />
conjectured by Dyer that A = Â. We now come to the main result:<br />
Theorem 8.1.4. Let (W, S) be any Coxeter system. The following are equivalent:<br />
1. Ω(W,A) is acyclic.<br />
2. ≤A is a partial order.<br />
3. Ω(W,A) has no cycle of length four.<br />
4. A ∈ Â.<br />
5. lA(xt) < lA(x) for all x ∈ W and t ∈ N(x −1 ) + A.<br />
8.2 Proof of the Classification<br />
In the following proofs, for any positive root α ∈ Φ + , let tα ∈ T be the<br />
corresponding reflection, and for any reflection t ∈ T , let αt ∈ Φ + be the cor-<br />
responding positive root. To begin with we investigate the dihedral case. Sup-<br />
pose (W, S) is dihedral, i.e. S = {r, s}. There is a bijection between subsets<br />
92
A ⊆ T and subsets Ψ ⊂ Φ such that Ψ ∪ −Ψ = Φ and Ψ ∩ −Ψ = ∅ given by<br />
A = AΨ = {tα| α ∈ Ψ ∩ Φ + }. We note that A−Ψ = AΨ + T .<br />
Lemma 8.2.1. For any AΨ ⊆ T , w · AΨ = Aw(Ψ).<br />
Proof. It is enough to show that this is true for r ∈ S. Now we know that<br />
r · AΨ = {r} + rAΨr = {r} + {rtαr| α ∈ Ψ ∩ Φ + } = {r} + {tr(α)| α ∈ Ψ ∩ Φ + } =<br />
{r} + {tα| α ∈ r(Ψ) ∩ r(Φ + )}. If αr ∈ Ψ then −αr ∈ r(Ψ) ∩ r(Φ + ) and so r ∈<br />
{tα| α ∈ r(Ψ) ∩ r(Φ + )}. This implies that r · AΨ = {tα| α ∈ r(Ψ) ∩ Φ + }. If αr �∈ Ψ<br />
then r �∈ {tα| α ∈ r(Ψ)∩r(Φ + )}. But αr ∈ r(Ψ) so {r}+{tα| α ∈ r(Ψ)∩r(Φ + )} =<br />
{tα| α ∈ r(Ψ) ∩ Φ + }. Thus, in both cases r · AΨ = {tα| α ∈ r(Ψ) ∩ Φ + }.<br />
Since we are considering (W, S) dihedral, we choose an orientation of the plane<br />
spanned by Φ. Then for any root α ∈ Φ we can define the neighbor of α, denoted<br />
nbr(α), to be the root lying directly next to α if we traverse the root system<br />
clockwise. Recall that S = {r, s}. Interchanging r and s if necessary, we assume<br />
without loss of generality that αr = nbr(−αs). We note that this implies that<br />
nbr(αs) �∈ Φ + .<br />
Lemma 8.2.2. Let (W, S) be dihedral. Suppose α ∈ Φ + and γ := nbr(α) ∈ Φ + .<br />
Then tαtγ = sr.<br />
Proof. Let (rsr...)n denote an element with n simple reflections listed (we note<br />
that l((rsr...)n) is not necessarily n). With this terminology, t ∈ T can be written<br />
t = (rsr...)2i+1 or t = (srs...)2i+1 for some i ≥ 0. Under the chosen orientation<br />
for Φ + , if tα = (rsr...)2i+1 then we can write tγ = (rsr...)2i+3 so that tαtγ =<br />
(rsr...)2i+1(rsr...)2i+3 = sr. Otherwise, if tα = (srs...)2i+1 (i ≥ 1) then tγ =<br />
(srs...)2i−1 and so tαtγ = sr.<br />
93
Now, given A = AΨ, we introduce two conditions that a positive system, Γ + ,<br />
of Φ can have:<br />
C1: There are roots α, nbr(α) ∈ Γ + such that α �∈ Ψ and nbr(α) ∈ Ψ.<br />
C2: There are roots β, nbr(β) ∈ Γ + such that β ∈ Ψ with nbr(β) �∈ Ψ.<br />
Lemma 8.2.3. Let (W, S) be dihedral and let A = AΨ ⊆ T .<br />
1. If the positive system r(Φ + ) has condition C1, then there exists a path 1 →<br />
x → sr in Ω(W,A).<br />
2. If the positive system r(Φ + ) has condition C2, then there exists a path sr →<br />
x → 1 in Ω(W,A).<br />
Proof. Replacing A by A + T reverses the orientation of edges in Ω(W,A) and so 2<br />
follows from 1.<br />
We now prove 1. There are two cases to consider. If α �= αs, then both α<br />
and γ := nbr(α) ∈ Φ + . So tα �∈ A and tγ ∈ A. Also, since tα �∈ {r, s}, it<br />
follows that tγ ∈ N(tα). So we have that tγ �∈ N(tα) + A. Thus, we have a path<br />
1 → tα → tαtγ = sr, where the last equality follows from Lemma 8.2.2. Now, if<br />
α = αs then nbr(α) = −αr. Since −αr ∈ Ψ we know that αr �∈ Ψ which implies<br />
r �∈ A. Also, s �∈ A and clearly r �∈ N(s). Together, we see that there is a path<br />
1 → s → sr in Ω(W,A).<br />
Lemma 8.2.4. For (W, S) dihedral, let A = AΨ ⊆ T . If there are no 4-cycles in<br />
Ω(W,A) then there is no positive system, Γ + ⊂ Φ satisfying C1 and C2.<br />
Proof. Suppose there is a positive system Γ + satisfying C1 and C2. It is clear that<br />
−Γ + also satisfies C1 and C2. Since Γ + satisfies both C1 and C2, then w(Γ + )<br />
94
will also satisfy both conditions (w with even length respects both conditions and<br />
w with odd length interchanges the conditions). Thus, we can find a w ∈ W such<br />
that r(Φ + ) = w(Γ + ) or r(Φ + ) = w(−Γ + ). So we have that r(Φ + ) satisfies C1 and<br />
C2 with respect to Aw(Ψ) = w · AΨ. Since Ω(W,A) is isomorphic to Ω(W,w·A), we can<br />
assume without loss of generality that r(Φ + ) satisfies C1 and C2 with respect to<br />
AΨ. Thus, Lemma 8.2.3 implies that we have a path sr → u → 1 → v → sr in<br />
Ω(W,A).<br />
Lemma 8.2.5. Let (W, S) be dihedral and A = AΨ ⊆ T . If there is no positive<br />
system, Γ + , of Φ satisfying C1 and C2, then A ∈ A(W,S).<br />
Proof. Let A �∈ A(W,S). Recall that the only two reflection orders on (W, S) are<br />
≺R and ≺ ′ R described in Definition 8.1.2. Since A is not an initial section of either<br />
of these orders, without loss of generality (replacing A with T \ A if necessary)<br />
we can find t0 ∈ A and t1, t2 ∈ T \ A such that t1 ≺R t0 and t2 ≺ ′ R t0, i.e.<br />
t2 ≺R t0 ≺R t1.<br />
Now, if (W, S) is finite, we can list all of the reflections and thus can find<br />
t ′ , t ′′ ∈ T such that t2 �R t ′ ≺R t0 �R t ′′ ≺R t1 with αt ′ �∈ Ψ and nbr(αt ′) ∈ Ψ,<br />
and αt ′′ ∈ Ψ and nbr(αt ′′) �∈ Ψ. Thus Φ+ satisfies C1 and C2.<br />
If (W, S) is infinite, then T = Tr ∪ Ts (disjoint union) where Tr = {t ∈ T | r ∈<br />
N(t)} and Ts = {t ∈ T | s ∈ N(t)}. Suppose Φ + does not satisfy C1 and C2.<br />
We cannot have t1, t2, t0 ∈ Tu for some u ∈ {r, s} since this would imply, by<br />
the reasoning for (W, S) finite, that Φ + satisfies C1 and C2. So, we can assume<br />
that t2, t0 ∈ Tr and t1 ∈ Ts (the case t2 ∈ Tr and t0, t1 ∈ Ts is exactly similar).<br />
Additionally, since C1 and C2 aren’t both satisfied by Φ + , we may assume the<br />
following conditions hold (see Figure 8.1):<br />
1. αt0 = nbr(αt2),<br />
95
2. t ∈ T \ A if t �R t2,<br />
3. t ∈ T \ A if t ∈ Ts and t �R t1,<br />
4. t ∈ A if t ∈ Tr and t0 �R t,<br />
5. t ∈ A if t1 ≺R t.<br />
Now, consider the positive system Γ + with simple roots αt0 and −nbr(αt0). Then<br />
Γ + satisfies C1 using the roots αt2 and αt0 = nbr(αt2). Also, by above, αt1 �∈ Ψ<br />
and nbr(αt1) ∈ Ψ (note that even if t1 = s this is true since nbr(αs) = −αr ∈ Ψ<br />
because by above αr �∈ Ψ). Using this, we see that Γ + satisfies C2 using the roots<br />
−αt1 and nbr(−αt1) = −nbr(αt1) (again see Figure 8.1).<br />
With the dihedral case taken care of, we proceed to the general case. For the<br />
remainder of the chapter, we assume that (W, S) is a general Coxeter system.<br />
Proposition 8.2.6. For all w ∈ W and A ∈ Â(W,S), w · A ∈ Â(W,S).<br />
Proof. It suffices to check the condition for w = r ∈ S. Let W ′ be a dihedral<br />
reflection subgroup. If r ∈ W ′ , then (r · A) ∩ W ′ = N(W ′ ,χ(W ′ ))(r) + r(A ∩ W ′ )r ∈<br />
A(W ′ ,χ(W ′ )) by [12]. Now, if r �∈ W ′ , then (r · A) ∩ W ′ = r(A ∩ rW ′ r)r. However,<br />
conjugation by r defines an isomorphism (W ′ , χ(W ′ )) ∼ = (rW ′ r, rχ(W ′ )r) in this<br />
case, and by assumption A∩rW ′ r ∈ A(rW ′ r,rχ(W ′ )r), thus (r·A)∩W ′ ∈ A(W ′ ,χ(W ′ )).<br />
Before, we prove Theorem 8.1.4, we provide one more important proposition,<br />
which is proven in [11], but we include a proof here as well.<br />
Proposition 8.2.7. Let A ∈ Â(W,S), x ∈ W , t ∈ T . Then lA(x, xt) > 0 iff<br />
t �∈ N(x −1 ) + A.<br />
96
αr<br />
. . .<br />
. . .<br />
αt0<br />
. . .<br />
αt2 αt1<br />
. . . . . .<br />
Figure 8.1. This is a schematic diagram of the root system for (W, S) infinite.<br />
Φ + consists of the roots which are non-negative linear combinations<br />
of αr and αs. The dotted ray through the origin represents a limit line<br />
of roots. If C1 and C2 are not both satisfied by Φ + then Ψ is pictured<br />
above with the roots in Ψ labeled by •, and those not in Ψ labeled by ◦.<br />
97<br />
. . .<br />
. . .<br />
. . .<br />
αs
Proof. We begin by proving the statement for all A ∈ Â(W,S) and t ∈ T with x = 1,<br />
i.e. lA(1, t) = lA(t) > 0 if and only if t �∈ A. If (W, S) is dihedral with S = {r, s},<br />
then we see that, without loss of generality, t = (rsr...)2n+1 with l(t) = 2n + 1.<br />
Then, N(t) = {r, rsr, ..., (rsr...)2n+1, ..., (rsr...)4n+1}. Since A ∈ Â(W,S), then if<br />
(rsr...)2i+1 ∈ A either (rsr...)2j+1 ∈ A for all j ≤ i or (rsr...)2j+1 ∈ A for all<br />
i ≤ j ≤ 2n. In particular |N(t) ∩ A| ≥ n + 1 if and only if t = (rsr...)2n+1 ∈ A.<br />
Therefore, in this case lA(t) > 0 if and only if t �∈ A.<br />
Suppose that (W, S) is not dihedral. Then if W ′ is any dihedral reflection<br />
subgroup containing t, since A ∈ Â(W,S) then A ′ := A ∩ W ′ ∈ Â(W ′ ,χ(W ′ )), and so<br />
we have lA ′(t) makes sense in the Coxeter system (W ′ , χ(W ′ )). Now, by equation<br />
2.3.1, we get that<br />
and<br />
which implies that<br />
and<br />
N(t) \ {t} = ˙�<br />
N(t) \ {t} ∩ A = ˙�<br />
l(t) − 1 = |N(t) \ {t}| = �<br />
W ′ ∈Mt<br />
W ′ ∈Mt<br />
W ′ ∈Mt<br />
lA(t) − 1 = �<br />
N(W ′ ,χ(W ′ ))(t) \ {t}<br />
� N(W ′ ,χ(W ′ ))(t) \ {t} � ∩ A,<br />
|N(W ′ ,χ(W ′ ))(t) \ {t}| = �<br />
W ′ ∈Mt<br />
W ′ ∈Mt<br />
l(W ′ ,χ(W ′ ))(t) − 1<br />
(lA∩W ′(t) − 1). (8.2.1)<br />
Now, t �∈ A, if and only if t �∈ A ∩ W ′ for all W ′ ∈ Mt. This occurs if and only<br />
if lA∩W ′(t) > 0 for all W ′ ∈ Mt and then by equation (8.2.1) is true if and only if<br />
we get lA(t) > 0 as required.<br />
98
Now, if x ∈ W ,<br />
lA(x, xt) = l(xtx −1 ) − 2|N(xtx −1 ) ∩ x · A|<br />
= l(xtx −1 ) − 2|N(xtx −1 ) ∩ 1 · (x · A)| = lx·A(1, xtx −1 )<br />
so lA(x, xt) = lx·A(xtx −1 ). Therefore, lA(x, xt) > 0 if and only if lx·A(xtx −1 ) > 0.<br />
Also, t �∈ N(x −1 ) + A if and only if xtx −1 �∈ xN(x −1 )x −1 + xAx −1 = N(x) +<br />
xAx −1 = x · A. So t �∈ N(x −1 ) + A if and only if xtx −1 �∈ x · A.<br />
Putting together the previous statements, we have that the statement of the<br />
proposition is equivalent to the statement lx·A(xtx −1 ) > 0 if and only if t �∈ x · A.<br />
Following Proposition 8.2.6, we let A ′ = x · A ∈ Â(W,S) and t ′ = xtx −1 , then this<br />
case is equivalent to lA ′(t′ ) > 0 if and only if t ′ �∈ A ′ which is true due to the x = 1<br />
case above.<br />
At this point, we can use the previous lemmas and propositions to prove The-<br />
orem 8.1.4:<br />
Proof. (1)⇔(2)⇒(3) is clear.<br />
(3)⇒(4): According to section 2.3, we know that for any dihedral subgroup W ′<br />
of W , N(W ′ ,χ(W ′ ))(w) = N(W,S)(w) ∩ W ′ . This implies Ω(W ′ ,A∩W ′ ) is a subgraph of<br />
Ω(W,A). So, if A �∈ Â(W,S), Lemma 8.2.4 and Lemma 8.2.5 imply that there exists<br />
some dihedral subgroup W ′ of W with Ω(W ′ ,A∩W ′ ) containing a cycle with four<br />
edges.<br />
(4)⇒(5): Suppose that A ∈ Â(W,S), x ∈ W and t ∈ N(x −1 )+A. Then Proposition<br />
8.2.7 implies that lA(x, xt) < 0. But by [11], lA(1, x) + lA(x, xt) = lA(1, xt) and<br />
this implies lA(xt) − lA(x) = lA(x, xt) < 0.<br />
(5)⇒(1): Suppose Ω(W,A) has a cycle. This means that xt1...tn = x for some n > 0<br />
(all ti ∈ T and with ti �∈ N((xt1...ti−1) −1 ) + A for all i). By assumption, lA(x) <<br />
99
lA(xt1) < lA(xt1t2) < ... < lA(xt1t2...tn) = lA(x) which is a contradiction.<br />
Remark 8.2.8. In case (W, S) is finite, say of order m, we can give a much simpler<br />
proof of the equivalence of (1), (2), (4), and (5). By the proof above, we have<br />
(4)⇒(5)⇒(1)⇔(2). If (4) fails, then w · A �= T for all w ∈ W (or else A = w −1 · T<br />
which is an initial section). So we can choose t �∈ w · A. It follows that we can<br />
recursively choose t1, ..., tm such that t1 �∈ A and ti �∈ ti−1 · · · t1 · A for i = 2, ..., m.<br />
This gives us the following path in Ω(W,A): 1 → t1 → . . . → tm · · · t1. However, this<br />
path has m + 1 elements and W has m elements, so there must be two elements<br />
in the path that are the same thus creating a cycle.<br />
100
CHAPTER 9<br />
CLOSURE <strong>IN</strong> THE ROOT SYSTEM<br />
This chapter recalls a notion of closure on the positive roots that has been<br />
studied by Dyer [16]. A conjectural relation, due to Dyer, between these closures<br />
and the sets T≤m (m ∈ N), as defined in section 2.4 and generalized in Chapter<br />
3, would lead to a new and potentially interesting normal form for elements of<br />
Coxeter groups, via known special cases of other conjectures of Dyer ([17]) involv-<br />
ing initial sections and these closures. The results of the previous chapter afford<br />
the possibility of bringing the study of twisted Bruhat orders to bear on these<br />
conjectures.<br />
9.1 2-closure<br />
Let (W, S) be a fixed Coxeter system. Recall that we denote the roots and<br />
positive roots by Φ and Φ + respectively. We begin by describing a notion of<br />
closure on the subsets of positive roots, Φ + . This closure is similar to the notion<br />
of closure in the crystallographic root system of a finite Weyl group as described<br />
in [3].<br />
Definition 9.1.1. Let Γ ⊆ Φ + . We say that Γ is 2-closed if for all α, β ∈ Γ, we<br />
have (R≥0α + R≥0β) ∩ Φ + ⊆ Γ.<br />
101
Remark 9.1.2. Let TΓ ⊆ T be the subset of reflections corresponding to Γ via the<br />
canonical bijection. Then, 2-closure is the same as the following. TΓ is 2-closed<br />
if for all tα, tβ ∈ TΓ with W ′ = 〈tα, tβ〉, {t ∈ W ′ ∩ T | tα ≺R t ≺R tβ} ∪ {t ∈<br />
W ′ ∩ T | tβ ≺R t ≺R tα} ⊂ TΓ for all reflection orders ≺R on T (see Chapter 8).<br />
Definition 9.1.3. We say a subset Γ ⊆ Φ + is biclosed if Γ and Φ + \ Γ are both<br />
2-closed.<br />
In particular, initial sections of reflection orders are biclosed subsets. Moreover,<br />
Â(W,S) as defined in Chapter 8 is the set of all subsets of T corresponding to<br />
biclosed subsets of Φ + under the natural bijection Φ + ↔ T .<br />
9.2 More types of subsets of Φ +<br />
We introduce three more types of subsets of Φ + . These notions are due to<br />
Dyer, and further information about them can be found in [17].<br />
Definition 9.2.1. Let Γ ⊆ Φ + .<br />
1. We say Γ is balanced if for all α ∈ Γ and W ′ ∈ Msα, then β ∈ Φ +<br />
W ′ such<br />
that l(W ′ ,χ(W ′ ))(sβ) < l(W ′ ,χ(W ′ ))(sα) implies that β ∈ Γ.<br />
2. We say Γ is bipedal if for all α ∈ Γ and W ′ ∈ Msα with α �∈ ΠW ′, ΠW ′ ⊂ Γ.<br />
3. We say Γ is unipedal if for all α ∈ Γ and W ′ ∈ Msα, then ΠW ′ = {β, γ}<br />
implies that either β ∈ Γ or γ ∈ Γ.<br />
The following proposition is then clear from the definitions.<br />
Proposition 9.2.2. Suppose Γ ⊆ Φ + .<br />
1. If Γ is balanced then Γ is bipedal.<br />
102
2. If Γ is bipedal, then Γ is unipedal.<br />
3. If Φ + \ Γ is closed, then Γ is unipedal.<br />
4. If Γ ⊆ Φ + is bipedal and ∆ ⊆ Φ + is unipedal, then Γ ∩ ∆ is unipedal.<br />
For any subset Γ ⊆ Φ + , we let ¯ Γ denote the 2-closure of Γ, that is the smallest<br />
2-closed set containing Γ. The following is conjectured by Dyer in [17].<br />
Conjecture 9.2.3. If Γ ⊆ Φ + is unipedal, then ¯ Γ is biclosed.<br />
This would strengthen the following conjecture, also in [17], parts of which<br />
were raised in [13, Remark 2.1.4].<br />
Conjecture 9.2.4. Let A := {Γ ⊆ Φ + | Γ is biclosed}. Then A is a complete<br />
lattice with �<br />
i∈I Γi = �<br />
i∈I Γi for an arbitrary family {Γi}i∈I ⊂ A.<br />
The last conjecture extends the known fact that W under weak order is a meet<br />
semi-lattice (see [2] for example). In fact, [17] proves the following:<br />
Theorem 9.2.5. Let Φw := Φ + ∩ w(−Φ + ). If Γ ⊆ Φw for some w ∈ W and Γ is<br />
unipedal, then ¯ Γ = Φx ⊆ Φw for some x ∈ W .<br />
The previous theorem provides a new formula for the join (when defined) of<br />
elements of W in weak order. For x, y, z ∈ W , x∨y = z if and only if Φx ∪ Φy = Φz.<br />
There is a similar (proven) formula for meet, which we do not give.<br />
9.3 A Conjectural Normal Form for Elements of W<br />
Our conjectural normal form for elements of W depends on the following con-<br />
jecture of Dyer in [17].<br />
103
Conjecture 9.3.1. Let m ∈ N and let Φ +<br />
≤m = {α ∈ Φ+ | h ∞ (sα) ≤ m} be the<br />
subset of positive roots corresponding to T≤m. Then Φ +<br />
≤m is balanced.<br />
In fact, we will only need the weaker conjecture that Φ +<br />
≤m is bipedal (see part<br />
1 of Proposition 9.2.2). Then we have the following theorem.<br />
Theorem 9.3.2. Let m ∈ N such that Φ +<br />
≤m<br />
is bipedal.<br />
1. For any x ∈ W , there is a unique x ′ ∈ W such that Nm(x ′ ) = Nm(x),<br />
and any y ∈ W with Nm(y) ⊇ Nm(x) can be written y = x ′ y ′′ with l(y) =<br />
l(x ′ ) + l(y ′′ ).<br />
2. Any 1 �= w ∈ W can be uniquely written as w = w1 · · · wn for certain<br />
wi ∈ W \ {1} with i = 1, ..., n and n ≥ 1 such that l(w) = l(w1) + · · · + l(wn)<br />
and wi = (wi · · · wn) ′ for each i where x ↦→ x ′ is as in 1.<br />
Proof. Let x ∈ W . Then Φx is biclosed and thus unipedal by part 3 of Proposition<br />
9.2.2. By assumption, Φ +<br />
≤m is bipedal and so Φx ∩ Φ +<br />
≤m<br />
is unipedal by part 4 of<br />
9.2.2. Therefore, according to Theorem 9.2.5, we see that Φx ∩ Φ +<br />
≤m<br />
= Φx ′ for<br />
some x ′ ∈ W . If y ∈ W with Nm(y) ⊇ Nm(x), then Φx ∩ Φ +<br />
≤m ⊆ Φy ∩ Φ +<br />
≤m ⊆ Φy.<br />
Therefore we get that<br />
Φx ′ = Φx ∩ Φ +<br />
≤m ⊆ Φy = Φy,<br />
and so y = x ′ y ′′ with l(y) = l(x ′ ) + l(y ′′ ) due to well known properties of the weak<br />
order on (W, S), which can be found in [2]. This proves 1, and 2 follows from 1<br />
inductively.<br />
The next proposition shows that the previous theorem holds in many basic<br />
Coxeter systems.<br />
Proposition 9.3.3. Let (W, S) be a Coxeter system, and let m ∈ N.<br />
104
1. If (W, S) is finite or affine, then Φ +<br />
≤m<br />
is bipedal.<br />
2. If (W, S) is right-angled, that is m(s, s ′ ) ∈ {1, 2, ∞} for all (s, s ′ ) ∈ S × S,<br />
then Φ +<br />
≤m<br />
is bipedal.<br />
Proof. If (W, S) is finite, then Φ +<br />
≤m = Φ+ for all m ∈ N, and so the result is<br />
trivial. Suppose that (W, S) is affine. Then there is a well-known description of<br />
the root system of (W, S) in terms of the root system, Ψ, of the associated finite<br />
Weyl group, (W ◦ , S ◦ ). We remind the reader of this now (see [24]). We let U<br />
denote the span of Ψ, and let ∆ be the simple roots for Ψ. We define V := U +Rδ<br />
to be a vector space with U as a subspace of codimension one. Then, we have<br />
Φ = {α + nδ| α ∈ Ψ; n ∈ Z},<br />
Φ + := {α + nδ| α ∈ Ψ + ; n ≥ 0} ∪ {−α + nδ| α ∈ Ψ + ; n ≥ 1},<br />
and Π = ∆∪{δ − �α} is the set of simple roots for Φ where �α is the highest root for<br />
Ψ. It is well known that every reflection, sα+nδ (α ∈ Ψ), is contained in a unique<br />
infinite maximal dihedral reflection subgroup, namely the group generated by<br />
{sα, sδ−α}. It follows from inspection in the dihedral case then that for α ∈ Ψ + we<br />
get h ∞ (sα+nδ) = h ∞ (s−α+(n+1)δ) = n. Now, let m ∈ N, and suppose α+eδ ∈ Φ≤m.<br />
Let sα+eδ ∈ W ′ be a dihedral reflection subgroup with ΠW ′ = {β + nδ, γ + kδ}.<br />
Then, we have α + eδ = a(β + nδ) + b(γ + kδ) with a, b ∈ Z>0. Therefore,<br />
e = an + bk, and so an + bk ≤ m with a, b ∈ Z>0. This implies that n ≤ m<br />
and k ≤ m, and thus h ∞ (β + nδ) ≤ n ≤ m and h ∞ (γ + kδ) ≤ k ≤ m. Hence<br />
ΠW ′ ⊂ Φ≤m as required.<br />
To show 2, suppose that (W, S) is right-angled. We know that every dihedral<br />
reflection subgroup either has order 4 or ∞. Then for any t ∈ T and W ′ ∈ Mt<br />
105
with |W ′ | = 4, we must have t ∈ χ(W ′ ) and so h(W ′ ,χ(W ′ ))(t) = 0. Therefore, we<br />
must have<br />
h ∞ (t) = �<br />
W ′ ∈Mt<br />
|W ′ |=∞<br />
h(W ′ ,χ(W ′ ))(t) = �<br />
W ′ ∈Mt<br />
h(W ′ ,χ(W ′ )) = h(t).<br />
Now suppose α ∈ Φ +<br />
≤m and W ′ ∈ Msα. Let {β, γ} = ΠW ′. Then, by previous<br />
statement and the fact that sβ ≤∅ sα in Bruhat order, we get that<br />
h ∞ (sβ) = h(sβ) ≤ h(sα) = h ∞ (sα) ≤ m,<br />
and so β ∈ Φ +<br />
≤m , and similarly for γ as required.<br />
This normal form described by Theorem 9.3.2 could be called the m-automatic<br />
normal form for reasons we now describe. Suppose m ∈ N is fixed. Let p : T → N<br />
be given by p(t) = 1 for all t ∈ T . Then lp = l, and we can construct the m-<br />
canonical automata from Chapter 5, Mm. If we input a reduced expression for w ∈<br />
W in to the automata starting at (m, Nm(1)), we end on the node corresponding<br />
to (m, Nm(w)). According to the conjecture, there is a unique minimal element<br />
x ∈ W with (m, Nm(x)) = (m, Nm(w)). This element is w1. Then, we repeat this<br />
process with w −1<br />
1 w and so on until we are finished. Since this normal form comes<br />
from the m-canonical automata, the name is justified.<br />
Remark 9.3.4. Let p be as in Chapter 3 with associated set M. For any m ∈ M,<br />
we can define Φ +<br />
≤m := {α ∈ Φ+ | hW,p,M∞(sα) ≤ m}. Following Conjecture 9.3.1, it<br />
is reasonable to conjecture that Φ +<br />
≤m is balanced (or weaker, bipedal). We include<br />
the following example to show that this is in fact not the case in general.<br />
Example 9.3.5. Let (W, S) be the affine Weyl group of type ˜ B2. That is S =<br />
106
{r, s, t} with m(r, s) = m(s, t) = 4 and m(r, t) = 2. Due to well known facts,<br />
there are 8 elementary reflections. We list this set below.<br />
T0 = {r, rsr, srs, s, sts, tst, t, rtstr}.<br />
Now, there are three conjugacy classes of reflections. Define p : T → N 3 by<br />
p(u) = (1, 0, 0) if u is conjugate to r, p(u) = (0, 1, 0) if u is conjugate to s, and<br />
p(u) = (0, 0, 1) if u is conjugate to t; let lp : W → N 3 and hW,p,M∞ : T → N 3 be<br />
the corresponding length and infinite height functions.<br />
Next, let Φ +<br />
≤(0,1,0) be as defined in the previous remark. Define<br />
W ′ := 〈rstsr, rsrtstrsr, srstsrs, s〉<br />
so that χ(W ′ ) = {rstsr, s}. Then the reflections of W ′ have the following infinite<br />
heights: hW,p,M∞(s) = (0, 0, 0), hW,p,M∞(rstsr) = (1, 0, 0), hW,p,M∞(rsrtstrsr) =<br />
(0, 1, 0), and hW,p,M∞(srstsrs) = (1, 0, 0). Then clearly, αrsrtstrsr ∈ Φ +<br />
≤(0,1,0) and<br />
rsrtstrsr ∈ W ′ , but αrstsr ∈ ΠW ′ and αrstsr �∈ Φ +<br />
≤(0,1,0) . Therefore Φ+<br />
(0,1,0)<br />
is not<br />
bipedal. We note that (according to Proposition 9.3.3) this example does not<br />
provide a counterexample to Conjecture 9.3.1.<br />
107
CHAPTER 10<br />
HYPERBOLIC <strong>COXETER</strong> <strong>GROUPS</strong><br />
The standard classification of finite reflection groups (or finite Coxeter sys-<br />
tems) and affine Coxeter systems can be found in [23] or [24]. Affine Coxeter<br />
systems are in some natural sense the smallest infinite Coxeter groups (e.g. they<br />
have only polynomial growth, see Chapter 12). In this chapter, we will consider<br />
the “next” class of infinite Coxeter systems known as hyperbolic Coxeter systems.<br />
There is a very well known, case-by-case, classification of hyperbolic Coxeter sys-<br />
tems, which can be found in [23, chapter 6]. In this chapter, we discuss several<br />
characterizations of hyperbolic Coxeter systems by properties of their Coxeter<br />
graphs. According to [6], it is well known that any irreducible, infinite, non-affine<br />
Coxeter system contains a standard parabolic subsystem that is a hyperbolic Cox-<br />
eter system; however, we could not find a proof in the literature, and the proof of<br />
the characterizations we include allows us to provide a case-free proof of this fact.<br />
For the remainder of this chapter, we will assume that S is a finite set.<br />
10.1 Coxeter Graphs<br />
Fix a Coxeter system, (W, S). We let Γ := Γ(W,S) be the associated Coxeter<br />
graph; that is, Γ is the undirected, labeled graph with vertex set S and edges<br />
(si, sj) whenever m(si, sj) ≥ 3 and each edge labeled by the corresponding order<br />
108
m(si, sj). To this graph, we associate the matrix of the bilinear form defined in<br />
section 2.2. That is, A is a symmetric matrix with Ai,j := Asi,sj = (αi, αj). We<br />
say that (W, S) is irreducible if Γ is connected. For J ⊆ S, let ΓJ be the full<br />
subgraph of Γ with vertices corresponding to J. We may abuse notation and say<br />
that s ∈ S − J is connected to J if s is connected to ΓJ.<br />
10.2 Symmetric Matrices and Sylvester’s Law<br />
For any real symmetric matrix over a finite dimensional vector space, we define<br />
the inertia of A to be the triple (a, b, c) where a represents the number of positive<br />
eigenvalues of A, b represents the number of negative eigenvalues of A, and c is the<br />
number of zero eigenvalues of A. We will abuse notation and say that a Coxeter<br />
system has inertia (a, b, c) if the corresponding matrix A has inertia (a, b, c). We<br />
say that an n × n symmetric matrix A is positive definite if it has inertia (n, 0, 0).<br />
We say that a n × n symmetric matrix is positive semidefinite if it has inertia<br />
(a, 0, c), i.e. it has only non-negative eigenvalues. If A is positive definite or<br />
positive semi-definite then we say A is of positive type. We state the following<br />
fact of real symmetric matrices (see for example [22]):<br />
Proposition 10.2.1 (Sylvester’s law of inertia). If A is an n × n real symmetric<br />
matrix with inertia (a, b, c), then A is congruent to the diagonal matrix D =<br />
Ia ⊕ −Ib ⊕ 0c.<br />
10.3 Hyperbolic Coxeter Systems<br />
Let (W, S) be an irreducible Coxeter system with associated graph Γ (so Γ<br />
is connected) and matrix A (representing the bilinear form as in 2.2). Suppose<br />
S is finite and |S| = n. Following [23], we know that W is finite if and only<br />
109
if A is positive definite, that is, if A has inertia (n, 0, 0). Also, we know that<br />
W is affine if and only if the inertia is (n − 1, 0, 1). We define an irreducible<br />
Coxeter system to be hyperbolic if it has inertia (n − 1, 1, 0) and (v, v) < 0<br />
for all v ∈ C := {λ ∈ V | (λ, αs) > 0 ∀s ∈ S}. Also, we have the following<br />
characterization of hyperbolic Coxeter systems:<br />
Proposition 10.3.1. [23, Prop. 6.8] Let (W, S) be an irreducible Coxeter system,<br />
with graph Γ and associated bilinear form (−, −) with matrix A. Then (W, S) is<br />
hyperbolic if and only if the following two conditions are satisfied:<br />
1. (−, −) (or equivalently A) is non-degenerate, but not positive definite.<br />
2. For each s ∈ S, the Coxeter graph Γ ′ , obtained by removing s from Γ, is of<br />
positive type.<br />
10.4 Non-affine Coxeter Systems<br />
At this point, we prove a quick technical lemma that will be needed in what is to<br />
follow. From this point on, we may use (W, S), W and Γ = Γ(W,S) interchangeably<br />
to represent the corresponding Coxeter system. There is no confusion in doing this<br />
as every finite rank (W, S) has a corresponding Coxeter graph and every simple<br />
finite, undirected, edge-labeled graph (labels in N≥3 ∪ {∞}) gives rise to a finite<br />
rank Coxeter system. We say a graph Γ is of finite (resp. affine) type if the<br />
corresponding Coxeter system is finite (resp. affine).<br />
Lemma 10.4.1. Suppose that (W, S) is an infinite, irreducible, non-affine Coxeter<br />
system. Let |S| = n. Suppose that there exists s ∈ S such that the standard<br />
parabolic subsystem (WS\{s}, S \ {s}) is a connected affine Coxeter system. Then<br />
the inertia of (W, S) is (n − 1, 1, 0).<br />
110
Proof. Let (W, S) be as in the statement. Thus, there exists s ∈ S such that<br />
S \ {s} is of connected affine type. By removing another simple reflection s ′ we<br />
must get a finite Coxeter system of rank n − 2, and so (W, S) must have at least<br />
n − 2 positive eigenvalues.<br />
Now, we order S so that S = {s1, ..., sn−1, sn} where sn = s. Then, we let A ′<br />
be the (n − 1) × (n − 1) leading principal minor of the matrix A associated to<br />
(W, S) and Γ. Since A ′ is the matrix for (WS\{s}, S \ {s}) which is of affine type<br />
of rank n − 1, it has inertia (n − 2, 0, 1) and there exists an isotropic root called<br />
δ. Additionally, we know that δ = � n−1<br />
i=1 kiαsi with ki > 0 for all i (c.f. [24]). By<br />
Proposition 10.2.1, we know that there exists an invertible matrix C such that<br />
C tr A ′ C = In−2 ⊕ 01. We can lift this to a matrix D = C ⊕ I1 so that<br />
⎛<br />
B := D tr ⎜<br />
AD = ⎜<br />
⎝<br />
In−2 0<br />
.<br />
0<br />
a1<br />
.<br />
a n−2<br />
0 · · · 0 0 c<br />
a1 · · · an−2 c 1<br />
where c = (αs, δ) and ai ∈ R for all i. Since (W, S) is irreducible, Γ is con-<br />
nected and so there exists some i �= n such that (αsn, αsi ) < 0. Thus by the<br />
characterization of δ above we see that (αsn, δ) �= 0, i.e. c �= 0.<br />
Next, we want to find the determinant of B. To do this, we expand the matrix<br />
along the (n−1)’st row noting that we have to multiply c by −1 due to its position<br />
in the matrix. We thus get that det(B) = −c · det(B ′ ) where<br />
⎛<br />
B ′ ⎜<br />
:= ⎜<br />
⎝<br />
In−2 0<br />
.<br />
0<br />
a1 · · · an−2 c<br />
111<br />
⎞<br />
⎟<br />
⎠<br />
⎞<br />
⎟<br />
⎠
Now, we can find the determinant of B ′ easily by expanding along the (n − 1)’st<br />
column of B ′ (where we will multiply c by +1 due to its position in the matrix.<br />
This gives us that det(B ′ ) = c det(In−2). Putting the results together we see that<br />
det(B) = −c det(B ′ ) = −c 2 det(In−2) = −c 2 .<br />
Since B has at least n−2 positive eigenvalues and it has negative determinant (due<br />
to the fact that c �= 0), we know that exactly one of the remaining eigenvalues must<br />
be negative. Indeed, if both were negative or both were positive, the determinant<br />
of B would be positive. Thus B has inertia (n − 1, 1, 0). Using Proposition 10.2.1,<br />
we can conclude that A must have inertia (n − 1, 1, 0) as well.<br />
10.5 A Class of Coxeter Systems<br />
We define a class of Coxeter systems as follows.<br />
Definition 10.5.1. Let H be the class of all irreducible, finite rank, infinite,<br />
non-affine Coxeter systems, (W, S), with Coxeter graph Γ such that every proper<br />
subgraph Γ ′ ⊂ Γ is the disjoint union of graphs that are Coxeter graphs of only<br />
finite or affine type.<br />
Lemma 10.5.2. Every hyperbolic Coxeter system is in H.<br />
Proof. By Proposition 10.3.1 part (2) we see that any hyperbolic Coxeter system<br />
must be in H. In particular, the hyperbolic Coxeter systems are precisely those<br />
in H with inertia (n − 1, 1, 0).<br />
10.6 A Smaller Class of Coxeter Systems<br />
We now define a certain subclass of H that will be interesting to study.<br />
112
Definition 10.6.1. Let h ⊆ H be the class of all Coxeter systems, (W, S), in H<br />
that have Γ satisfying exactly one of the following (it will be clear that a graph<br />
cannot satisfy both):<br />
(h1) There exists s ∈ S such that Γ − {s} is a disjoint union of Coxeter graphs<br />
of finite type.<br />
(h2) Γ − {s} is a connected Coxeter graph of affine type for all s ∈ S.<br />
Example 10.6.2. Notice that any infinite rank 3 Coxeter system can be described<br />
by one of the two following graphs (note: we color a vertex ◦ if we intend to refer<br />
to it below):<br />
(a)<br />
•�<br />
���<br />
l �<br />
� k<br />
�<br />
�<br />
◦ •<br />
m<br />
(b)<br />
m k<br />
• ◦ •<br />
with l ≥ m ≥ k ≥ 3. For arbitrary m and k, if we remove vertex ◦ from graph<br />
(b) we will be left with the Coxeter graph for a dihedral Coxeter system of order<br />
4. Thus, graph (b) is clearly of type (h1) (provided it is infinite and non-affine).<br />
For graph (a), if k �= ∞, then removing vertex ◦ leaves us with the Coxeter<br />
graph for a finite dihedral Coxeter system, and thus (a) is of type (h1) (provided<br />
it is not affine). If k = ∞ then we necessarily have m = ∞ and l = ∞. Therefore,<br />
∞<br />
removing any vertex from (a) gives us the graph • • , which is clearly of affine<br />
type. Thus any infinite, non-affine irreducible rank 3 Coxeter system is in h.<br />
Lemma 10.6.3. Any Coxeter system in h is hyperbolic.<br />
Proof. Let (W, S) be in h ⊆ H and let |S| = n, i.e. the rank of (W, S) is n. Let Γ<br />
be the Coxeter graph of (W, S) and let A be the matrix associated to (−, −) (or<br />
similarly to Γ).<br />
113
Case 1: (W, S) is of type (h1). Then, since there exists s ∈ S such that (WS−{s}, S −<br />
{s}) is of finite type and so the associated bilinear form and matrix A ′ must<br />
have inertia (n−1, 0, 0). Thus A must have at least n−1 positive eigenvalues.<br />
If A has inertia (n, 0, 0) (resp. (n − 1, 0, 1)) then (W, S) would be of finite<br />
type (resp. affine type) and thus (W, S) �∈ H, which is a contradiction (since<br />
h ⊆ H by definition). Therefore A must have inertia (n − 1, 1, 0) forcing<br />
(W, S) to be hyperbolic by Proposition 10.3.1.<br />
Case 2: (W, S) is of type (h2). Then since S −{s} is of connected affine type Lemma<br />
10.4.1 implies that the inertia of (W, S) is (n − 1, 1, 0). In particular (−, −)<br />
is non-degenerate. Furthermore, since for each r ∈ S we have Γ − {r} is of<br />
positive type, Proposition 10.3.1 implies that (W, S) is hyperbolic.<br />
10.7 Parabolic Subsystems of Non-affine Coxeter Systems<br />
Lemma 10.7.1. Any infinite, irreducible, finite rank, non-affine Coxeter system<br />
(W, S) contains a standard parabolic subsystem in the class H.<br />
Proof. Let (W ′ , S ′ ) be a minimal rank, irreducible, infinite, non-affine parabolic<br />
subsystem of (W, S), which must exist. Then since (W ′ , S ′ ) is minimal rank, if we<br />
remove any vertex s then (WS ′ \{s}, S ′ \ {s}) cannot contain an infinite, non-affine<br />
parabolic subsystem. Thus, (WS ′ \{s}, S ′ \ {s}) must be of finite or affine type.<br />
10.8 Characterizations of Hyperbolic Coxeter Systems<br />
Lemma 10.8.1. Let (W, S) be in the class H. Then (W, S) is in the class h.<br />
114
Proof. Suppose that (W, S) is not of type (h1) and let Γ := ΓS. Then because<br />
(W, S) is in class H, we know that (WS\{s}, S \ {s}) is of affine type for every<br />
s ∈ S. Suppose that s ∈ S and ΓS\{s} is not connected. Let Γ0, Γ1, ..., Γn be<br />
the connected components (so n ≥ 1). Now, since ΓS\{s} is of affine type we can<br />
assume without loss of generality that Γ0 is of affine type. Then Γ ′ = Γ0 ∪ {s} is<br />
irreducible (since (W, S) is irreducible) and is properly contained in Γ0 ∪{s}∪{s1}<br />
where s1 ∈ Γ1. Thus Γ ′ is properly contained in Γ. However, Γ ′ cannot be of finite<br />
or affine type or else its subgraph Γ0 would be finite, contradicting that Γ0 is<br />
affine. Therefore if ΓS\{s} is not connected then Γ properly contains Γ ′ which is<br />
infinite and non-affine, contradicting the assumption that (W, S) is in H. Thus,<br />
we have shown that if (W, S) is not of type (h1), then for every s ∈ S, Γ \ {s} is<br />
of connected affine type, i.e. (W, S) is of type (h2).<br />
Theorem 10.8.2. The following hold:<br />
1. The classes h and H both coincide with the class of hyperbolic Coxeter sys-<br />
tems.<br />
2. Any finite rank, irreducible, infinite, non-affine Coxeter system contains a<br />
standard parabolic subsystem of hyperbolic type.<br />
Proof. By Lemma 10.5.2, we know that all hyperbolic Coxeter systems are con-<br />
tained in H. Then, by Lemma 10.8.1 we know that H is contained in h (so clearly<br />
h and H are equal). Finally, by Lemma 10.6.3 we know that every element of h<br />
is a hyperbolic Coxeter system. Thus we have equality throughout and 1 follows.<br />
For 2, we use part 1 of the theorem just proved along with Lemma 10.7.1.<br />
115
CHAPTER 11<br />
LARGE REFLECTION SUB<strong>GROUPS</strong><br />
By a universal Coxeter system, we mean one with no braid relations (the un-<br />
derlying group is a free product of cyclic groups of order 2). The main result<br />
of this chapter is that all irreducible, non-affine Coxeter systems have universal<br />
reflection subgroups of arbitrarily large rank. To prove this, we use the imagi-<br />
nary cone, which was introduced by Dyer ([14]) in order to prove a conjectured<br />
characterization of coverings in dominance order (see Remark 2.5.3). Dyer raised<br />
the question of whether the imaginary cone of an infinite, irreducible, non-affine<br />
Coxeter system can be approximated by imaginary cones of universal reflection<br />
subgroups. We show this is the case for hyperbolic Coxeter groups, and then our<br />
main result follows from Chapter 10. Many of the auxiliary results in this chapter<br />
can be found in [15]. Since we will define a few different cones, we need some<br />
terminology regarding convexity and polyhedral cones. For a reference on such<br />
terms, consult [1].<br />
11.1 Convex Sets<br />
In this section, we describe our terminology involved with convex sets. Let V<br />
be a real vector space.<br />
Definition 11.1.1. We say that x is a convex combination of x1, ...xn ∈ V if there<br />
exists λ1, ..., λn ∈ R such that<br />
116
1. x = λ1x1 + · · · + λnxn<br />
2. λ1 + · · · + λn = 1<br />
3. λi ≥ 0 for all i.<br />
For any set M we call the set of all convex combinations of elements of M the<br />
convex hull of M and we denote it by conv(M).<br />
For any two elements x, y ∈ V , we denote<br />
[x, y] := {λx + (1 − λ)y| 0 ≤ λ ≤ 1}.<br />
We say a set C ⊂ V is convex if, for all x, y ∈ C (x �= y), [x, y] ⊂ C.<br />
11.2 A Metric on Convex Sets<br />
Assume that V is finite dimensional, so that V ∼ = R n . Then V has the usual<br />
distance function as a metric d : V × V → R. Let Y ⊆ V and x ∈ V . Define<br />
Then for any X ⊆ V and Y ⊆ V , we define<br />
d(x, Y ) = inf d(x, y) (11.2.1)<br />
y∈Y<br />
d(X, Y ) = sup d(x, Y ) (11.2.2)<br />
x∈X<br />
Definition 11.2.1. Let K := {X ⊆ V | X is compact}.<br />
The following proposition can be found in [25, §21 VII].<br />
117
Proposition 11.2.2. Consider the distance function defined in (11.2.2). We<br />
create a new distance function on K given by dK(X, Y ) = max{d(X, Y ), d(Y, X)}.<br />
Then dK defines a metric on K.<br />
The metric defined above is known as the Hausdorff metric (or Hausdorff<br />
distance).<br />
11.3 Approximation of a Sphere by Polytopes<br />
Now, suppose that S d is the d-sphere sitting inside of R d+1 . We want to show<br />
that S d can be approximated by the boundary of a convex hull of a finite number<br />
of points. For any set A ⊆ R n , let ∂A denote the boundary of A. To make the<br />
previous statement more concrete, we have the following lemma.<br />
Lemma 11.3.1. There exists a sequence of compact, convex sets Pm with Pm =<br />
conv({x1, ..., xm}) where {x1, ..., xm} ⊂ S d and m < ∞ such that dK(∂Pm, S d ) → 0<br />
as m → ∞, i.e. ∂Pm → S d in the metric space K.<br />
Proof. Let ɛ > 0 be given. Now, for any point z ∈ R d+1 , define Ba(z) to be the<br />
open ball of radius a centered at z. Then we define<br />
U := {B ɛ<br />
2 (z)| z ∈ Sd }.<br />
Clearly S d ⊂ �<br />
U∈U U. Since Sd is compact, we can find a finite open subcover of<br />
U call it U ′ := {U1, ..., Um} (where m = mɛ is dependent on ɛ).<br />
Now, each Ui is a ball of radius ɛ/2 centered at xi ∈ S d . Thus, we have a finite<br />
set of points {x1, ..., xm} ⊂ S d . Let Pm := conv({x1, ..., xm}). Let ∂Pm denote the<br />
boundary of Pm.<br />
118
Suppose y ∈ ∂Pm. Now, there exists a minimal set I := {xi1, ..., xik } ⊂<br />
{x1, ..., xm} such that y ∈ conv({xi1, ..., xik } = conv(I) ⊂ ∂Pm. Then, we can<br />
choose a supporting affine hyperplane H of Pm such that I ⊂ Pm ∩ H. Let H +<br />
and H − represent the positive and negative half-spaces associated to H. Suppose,<br />
without loss of generality, that Pm ⊂ H − . Let H ′ := int(H + ) and let u be an<br />
outer unit normal vector of H (u points into H + ). By assumption, Pm ∩ H ′ = ∅<br />
and so ∂Pm ∩ H ′ = ∅ and {x1, ..., xm} ∩ H ′ = ∅.<br />
Next, for the sake of contradiction, we suppose that d(y, S d ) ≥ ɛ. Then let<br />
L = H + ɛ<br />
ɛ<br />
u be the affine hyperplane parallel to H with d(H, L) = 2 2 and L ⊂ H′ .<br />
It follows that S d ∩ L + �= ∅, where L + is the positive half-space associated to L.<br />
Indeed, if S d ∩L + = ∅, then S d ⊆ L − and so d(Pm ∩H, S d ) ≤ d(H, L). This would<br />
imply that<br />
d(y, S d ) ≤ sup d(x, S<br />
x∈Pm∩H<br />
d ) = d(Pm ∩ H, S d ) ≤ d(H, L) = ɛ<br />
2<br />
which is not true by assumption. Thus, we let z ∈ Sd ∩ L + . Then B ɛ (z) ⊂ H′<br />
2<br />
by construction. However, z ∈ Sd so z ∈ Ui for some Ui ∈ U ′ so xi ∈ B ɛ (z) ⊂ H′<br />
2<br />
which is a contradiction. Therefore, d(y, S d ) < ɛ.<br />
< ɛ,<br />
Since y is arbitrary, we can conclude that d(∂Pm, S d ) < ɛ. Now, suppose<br />
y ∈ S d , then y ∈ Ui for some i. Thus,<br />
d(y, ∂Pm) = inf d(y, z) ≤ d(y, xi) <<br />
z∈∂Pm<br />
ɛ<br />
2 .<br />
Again, since y is arbitrary, we can conclude that d(Sd , ∂Pm) < ɛ . Thus we have<br />
2<br />
determined that dK(∂Pm, Sd ) = max{d(Sd , ∂Pm), d(∂Pm, Sd )} < max{ ɛ , ɛ} = ɛ<br />
2<br />
as desired.<br />
119
11.4 Cones in Coxeter Systems<br />
Recall, for any reflection subgroup W ′ , we let χ(W ′ ) be the set of canonical<br />
generators of W ′ as a Coxeter system. Then, according to section 2.3, we have a<br />
corresponding set of roots, positive roots, and simple roots for the Coxeter system<br />
(W ′ , χ(W ′ )) given by<br />
ΦW ′ = {α ∈ Φ| sα ∈ W ′ }, Φ +<br />
W ′ := ΦW ′ ∩ Φ+ , ΠW ′ = {α ∈ Φ+ | sα ∈ χ(W ′ )}<br />
respectively.<br />
We now introduce several cones that appear inside of the vector space V in-<br />
troduced in section 2.2. For any W ′ a reflection subgroup of W , we define the<br />
fundamental chamber for W ′ on the vector space V by<br />
CW ′ = {v ∈ V | (v, α) ≥ 0 for all α ∈ ΠW ′}.<br />
Then we denote the Tits cone of W ′ by XW ′ where this is defined by<br />
�<br />
XW ′ =<br />
w∈W ′<br />
w(CW ′).<br />
In addition to these we define the following:<br />
�<br />
KW ′ := R≥0ΠW ′ ∩ −CW ′, ZW ′ :=<br />
w∈W ′<br />
w(KW ′)<br />
where ZW ′ is known as the imaginary cone of W ′ on V (we only define it if S is<br />
finite). The definition of the imaginary cone here is taken from [15], which models<br />
it loosely on one characterization of imaginary cones of Kac-Moody Lie algebras<br />
120
([24]). The results we use are analogs for general W of results proved by Kac for<br />
imaginary cones of Kac-Moody Lie algebras (for which, however, the proofs use<br />
imaginary roots, which have no counterpart here). We will refer to CW , XW , KW<br />
and ZW simply by C , X , K and Z .<br />
11.5 Topology on Rays in Cones<br />
By a ray through v in V we mean a set {λv| λ ∈ R≥0}. We let R denote the<br />
set of rays of V . We can give R a topology and a metric in the following way. Let<br />
B be compact convex body containing 0 in V , and let ∂B denote the boundary of<br />
B. Define the map ϕ : R → ∂B by ϕ(x) = x∩∂B. It is clear that ϕ is a bijection.<br />
We thus declare ϕ to be a homeomorphism between R and ∂B, and this gives R a<br />
topology and a metric corresponding to ∂B in V , with the topology independent<br />
of B. We also introduce the following subsets of R. Let R+ := {R≥0α| α ∈ Φ + }<br />
and let R0 := R+ \ R+. We say a ray, R≥0α, is positive if (α, α) > 0 and isotropic<br />
if (α, α) = 0. We recall some facts about these sets (both proven in [15]).<br />
Proposition 11.5.1. 1. R+ consists of positive rays and is discrete in the<br />
subspace topology.<br />
2. R0 consists of isotropic rays and is closed in R.<br />
3. R0 is the set of limit rays of R+.<br />
In addition, we have the following theorem.<br />
Theorem 11.5.2. The cone Z is the convex closure of �<br />
This theorem has the following immediate corollary.<br />
121<br />
r∈R0<br />
r ∪ {0}.
Corollary 11.5.3. If W ′ is a finitely generated reflection subgroup of W , then<br />
ZW ′ ⊆ ZW .<br />
Remark 11.5.4. A much stronger fact, proven in [15], which we will not use, is<br />
that in the situation of the corollary above we actually have ZW ′ ⊆ ZW .<br />
11.6 Cones in Hyperbolic Coxeter Systems<br />
Recall that we say (W, S) is a hyperbolic Coxeter system if the bilinear form<br />
(−, −) on V has inertia (n − 1, 1, 0) and every proper standard parabolic Coxeter<br />
system is of finite or affine type (see Chapter 10).<br />
Suppose we fix a hyperbolic Coxeter system (W, S). We know that V has<br />
a basis x0, ..., xn such that (xi, xj) = δi,j, (xi, xi) = 1 for all i ∈ {1, ..., n}, and<br />
(x0, x0) = −1. Then, according to [15], we have (after replacing x0 by −x0 if<br />
necessary) that<br />
Z = L<br />
where L = { � n<br />
i=0 λixi| λ0 ≥ �� n<br />
i=1 λ2 i }.<br />
Now, consider the setup as in section 11.5. Let RΠ ⊂ R be the set of rays<br />
spanned by non-zero elements of R≥0Π. Since (W, S) is hyperbolic, the bilinear<br />
form is non-degenerate, and thus the interior of C is non-empty (see [15]). Take<br />
ρ ∈ int(C ) so that (ρ, α) > 0 for all α ∈ Π. Therefore, we may define a map<br />
τ : R≥0Π \ {0} → H := {v ∈ V | (v, ρ) = 1}<br />
given by τ(v) = v/(ρ, v). Then the image of τ is the set P := {v ∈ V | (ρ, v) = 1},<br />
which is a convex polytope in the affine hyperplane H with points (ρ, α) −1 α for<br />
α ∈ Π as vertices. From τ, we introduce a map on R as follows. For r ∈ R, we<br />
122
let ˆτ(r) = τ(x) for all x ∈ r \ {0} (note ˆτ is independent of choice of x ∈ r \ {0}).<br />
With terminology from section 11.5 we may take B so that P ⊂ ∂B.<br />
Now, we can see that Z ∩ H is a disk. Also, by Theorem 11.5.2 we see that<br />
Z ∩ H = conv ��<br />
r∈R0 r� ∩ H.<br />
Lemma 11.6.1. R0 is the set of rays in the boundary of Z .<br />
Proof. First, a ray r ∈ R0 is isotropic by Proposition 11.5.1, and thus it is in<br />
±∂(L )) = ±∂(Z ) due to the description of L above. However, r must also be<br />
in Z , so r ∈ ∂(Z ).<br />
For the reverse implication, the statements above imply it is enough to show<br />
that the result is true inside of H, that is the convex hull of limit points of<br />
(images of) roots in H includes the entire boundary sphere of � Z = L � ∩ H.<br />
We let the boundary of Z ∩ H be denoted by S := ∂ˆτ(Z ), and we let A :=<br />
ˆτ � conv ��<br />
r∈R0 r�� . Then A is the convex hull of the points �<br />
ˆτ(r). But then<br />
the statement is clear since S is a sphere and is contained in A since Z ∩ H =<br />
conv ��<br />
r∈R0 r� ∩H. So every point in S must also be in A or else we would not have<br />
r∈R0<br />
that the boundary of A is a sphere (which is true since (W, S) is hyperbolic).<br />
11.7 Universal Coxeter Systems<br />
Suppose we have a hyperbolic Coxeter system (W, S) as in section 11.6. Then<br />
with the same setup there, we know that Z = L and intersecting with H we get<br />
that S = ∂ˆτ(Z ) is a sphere in H.<br />
Lemma 11.7.1. Suppose that x, y ∈ S. Then there exist αx, αy ∈ Φ + such that<br />
τ(αx), τ(αy) ∈ ˆτ (R+) are sufficiently close to x and y respectively; furthermore,<br />
αx and αy satisfy (αx, αy) < −1. In particular, if S ′ := {sαx, sαy} then (W ′ , S ′ ) is<br />
an indefinite infinite dihedral reflection subgroup of (W, S) and χ(W ′ ) = S ′ .<br />
123
Proof. Since x, y ∈ S then x and y are limit points of ˆτ(R+) and therefore we can<br />
find (images of) positive roots arbitrarily close to x and y. Thus, we pick positive<br />
roots αx ∈ Φ + and αy ∈ Φ + such that τ(αx) and τ(αy) are close enough to x<br />
and y respectively to ensure that {aτ(αx) + bτ(αy)| a, b ∈ R>0} ∩ (Z ∩ H) �= ∅.<br />
Again, this is possible since S is a sphere and x and y are two points (x �= y)<br />
on the sphere and so the relative interior of the line connecting x and y must be<br />
included in the relative interior of the disk bounded by the sphere which is Z ∩H.<br />
Now, we pick a point in the interior of the imaginary cone, v = aτ(αx) + bτ(αy) ∈<br />
int(Z ∩ H) ⊂ int(Z ) with a, b ∈ R>0. Then (v, v) < 0 necessarily and so (−, −)<br />
restricted to Span{αx, αy} cannot be of finite type or affine type. Thus, (W ′ , S ′ )<br />
cannot be a finite dihedral subgroup. Therefore, we see that |(αx, αy)| > 1. Now<br />
if (αx, αy) > 1 then by [4, Proposition 2.2] (and without loss of generality) we<br />
must have αx dominates αy. However, this is impossible since, by investigation<br />
of dominance in infinite dihedral groups ([14]), we know that αx can dominate αy<br />
only if [αx, αy] ∩ ZW ′ = ∅, which is not the case here. Thus, (αx, αy) < −1 and<br />
so S ′ = {sαx, sαy} is the canonical set of generators for the infinite dihedral group<br />
generated by S ′ , (W ′ , S ′ ).<br />
Now, by Lemma 11.3.1 recall that for arbitrary m we can find m points<br />
{x1, ..., xm} ⊂ S such that Pm := conv{x1, ..., xm} along with the property that<br />
dK(Pm, S) → 0 as m → ∞. Then, by Lemma 11.7.1 above, since m is finite, we<br />
can find αxi ∈ Φ+ with τ(αxi ) sufficiently close to xi such that S ′ i,j := {sαx i , sαx j }<br />
is the set of canonical generators for the (infinite) dihedral group it generates,<br />
(W ′<br />
i,j, S ′ i,j). Now, let S ′ := {sαx 1 , ..., sαxm } and W ′ be the reflection subgroup<br />
generated by S ′ .<br />
Proposition 11.7.2. Then (W ′ , S ′ ) is a universal Coxeter system with χ(W ′ ) =<br />
124
S ′ .<br />
Proof. Consider the Coxeter system (W ′ , S ′ ). Then by [19, Proposition 3.5], S ′ =<br />
χ(W ′ ) if and only if {s, s ′ } = χ(〈s, s ′ 〉) for all s, s ′ ∈ S ′ and this follows from<br />
Lemma 11.7.1. Finally, since m(s, s ′ ) = ∞ for all s �= s ′ ∈ S ′ then (W ′ , S ′ ) is a<br />
universal Coxeter system.<br />
11.8 Large Reflection Subgroups<br />
With the terminology as in section 11.6, we have H containing P . Thus, we<br />
can view ˆτ(Z ) as a subset of P . Additionally, we can view any subset of RΠ as a<br />
subset of P . Thus, we will abuse notation by defining dK on any compact subset<br />
of RΠ in the following way: for X, Y ⊂ RΠ, dK(X, Y ) = dK(ˆτ(X), ˆτ(Y )). We put<br />
the previous results together to obtain the following result.<br />
Proposition 11.8.1. Suppose (W, S) is hyperbolic. There exists a sequence of fi-<br />
nite rank, universal reflection subgroups, (W ′ m, S ′ m), of W with dK(R≥0ΠW ′ , Z ) →<br />
m<br />
0 as m → ∞ and dK(Z W ′ , Z ) → 0 as m → ∞.<br />
m<br />
Proof. Let ɛ > 0 be given. Then we can pick m large enough so that dK(Pm, S) <<br />
ɛ/2 by Lemma 11.3.1 where Pm = conv{x1, ..., xm} for some {x1, ..., xm} ⊂ S.<br />
Thus, dK(Pm, Z ∩ H) < ɛ/2. According to Lemma 11.7.2 and the discussion<br />
before it, we can choose {αx1, ..., αxm} ⊂ Φ + so that τ(αxi ) is sufficiently close to<br />
xi (in particular so that dK(τ(αxi ), xi) < ɛ/2) with the property that (W ′ m, S ′ m) is<br />
a universal Coxeter system where S ′ m := {sαx 1 , ..., sαxm } and W ′ m = 〈S ′ m〉. Now, let<br />
Qm = conv{τ(αx1), ..., τ(αxm)}. Then Qm = R≥0ΠW ′ m ∩ H and dK(Qm, Pm) < ɛ/2<br />
125
since dK(τ(αxi ), xi) < ɛ/2 (by assumption above). Therefore, we see that<br />
dK(R≥0ΠW ′, Z ) = dK(Qm, Z ∩ H)<br />
≤ dK(Qm, Pm) + dK(Pm, Z ∩ H)<br />
= dK(Qm, Pm) + dK(Pm, S) < ɛ/2 + ɛ/2 = ɛ<br />
as desired. It remains to show that we can take dK(ZW ′ , Z ) → 0 also.<br />
m<br />
We temporarily fix i �= j. Then since (αxi , αxj ) < −1, we have<br />
Z 〈sαx i ,sαx j 〉 = {z ∈ R≥0αxi<br />
= R≥0βxi,xj + R≥0βxj,xi<br />
+ R≥0αxj | (z, z) ≤ 0}<br />
(which is a two-dimensional cone) for some isotropic linearly independent vectors<br />
βxi,xj<br />
R≥0βxj,xi<br />
and βxj,xi . We choose our previous notation so that βxi,xj ∈ R≥0αxi +<br />
and βxj,xi ∈ R≥0βxi,xj + R≥0αxj .<br />
Then, from [14], since 〈sαx i , sαx j 〉 is dihedral, we have<br />
Z 〈sαx i ,sαx j 〉 = Z ∩ � R≥0αxi<br />
+ R≥0αxj<br />
Thus, τ(βxi,xj ) and τ(βxj,xi ) are the points given by the intersection<br />
This implies that we can choose αxi<br />
[τ(αxi ), τ(αxj )] ∩ S = {τ(βxi,xj ), τ(βxj,xi )}.<br />
� .<br />
and αxj so that all of the following hold:<br />
dK(τ(αxi ), xi) < ɛ/4 dK(τ(αxj ), xj) < ɛ/4<br />
dK(τ(βxi,xj ), xi) < ɛ/4 dK(τ(βxj,xi ), xj) < ɛ/4.<br />
126<br />
(11.8.1)
Next, by Corollary 11.5.3, we know that<br />
Z 〈sαx i ,sαx j 〉 ⊆ Z W ′ m<br />
⊆ Z<br />
for all pairs i �= j. Each of these are convex cones, and so we also have<br />
�<br />
i�=j<br />
� �<br />
R≥0 R≥0βxi,xj + R≥0βxj,xi ⊆ Z W ′ ⊆ Z . (11.8.2)<br />
m<br />
Now, let ɛ > 0. We choose m large enough and αx1, ..., αxm sufficiently close to<br />
x1, ..., xm respectively so that (11.8.1) holds and so that dK(R≥0ΠW ′ , Z ) < ɛ/2<br />
m<br />
where W ′ m = 〈sαx , ..., sαxm 〉 (which is possible by the first part of the proposition).<br />
1<br />
Then (11.8.1) implies that dK(αxi , βxi,xj ) < ɛ/2 for all j �= i so that<br />
and thus<br />
dK<br />
dK([τ(βxi,xj ), τ(βxj,xi )], [τ(αxi ), τ(αxj )]) < ɛ/2<br />
�<br />
� � �<br />
R≥0βxi,xj + R≥0βxj,xi ,<br />
i�=j<br />
m�<br />
k=1<br />
Together, since �m k=1 R≥0αxk = R≥0ΠW ′ , we get that<br />
m<br />
dK<br />
dK<br />
� �<br />
� � �<br />
R≥0βxi,xj + R≥0βxj,xi , Z ≤<br />
i�=j<br />
�<br />
� � �<br />
R≥0βxi,xj + R≥0βxj,xi ,<br />
i�=j<br />
m�<br />
k=1<br />
R≥0αxk<br />
�<br />
R≥0αxk<br />
�<br />
< ɛ<br />
2 .<br />
+ dK(R≥0ΠW ′ ɛ ɛ<br />
, Z ) < + m 2 2<br />
Then, equation (11.8.2) implies that dK(Z W ′ , Z ) < ɛ as well, finishing the proof.<br />
m<br />
The next theorem follows from the previous propositions and Chapter 10.<br />
127<br />
= ɛ.
Theorem 11.8.2. Let (W, S) be a finite rank, irreducible, infinite, non-affine<br />
Coxeter system. Then, for any m ∈ N, W contains a reflection subgroup W ′ with<br />
|χ(W ′ )| = m such that (W ′ , χ(W ′ )) is a universal Coxeter system.<br />
Proof. Let m ∈ N. By Theorem 10.8.2, (W, S) contains a hyperbolic parabolic<br />
subsystem (WJ, J). By Proposition 11.7.2, we have that (WJ, J) contains a uni-<br />
versal Coxeter system (W ′ , S ′ ) of rank m and thus (W, S) also contains (W ′ , S ′ )<br />
proving the theorem.<br />
128
CHAPTER 12<br />
EXPONENTIAL GROWTH <strong>IN</strong> <strong>COXETER</strong> <strong>GROUPS</strong><br />
For any finitely generated group, we can define the growth function of the group<br />
by determining the number of elements of the group of all lengths (with respect<br />
to the generating set). In [6], de la Harpe demonstrates that irreducible, infinite,<br />
non-affine Coxeter systems have what is known as exponential growth by showing<br />
that any such Coxeter system must contain a free non-abelian subgroup. Also, in<br />
[27], Margulis and Vinberg prove the stronger result that such a W must contain<br />
a finite index subgroup which surjects onto a non-abelian free group. Finite and<br />
affine Coxeter systems are known to have polynomial growth.<br />
Even more recently, in [29], Viswanath uses an alternate method to show that<br />
any irreducible, infinite, non-affine simply-laced Coxeter system has exponential<br />
growth. He also obtains the stronger result that for such a system (W, S) and<br />
any J � S the quotient W/WJ also has exponential growth. His method involves<br />
finding a particular universal rank 3 reflection subgroup inside of (W, S). In [29,<br />
Remark 3], the author notes that it would be interesting to know if this result<br />
holds in the non-simply laced case as well. We intend to answer this question<br />
using our results from the previous two chapters.<br />
129
12.1 Growth Types<br />
To begin with, we introduce the basic notions of growth types in finitely gen-<br />
erated groups. Let (W, S) be a finitely generated Coxeter system. We follow [7]<br />
or [29] for general terminology and results.<br />
Definition 12.1.1. Let a := (ak)k≥0 be a sequence of non-decreasing natural<br />
numbers. We define the exponential growth rate of the sequence to be ω(a) :=<br />
lim sup k→∞ a 1/k<br />
k .<br />
For each k ∈ N, we define a subset W≤k := {w ∈ W | l(w) ≤ k}. Then, for any<br />
subset F ⊆ W with 1 ∈ F , we define F≤k := F ∩ W≤k. Finally, for any k ∈ N and<br />
F ⊆ W , we define the number γF,k := |F≤k| and the sequence γF := (γF,k)k≥0.<br />
Then, let ω(F ) := ω(γF ) = lim sup k→∞ γ 1/k<br />
F,k .<br />
Then, we say that a subset F has exponential growth if ω(F ) > 1 and subex-<br />
ponential growth otherwise. If a subset F has subexponential growth, and there is<br />
C ∈ R>0 and d ∈ Z≥0 with γF,k ≤ Ck d for all k ≥ 0, then we say F has polynomial<br />
growth. If F is of subexponential growth and not of polynomial growth, F has<br />
intermediate growth.<br />
We note that if P (F, W ) = �<br />
w∈F ul(w) = �<br />
k≥0 aku k is the Poincaré series of<br />
F , then the coefficients ak = γF,k − γF,k−1 for k ≥ 1 and a0 = γF,0.<br />
12.2 Properties of W(3)<br />
The proof that quotients have exponential growth requires some properties of<br />
the universal Coxeter system W(3) = 〈a, b, c| a 2 = b 2 = c 2 = 1〉 with Coxeter<br />
diagram<br />
•�<br />
∞ ���∞<br />
• •<br />
�<br />
�<br />
�<br />
�<br />
∞<br />
130
These properties are all known and listed in [29], but we remind the reader here<br />
for completeness.<br />
W(3) has the following properties:<br />
1. The Poincaré series of W(3) is P (W(3)) = 1+q<br />
1−2q .<br />
2. Each w ∈ W(3) has a unique reduced expression.<br />
3. The conjugacy classes of a, b, and c are all distinct.<br />
4. For x �= y ∈ W(3)/(W(3)){a} =: W {a}<br />
(3) , we know that xax−1 �= yay −1 . Also,<br />
P (W {a}<br />
(3) , W(3)) = P (W (3))<br />
1+q<br />
1 = 1−2q so there are exactly 2k elements x ∈ W {a}<br />
(3)<br />
with l(x) = k. For each such x, we have l(xax −1 ) ≤ 2l(x) + 1 = 2k + 1.<br />
5. Property 4 implies that for k ≥ 0 there are more than 2 k elements t ∈<br />
{xax−1 | x ∈ W {a}<br />
(3) } with l(t) ≤ 2k + 1.<br />
6. For any reflection t ∈ {xax−1 | x ∈ W {a}<br />
(3) }, properties 2 and 3 imply that a<br />
appears in any reduced expression for t.<br />
12.3 Parabolic Quotients<br />
We now prove a generalization of [29, Theorem 1].<br />
Theorem 12.3.1. Let (W, S) be an irreducible, infinite, non-affine Coxeter sys-<br />
tem. Then for all J � S, W J = W/WJ has exponential growth.<br />
Our proof is similar to the one found in [29], however, we need to make some<br />
changes so that proof works for all Coxeter systems (as opposed to only simply-<br />
laced Coxeter systems).<br />
We first recall an important result due to Deodhar, [9].<br />
131
Theorem 12.3.2. Let (W, S) be a Coxeter system, T the set of reflections, and<br />
J ⊆ S. If t1, t2 ∈ T \ WJ with t1 �= t2, then t1WJ �= t2WJ, that is, distinct<br />
reflections in T \ WJ lie in distinct left cosets of WJ.<br />
According to [29], to prove Theorem 12.3.1, we need to prove the following<br />
proposition.<br />
Proposition 12.3.3. Let (W, S) be an irreducible, infinite, non-affine Coxeter<br />
system and let J � S. Then there exists M ∈ N such that for all k ≥ 0 there are<br />
at least 2 k elements t ∈ T \ WJ such that l(t) ≤ M(2k + 1).<br />
[29].<br />
If this proposition holds, Theorem 12.3.1 holds by the same argument given in<br />
12.4 Proof of Proposition 12.3.3<br />
Proof. For all x ∈ W , any reduced expression for x uses the same simple reflec-<br />
tions. We write supp(x) = {r ∈ S| r appears in all reduced expression for x}. It<br />
is also well known that x ∈ WJ if and only if supp(x) ⊆ J.<br />
Now, let (W, S) be an irreducible, infinite, non-affine Coxeter system and let<br />
s ′ �∈ J. According to Theorem 11.8.2, (W, S) contains a reflection subgroup W ′<br />
such that W ′ ∼ = W(3). Suppose χ(W ′ ) = {t1, t2, t3}. Without loss of generality,<br />
we may assume that s ′ ∈ supp(t1). Indeed, if s ′ �∈ supp(ti) for all i, then let<br />
Z ⊂ S be the smallest subset such that W ′ ⊂ WZ. Then, take a minimal path<br />
s ′ = sn − sn−1 − · · · − s1 − z in the Coxeter diagram with si �∈ Z for all i and<br />
z ∈ Z; let the word corresponding to the path be x = s1 · · · sn ∈ W , and by<br />
reordering we assume z ∈ supp(t1). Now, we define a sequence of reflections in<br />
the following way. Let r0 = t1 and ri = si · · · s1t1s1 · · · si for all i ≥ 1. Let αri<br />
132<br />
∈ Φ+
e the root corresponding to ri. Now, since ({s1, ..., si−1} ∪ Z) ∩ ({si}) = ∅ for<br />
all i ≥ 1 and si is connected to si−1, we have (αri−1 , αsi ) < 0 for all i and so<br />
l(ri) = l(ri−1) + 2 for all i ≥ 1. This implies that l(x −1 t1x) = 2l(x) + l(t1);<br />
hence, s ′ = sn ∈ supp(x −1 tx). Then, xW ′ x −1 is a Coxeter system, isomorphic to<br />
W(3) with χ(xW ′ x −1 ) = {xtx −1 , xt2x −1 , xt3x −1 } = {t ′ 1, t ′ 2, t ′ 3} with s ′ ∈ supp(t ′ 1)<br />
as required.<br />
Next, by the properties listed in Section 12.2, we know that there are at least<br />
2 k elements t ∈ W ′ ∩ T such that l(W ′ ,χ(W ′ ))(t) ≤ 2k + 1 such that t1 ∈ supp(t).<br />
Define M := max{l(ti), l(t2), l(t3)}. Then, we have l(w) ≤ M(l(W ′ ,χ(W ′ ))(w)) for<br />
all w ∈ W ′ . This implies that for each t ∈ W ′ ∩T above, we have l(t) ≤ M(2k+1).<br />
For each t ∈ W ′ ∩ T described above, t1 appears in a reduced expression for t;<br />
in terms of roots this means αt = �3 i=1 ciαti with c1 > 0. Since s ′ ∈ supp(t1), we<br />
have αt1 = �<br />
s∈S csαs with cs ′ > 0. Therefore, it follows that αt = �<br />
s∈S dsαs with<br />
ds ′ > 0 so that s′ ∈ supp(t). Thus, we have t �∈ WJ. This implies that t ∈ T \ WJ.<br />
Therefore, by Theorem 12.3.2, we get that there are at least 2 k elements t ∈ T \WJ<br />
such that l(t) ≤ M(2k + 1). The proposition follows.<br />
133
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