Graduate Texts in Mathematics

231

Editorial Board
S. Axler F.W. Gehring K.A. Ribet


Anders Bjorner
Francesco Brenti

Combinatorics of
Coxeter Groups
With 81 Illustrations


Anders Bjorner
Department of Mathematics
Royal Institute of Technology
Stockholm 100 44
Sweden


Editorial Board:
S. Axler
Mathematics Department
San Francisco State
University
San Francisco, CA 94132
USA

Francesco Brenti
Dipartimento di Matematica
Universita di Roma
Via della Ricerca Scientifica, 1
Roma 00133
Italy


F.W. Gehring
Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109
USA


K.A. Ribet
Mathematics Department
University of California,
Berkeley
Berkeley, CA 94720-3840
USA


Mathematics Subject Classification (2000): 20F55, 05C25
Library of Congress Control Number: 2005923334
ISBN-10: 3-540-44238-3
ISBN-13: 978-3540-442387

Printed on acid-free paper.

© 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New
York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if
they are not identified as such, is not to be taken as an expression of opinion as to whether or not
they are subject to proprietary rights.
Printed in the United States of America.
9 8 7 6 5 4 3 2 1
springeronline.com

(EB)


To Annamaria and Christine


Contents

Foreword
Notation

xi
xiii

Part I
Chapter 1

1.1
1.2
1.3
1.4
1.5

The basics
Coxeter systems
Examples
A permutation representation
Reduced words and the exchange property
A characterization
Exercises
Notes

1
1
4
11
14
18
22
24

Chapter 2
2.1
2.2
2.3
2.4
2.5

2.6
2.7
2.8

Bruhat order
Definition and first examples
Basic properties
The finite case
Parabolic subgroups and quotients
Bruhat order on quotients
A criterion
Interval structure
Complement: Short intervals
Exercises
Notes

27
27
33
36
38
42
45
48
55
57
63


viii

Contents

Chapter 3
3.1
3.2
3.3
3.4

Weak order and reduced words
Weak order
The lattice property
The word property
Normal forms
Exercises
Notes

65
65
70
75
78
84
87

Chapter 4
4.1
4.2
4.3
4.4

4.5
4.6
4.7
4.8
4.9

Roots, games, and automata
A linear representation
The geometric representation
The numbers game
Roots
Roots and subgroups
The root poset
Small roots
The language of reduced words is regular
Complement: Counting reduced words and
small roots
Exercises
Notes

121
125
130

Chapter 5
5.1
5.2
5.3
5.4
5.5

5.6

Kazhdan-Lusztig and R-polynomials
Introduction and review
Reflection orderings
R-polynomials
Lattice paths
Kazhdan-Lusztig polynomials
Complement: Special matchings
Exercises
Notes

131
131
136
140
149
152
158
162
170

Chapter 6
6.1
6.2
6.3
6.4
6.5
6.6
6.7

Kazhdan-Lusztig representations
Review of background material
Kazhdan-Lusztig graphs and cells
Left cell representations
Knuth paths
Kazhdan-Lusztig representations for Sn
Left cells for Sn
Complement: W -graphs
Exercises
Notes

173
174
175
180
185
188
191
196
198
200

89
89
93
97
101
105
108

113
117

Part II


Contents

ix

Chapter 7
7.1
7.2
7.3
7.4
7.5

Enumeration
Poincar´e series
Descent and length generating functions
Dual equivalence and promotion
Counting reduced decompositions in Sn
Stanley symmetric functions
Exercises
Notes

201
201
208
214

222
232
234
242

Chapter 8
8.1
8.2
8.3
8.4
8.5
8.6

Combinatorial descriptions
Type B
Type D
Type A˜
Type C˜
˜
Type B
˜
Type D
Exercises
Notes

245
245
252
260
267

275
281
286
293

Classification of finite and affine Coxeter groups
Graphs, posets, and complexes
Permutations and tableaux
Enumeration and symmetric functions

295
299
307
319

Appendices
A1
A2
A3
A4
Bibliography
Index of notation
Index

323
353
359


Foreword

Coxeter groups arise in a multitude of ways in several areas of mathematics. They are studied in algebra, geometry, and combinatorics, and certain
aspects are of importance also in other fields of mathematics. The theory
of Coxeter groups has been exposited from algebraic and geometric points
of view in several places, also in book form. The purpose of this work is to
present its core combinatorial aspects.
By “combinatorics of Coxeter groups” we have in mind the mathematics
that has to do with reduced expressions, partial order of group elements,
enumeration, associated graphs and combinatorial cell complexes, and connections with combinatorial representation theory. There are some other
topics that could also be included under this general heading (e.g., combinatorial properties of reflection hyperplane arrangements on the geometric
side and deeper connections with root systems and representation theory
on the algebraic side). However, with the stated aim, there is already more
than plenty of material to fill one volume, so with this “disclaimer” we limit
ourselves to the chosen core topics.
It is often the case that phenomena of Coxeter groups can be understood
in several ways, using either an algebraic, a geometric, or a combinatorial
approach. The interplay between these aspects provides the theory with
much of its richness and depth. When alternate approaches are possible,
we usually choose a combinatorial one, since it is our task to tell this side
of the story. For a more complete understanding of the subject, the reader
is urged to study also its algebraic and geometric aspects. The notes at the
end of each chapter provide references and hints for further study.


xii

Foreword

The book is divided into two parts. The first part, comprising Chapters 1
– 4, gives a self-contained introduction to combinatorial Coxeter group theory. We treat the combinatorics of reduced decompositions, Bruhat order,

weak order, and some aspects of root systems. The second part consists of
four independent chapters dealing with certain more advanced topics. In
Chapters 5 – 7, some external references are necessary, but we have tried
to minimize reliance on other sources. Chapter 8, which is elementary, discusses permutation representations of the most important finite and affine
Coxeter groups.
Exercises are provided to all chapters — both easier exercises, meant
to test understanding of the material, and more difficult ones representing
results from the research literature. Open problems are marked with an
asterisk. Thus, the book is meant to have a dual character as both graduate
textbook (particularly Part I) and as research monograph (particularly
Part II).
Acknowledgments: Work on this book has taken place at highly irregular
intervals during the years 1993–2004. An essentially complete and final
version was ready in 1999, but publication was delayed due to unfortunate
circumstances. During the time of writing we have enjoyed the support of
the Volkswagen-Stiftung (RiP-program at Oberwolfach), of the Fondazione
San Michele, and of EC grants Nos. CHRX-CT93-0400 and HPRN-CT2001-00272 (Algebraic Combinatorics in Europe).
Several people have offered helpful comments and suggestions. We particularly thank Sergey Fomin and Victor Reiner, who used preliminary
versions of the book as course material at MIT and University of Minnesota and provided invaluable feedback. Useful suggestions have been
given also by Christos Athanasiadis, Henrik Eriksson, Axel Hultman, and
Federico Incitti. Gă
unter Ziegler provided much needed help with the mysteries of LATEX. Special thanks go to Annamaria Brenti and Siv Sandvik,
who did much of the original typing of text, and to Federico Incitti, who
helped us create many of the figures and improve some of the ones created
by us. Figure 1.1 was provided by Frank Lutz and Figure 1.3 by Jă
urgen
Richter-Gebert.

Stockholm and Rome, September 2004
Anders Bjăorner and Francesco Brenti

Notation

We collect here some notation that is adhered to throughout the book.
Z
N
P
Q, R, C
[n]
[a, b]
[±n]
{a1 , . . . , an }<
a
a
sgn(a)
δij or δ(i, j)
|A|, #A,
or card(A)
A B
A∆B
2A
A
k
∗

A

the
the

the
the
the
the
the
the
the
the

integers
non-negative integers
positive integers
rational, real, and complex numbers
set {1, 2, . . . , n} (n ∈ N), in particular [0] = ∅
set {n ∈ Z : a ≤ n ≤ b} (a, b ∈ Z)
set [−n, n] \ {0}
set {a1 , . . . , an } with total order a1 < · · · < an
largest integer ≤ a (a ∈ R)
smallest integer ≥ a (a ∈ R)
⎧
⎨ 1, if a > 0,
def
0, if a = 0,
the sign of a real number: sgn(a) =
⎩
−1, if a < 0.
1, if i = j,
def
the Kronecker delta: δij =
0, if i = j.

the
the
the
the
the
the

cardinality of a set A
union of two disjoint sets
symmetric difference A ∪ B \ (A ∩ B)
family of all subsets of a finite set A
family of all k-element subsets of a finite set A
set of all words with letters from an alphabet A

Each result (theorem, corollary, proposition, or lemma) is numbered consecutively within sections. So, for example, Theorem 2.3.3 is the third result
in the third section of Chapter 2 (i.e., in Section 2.3). The symbol ✷ denotes the end of a proof or an example. A ✷ appearing at the end of the
statement of a result signifies that the result should be obvious at that
stage of reading, or else that a reference to a proof is given.


1
The basics

Coxeter groups are defined in a simple way by generators and relations.
A key example is the symmetric group Sn , which can be realized as permutations (combinatorics), as symmetries of a regular (n − 1)-dimensional
simplex (geometry), or as the Weyl group of the type An−1 root system or
of the general linear group (algebra). The general theory of Coxeter groups
expands and interweaves the many mathematical themes and aspects
suggested by this example.
In this chapter, we give the basic definitions, present some examples, and

derive the most elementary combinatorial facts underlying the rest of the
book. Readers who already know the fundamentals of the theory can skim
or skip this chapter.

1.1 Coxeter systems
Let S be a set. A matrix m : S × S → {1, 2, . . . , ∞} is called a Coxeter
matrix if it satisfies
m(s, s ) = m(s , s) ;
m(s, s ) = 1 ⇔ s = s .

(1.1)
(1.2)

Equivalently, m can be represented by a Coxeter graph (or Coxeter diagram) whose node set is S and whose edges are the unordered pairs {s, s }
such that m(s, s ) ≥ 3. The edges with m(s, s ) ≥ 4 are labeled by that


2

1. The basics

number. For instance,
⎛

1
⎜ 2
⎜
⎝ 3
2

s4

⎞

2 3 2
1 4 2 ⎟
⎟
4 1 ∞ ⎠
2 ∞ 1

∞

←→

s1

s3

4

s2
2
Let Sfin
= {(s, s ) ∈ S 2 : m(s, s ) = ∞}. A Coxeter matrix m determines
a group W with the presentation

Generators: S
2
.
Relations: (ss )m(s,s ) = e, for all (s, s ) ∈ Sfin

(1.3)

Here, and in the sequel, “e” denotes the identity element of any group
under consideration. Since m(s, s) = 1, we have that
s2 = e,

for all s ∈ S,
m(s,s )

which, in turn, shows that the relation (ss )

(1.4)
= e is equivalent to

ss ss s... = s ss ss ... .
m(s,s )

(1.5)

m(s,s )

In particular, m(s, s ) = 2 (i.e., two distinct nodes s and s are not neighbors
in the Coxeter graph) if and only if s and s commute.
For instance, the group determined by the above Coxeter diagram is
generated by s1 , s2 , s3 , and s4 subject to the relations
⎧ 2
s1 = s22 = s23 = s24 = e
⎪
⎪

⎪
⎪
s1 s2 = s2 s1
⎪
⎪
⎨
s1 s3 s1 = s3 s1 s3
s1 s4 = s4 s1
⎪
⎪
⎪
⎪
s2 s3 s2 s3 = s3 s2 s3 s2
⎪
⎪
⎩
s2 s4 = s4 s2 .
If a group W has a presentation such as (1.3), then the pair (W, S) is
called a Coxeter system. The group W is the Coxeter group and S is the
set of Coxeter generators. The cardinality of S is called the rank of (W, S).
Most groups of interest will be of finite rank. The system is irreducible if
its Coxeter graph is connected.
When referring to an abstract group as a Coxeter group, one usually
has in mind not only W but the pair (W, S), with a specific generating set
S tacitly understood. Some caution is necessary in such cases, since the
isomorphism type of (W, S) is not determined by the group W alone; see
Exercise 2.
The following three statements are equivalent and make explicit what it
means for W to be determined by m via the presentation (1.3):

1.1. Coxeter systems

3

1. (Universality Property) If G is a group and f : S → G is a mapping
such that
(f (s)f (s ))m(s,s ) = e
2
for all (s, s ) ∈ Sfin
, then there is a unique extension of f to a group
homomorphism f : W → G.
2. W ∼
= F/N , where F is the free group generated by S and N is the
2
normal subgroup generated by {(ss )m(s,s ) : (s, s ) ∈ Sfin
}.

3. Let S ∗ be the free monoid generated by S (i.e., the set of words in the
alphabet S with concatenation as product). Let ≡ be the equivalence
relation generated by allowing insertion or deletion of any word of
the form
(ss )m(s,s ) = s s s s s . . . s s s
2m(s,s )

for (s, s ) ∈

2
Sfin
.

∗

Then, S / ≡ forms a group isomorphic to W .

It might seem that to be precise we should use different symbols for the
elements of S and for their images in W ∼
= S ∗ / ≡ under the surjection
ϕ : S ∗ → W.

(1.6)

However, this is needlessly pedantic since, in practice, the possibility of
confusion is negligible. It will be shown (Proposition 1.1.1) that s = s in
S implies ϕ(s) = ϕ(s ) in W and (Corollary 1.4.8) that S is a minimal
generating system for W .
Let (W, S) be a Coxeter system. Definition (1.3) leaves some uncertainty
about the orders of pairwise products ss as elements of W (s, s ∈ S). All
that immediately follows is that the order of ss divides m(s, s ) if m(s, s )
is finite. This leaves open the possibility that distinct Coxeter graphs might
determine isomorphic Coxeter systems. However, this is not the case.
Proposition 1.1.1 Let (W, S) be the Coxeter system determined by a Coxeter matrix m. Let s and s be distinct elements of S. Then, the following
hold:
(i) (The classes of ) s and s are distinct in W .
(ii) The order of ss in W is m(s, s ).
The proof is postponed to Section 4.1, where it is obtained for free as a
by-product of some other material. Section 4.1 makes no use of (or even
mention of) any material in the intermediate sections, so it is possible for
a systematic reader, who wants to see a proof for Proposition 1.1.1 at this
stage of reading, to go directly from here to Section 4.1.

It is a consequence of Proposition 1.1.1 that the Coxeter matrix
(m(s, s ))s,s ∈S can be fully reconstructed from the group W and the
generating set S. This leads to an important conclusion.


4

1. The basics

Theorem 1.1.2 Up to isomorphism there is a one-to-one correspondence
between Coxeter matrices and Coxeter systems. ✷
The finite irreducible Coxeter systems, as well as certain classes of infinite
ones, have been classified. See Appendix A1 for the classification of the
finite and so-called affine groups and [306] for additional information. From
now on, we will every now and then refer to these Coxeter groups by their
conventional names mentioned in Appendix A1, but the classification as
such will not play any significant role in the book. There is no essential
restriction in confining attention to the irreducible case, since reducible
Coxeter groups decompose uniquely as a product of irreducible ones (see
Exercise 2.3).
The finite Coxeter groups for which m(s, s ) ∈ {2, 3, 4, 6} for all (s, s ) ∈
S 2 , s = s are called Weyl groups, a name motivated by Lie theory (see
Example 1.2.10). The Coxeter groups for which m(s, s ) ∈ {2, 3} for all
(s, s ) ∈ S 2 , s = s are called simply-laced.

1.2 Examples
Let us now look at a few examples. The following list is not intended to be
systematic — the aim is merely to acquaint the reader with some of the
groups that play an important role in the combinatorial theory of Coxeter
groups and to exemplify some of the diverse ways in which Coxeter groups

arise. More examples can be found in Chapter 8.
Example 1.2.1 The graph with n isolated vertices (no edges) is the
Coxeter graph of the group Z2 × Z2 × · · · × Z2 of order 2n . ✷
Example 1.2.2 The universal Coxeter group Un of rank n is defined by
the complete graph with all (n2 ) edges marked by “∞.” Equivalently, it is
the group having n generators of order 2 and no other relations. Each group
element can be uniquely expressed as a word in the alphabet of generators,
and these words are precisely the ones where no adjacent letters are equal. ✷
Example 1.2.3 The path
s1

s2

s3

sn−2

sn−1

is the Coxeter graph of the symmetric group Sn with respect to the generating system of adjacent transpositions si = (i, i + 1), 1 ≤ i ≤ n − 1. This is
proved in Proposition 1.5.4. An understanding of this particular example
is very valuable, both because of the importance of the symmetric group
as such and its role as the most accessible nontrivial example of a Coxeter group. We will frequently return to Sn in order to concretely illustrate
various general concepts and constructions. ✷


1.2. Examples

5

Example 1.2.4 The graph
4

s0

s1

s2

s3

sn−2

sn−1

is the Coxeter graph of the group SnB of all signed permutations of the
set [n] = {1, 2, . . . , n}. See Section 8.1 for a detailed description of this
group. It can be thought of in terms of the following combinatorial model.
Suppose that we have a deck of n cards, such that the j-th card has “+j”
written on one side and “−j” on the other. The elements of SnB can then
be identified with the possible rearrangements of stacks of cards; that is,
a group element is a permutation of [n] = {1, 2, . . . , n} (the order of the
cards in the stack) together with the sign information [n] → {+, −} (telling
which side of each card is up). The Coxeter generators si , 1 ≤ i ≤ n − 1,
interchange the card in position i with that in position i + 1 in the stack
(preserving orientation), and s0 flips card 1 (the top card).
The group SnB has a subgroup, denoted SnD , with Coxeter graph
s0

s1

s2

s3

s4

sn−2

sn−1

Here, s0 = s0 s1 s0 . In terms of the card model this group consists of the
stacks with an even number of turned-over cards (i.e., with minus side up).
The generators si , 1 ≤ i ≤ n − 1, are adjacent interchanges as before, and
s0 flips cards 1 and 2 together (as a package). See Section 8.2 for more
about this group. ✷
Example 1.2.5 The circuit
sn

s1

s2

s3

sn−2

sn−1

is the Coxeter graph of the group Sn of affine permutations of the integers.

This is the group of all permutations x of the set Z such that
x(j + n) = x(j) + n,

for all j ∈ Z,

and
n

x(i) =
i=1

n+1
,
2

with composition as group operation. The Coxeter generators are the periodic adjacent transpositions si = j∈Z (i + jn, i + 1 + jn) for i = 1, . . . , n.
See Section 8.3 for more about these infinite permutation groups. ✷


6

1. The basics

Example 1.2.6 The one-way infinite path
s1

s2

s3

s4

is the Coxeter graph of the group of permutations with finite support of the
positive integers (i.e., permutations that leave all but a finite subset fixed).
The generators are the adjacent transpositions si = (i, i + 1), 1 ≤ i. ✷
Example 1.2.7 Dihedral groups. Let L1 and L2 be straight lines through
the origin of the Euclidean plane E2 . Assume that the angle between them
π
is m
, for some integer m ≥ 2. Let r1 be the orthogonal reflection through
L1 , and similarly for r2 . Then, r1 r2 is a rotation of the plane through the
m
angle 2π
m and, hence, (r1 r2 ) = e.
r1 (L2 )

[ = r1 r2 (L2 ) ]

π/m

L1

π/m

L2

Let Gm be the group generated by r1 and r2 . Simple geometric considerations show that Gm consists of the m rotations of the plane through
angles 2πk
m , 0 ≤ k < m, and these m rotations followed by the reflection r1 .
Hence, |Gm | = 2m.

Now, define I2 (m) to be the Coxeter group given by the Coxeter graph
m
s1

s2

Directly from the definition, one sees that every element of I2 (m) can
be represented as an alternating word s1 s2 s1 s2 s1 . . . or s2 s1 s2 s1 s2 . . . of
length ≤ m. (This includes the identity element represented by the empty
word.) Since there are two such words of each positive length and the
two words of length m represent the same group element, it follows that
|I2 (m)| ≤ 2m.
Since r12 = r22 = (r1 r2 )m = e, there is a surjective homomorphism f :
I2 (m) → Gm extending si → ri , i = 1, 2. We have seen that |I2 (m)| ≤ |Gm |.
Consequently, f must be an isomorphism.
The group I2 (m) is called the dihedral group of order 2m. Similarly, the
group I2 (∞) (which is easily seen to be of infinite order) is called the infinite
dihedral group. It arises as the group generated by orthogonal reflections r1
and r2 in lines whose angle is a nonrational multiple of π. ✷


1.2. Examples

7

Example 1.2.8 Symmetry groups of regular polytopes. The symmetries of
a regular m-gon in the plane are the m rotations and the m orthogonal
reflections through a line of symmetry. Thus, the symmetry group is the
dihedral group I2 (m) discussed in the previous example.
The 2-dimensional regular polygons have their counterparts in higher

dimensions among the regular polytopes. A d-dimensional convex polytope
is regular if given two nested sequences of faces F0 ⊆ F1 ⊆ · · · ⊆ Fd−1
(dim Fi = i), there is some isometry of d-space that maps the polytope
onto itself and maps the first sequence of faces to the other one. It turns
out that the symmetry groups of regular polytopes are always Coxeter
groups.
The 3-dimensional regular polytopes are known since antiquity. They
are the five Platonic solids. The higher-dimensional regular poytopes
were classified by Schlă
ai in the mid-1800s. The full classication, with
corresponding Coxeter groups as symmetry groups, is as follows:
Dimension

Regular Polytope

Coxeter group

d
d
d
2
3
3
4
4
4

simplex
cube
hyperoctahedron

m-gon
dodecahedron
icosahedron
24-cell
120-cell
600-cell

Ad
Bd
Bd
I2 (m)
H3
H3
F4
H4
H4

Certain of these polytopes appear in pairs of dual polytopes that share the
same symmetry group. The rest are self-dual.
Let us illustrate the link between polytope and group by having a look
at the geometry of the dodecahedron. As illustrated in Figure 1.1, the
dodecahedron has 15 planes of symmetry, and these planes subdivide its
boundary into 120 congruent triangles. The orthogonal reflections through
the planes generate the full symmetry group W , and this group acts simply
transitively on the triangles.
Seen from this geometric perspective, what are the Coxeter generators?
Fix any one of the 120 triangles and call this the “fundamental region.”
Take as S the three reflections in the “walls” of this triangle. Then, (W, S)
is a Coxeter system.
The Coxeter matrix can be read from the geometry in the following

way. Notice in Figure 1.1 that the dihedral angles in the corners of any
triangle (in particular, of the fundamental region) are π/2, π/3, and π/5.
The denominators are the defining numbers m(s, s ) of the Coxeter system
of type H3 . ✷


8

1. The basics

Figure 1.1. Symmetries of the dodecahedron.

Example 1.2.9 Reflection groups. The example of the dodecahedron
shows how a certain finite Coxeter group can be realized as a group of
geometric transformations generated by reflections. This is, in fact, true of
all finite Coxeter groups, not only the ones related to regular polytopes. It
is also true for the infinite Coxeter groups, although here one may need to
relax the concept of reflection.
The two most important classes of infinite Coxeter groups are defined
in terms of their realizations as reflection groups. These are the affine and
hyperbolic Coxeter groups. We will not discuss the precise definitions here;
suffice it to say that they arise from suitably defined reflections in affine
(resp. hyperbolic) space. The irreducible groups of both types have been
classified.
Here are a few low-dimensional examples that should convey the general
idea. There are three affine irreducible Coxeter systems of rank 3: A2 , C2 ,
and G2 (cf. Appendix A1). The corresponding arrangements of reflecting
lines are shown in Figure 1.2. There are infinitely many hyperbolic irreducible Coxeter systems of rank 3 (but only finitely many in higher ranks);
the system of reflecting lines for one of them is shown in Figure 1.3.
Just as for the dodecahedron, the Coxeter generators for these affine

and hyperbolic groups can be taken to be the reflections in the three lines
that border a fundamental region. Furthermore, the Coxeter matrix of the
group can be read off from the angles at which these lines pairwise meet.
For instance, these angles are, in the case of Figure 1.3, respectively π/2,
π/3, and 0 = π/∞. Again, the denominators are the edge labels of the
∞
. ✷
Coxeter diagram


1.2. Examples

9

e2 , C
e2 , and G
e 2 tesselations of the affine plane.
Figure 1.2. The A

Figure 1.3. The

∞

tesselation of the hyperbolic plane.

Example 1.2.10 Weyl groups of root systems. This example concerns a
special class of groups generated by reflections, which is of great importance
in the theory of semisimple Lie algebras. In that context, the following finite
vector systems in Euclidean space Ed naturally arise. (Recall that Ed is the
same as Rd endowed with a positive definite symmetric bilinear form.)

For α ∈ Ed \{0}, let σα denote the orthogonal reflection in the hyperplane
orthogonal to α. In particular, σα (α) = −α.


10

1. The basics

Definition. A finite set Φ ⊂ Ed \{0} is called a crystallographic root system
if it spans Ed and for all α, β ∈ Φ, the following hold:
(1) Φ ∩ Rα = {α, −α}.
(2) σα (Φ) = Φ.
(3) σα (β) is obtained from β by adding an integral multiple of α.
The group W generated by the reflections σα , α ∈ Φ, is called the Weyl
group of Φ. It is (with a natural choice of generators) a Coxeter group. It
is known that every finite irreducible Coxeter group, with the exception of
H3 , H4 , and I2 (m) for m = 2, 3, 4, 6, can appear as the Weyl group of a
crystallographic root system. The classification of semisimple Lie algebras
proceeds via the classification of their root systems and is thus closely
linked to the classification of finite Coxeter groups.
In Chapter 4, we consider a more general concept of root system, available
for every Coxeter group. The more restrictive crystallographic root systems
will not reappear in this book. ✷
Example 1.2.11 Matrix groups and BN-pairs. Coxeter groups can arise
∞
∼
as groups of matrices. For instance,
= PGL2 (Z), the projective
general linear group (consisting of invertible 2 × 2 matrices with integer
entries where a matrix and its negative are identified).

However, the classical matrix groups over fields are not themselves Coxeter groups. Nevertheless, Coxeter groups play an important role in their
theory. In a precise sense, there sits “inside” such a matrix group G a
certain Coxeter group, called its “Weyl group,” which controls important
features of the structure of G.
We now sketch the connection with matrix groups, via the axiomatization
as “groups with BN-pair,” due to Tits.
Definition. A pair B, N of subgroups of a group G is called a BN-pair (or
Tits system) if the following hold:
(1) B ∪ N generates G, and B ∩ N is normal in N .
def

(2) W = N/(B ∩ N ) is generated by some set S of involutions.
(3) s ∈ S, w ∈ W ⇒ BsB · BwB ⊆ BswB ∪ BwB.
(4) s ∈ S ⇒ BsB · BsB = B.
It can be shown to follow from these axioms that the set S is uniquely
determined and that the pair (W, S) is a Coxeter system. The group W
is called the Weyl group and the number |S| is the rank of the BN-pair
(G; B, N ).
Notice that the double coset BwB is well-defined by the coset w ∈
N/(B ∩ N ). Axioms (3) and (4) suggest the possibility of an induced alge-


1.3. A permutation representation

11

braic structure on the set {BwB}w∈W . This leads to the so-called “Hecke
algebra” of (W, S), underlying Chapters 5 and 6.
The simplest example of a BN-pair is that of a group G acting doubly
def

transitively on a set E of size ≥ 3. Let x = y in E, and let B = Stab({x})
def
and N = Stab({x, y}). This is a BN-pair of rank 1. Conversely, given a
BN-pair of rank 1, one can show that G acts doubly transitively on G/B.
A more instructive example is that of the general linear group GLn (F),
consisting of invertible n×n matrices over a field F. Here, there is a “canonical” BN-pair consisting of the group B of upper-triangular matrices and
the group N of monomial matrices (having exactly one nonzero element
in each row and each column). In this case, as is easy to see, the Weyl
group is the group of permutation matrices. Hence, GLn (F) has a BN-pair
of type A and of rank n − 1. The other classical matrix groups (orthogonal,
symplectic, etc.) have BN-pairs, as do all groups “of Lie type.” Hence, they
can be classified according to the type of their Weyl group.
It is known that every finite irreducible Coxeter group, with the exception
of H3 , H4 , and I2 (m) for m = 2, 3, 4, 6, 8, can appear as the Weyl group of
a finite group with a BN-pair. The finite groups with BN-pairs of rank ≥ 3
have been classified by Tits. They include all of the finite simple groups
except the cyclic groups of prime order, the alternating groups An (n ≥ 5),
and the 26 sporadic groups.
One of the most important features of a group G with a BN-pair is the
following:
Bruhat decomposition: G =

w∈W BwB.

Thus, the Weyl group W acts as indexing set for a partition of the group
G into pairwise disjoint subsets (double cosets w.r.t. the subgroup B). In
the cases of classical matrix groups, this partition induces a partial order
on the set W . This partial order is the topic of Chapter 2. ✷

1.3 A permutation representation

We now return to the program of deriving the basics of combinatorial Coxeter group theory “from scratch,” and we continue the discussion where we
left off in Section 1.1. Our immediate goal is to get as quickly as possible
to the core combinatorial properties of a Coxeter group, such as the exchange property that is discussed in the next section. It turns out that the
description of the group via its defining presentation (1.3) is ill suited for
this purpose — one needs the added structure coming from some suitable
concrete realization of the group.
This section describes a realization as a permutation group that leads
quite quickly to the goal. This permutation representation is introduced
here for the sole purpose of proving Theorem 1.4.3 of the following section;


12

1. The basics

it will not reappear in this form after that. (However, it will reappear in
the guise of permutations of the root system of (W, S) in Section 4.4, for
the connection see Exercise 4.7.)
def

Throughout this section, (W, S) denotes a Coxeter system. Let T =
{wsw−1 : s ∈ S, w ∈ W }. The elements of T (i.e., the elements conjugate
to some Coxeter generator) are called reflections. The definition shows that
S ⊆ T and that
t2 = e,

for all t ∈ T .

(1.7)

The elements of S are sometimes called simple reflections.
Given a word s1 s2 . . . sk ∈ S ∗ , define
def

ti = s1 s2 . . . si−1 si si−1 . . . s2 s1 ,

for 1 ≤ i ≤ k,

(1.8)

and the ordered k-tuple
def

T (s1 s2 . . . sk ) = (t1 , t2 , . . . , tk ).

(1.9)

For instance,
T (abca) = (a, aba, abcba, abcacba).
As stated earlier, we consider words in S ∗ also as elements in W (reading
them as a product) without change of notation. Note that
ti = (s1 . . . si−1 ) si (s1 . . . si−1 )−1 ∈ T,

(1.10)

ti s1 s2 . . . sk = s1 . . . si . . . sk (si omitted),

(1.11)

s1 s2 . . . si = ti ti−1 . . . t1 .

(1.12)

and

Lemma 1.3.1 If w = s1 s2 . . . sk , with k minimal, then ti = tj for all
1 ≤ i < j ≤ k.
Proof. If ti = tj for some i < j then w = ti tj s1 s2 . . . sk =
s1 . . . si . . . sj . . . sk (i.e., si and sj deleted), which contradicts the minimality of k. ✷
For s1 s2 . . . sk ∈ S ∗ and t ∈ T , let
def

n(s1 s2 . . . sk ; t) = the number of times t appears in T (s1 s2 . . . sk ).
(1.13)
Furthermore, for s ∈ S and t ∈ T , let
def

η(s ; t) =

−1, if s = t,
+1, if s = t.

(1.14)

Note that
k

(−1)n(s1 s2 ...sk ;t) =

η(si ; si−1 . . . s1 t s1 . . . si−1 ).

i=1

(1.15)


1.3. A permutation representation

13

We will consider the group S(R) of all permutations of the set
R = T × {+1, −1}.
For s ∈ S, define a mapping πs of R to itself by
def

πs (t, ε) = (sts, ε η(s ; t)).
The computation
πs2 (t, ε) = πs (sts, ε η(s ; t)) = (sstss, ε η(s ; t) η(s ; sts)) = (t, ε)
shows that πs ∈ S(R).
Theorem 1.3.2 (i) The mapping s → πs extends uniquely to an
injective homomorphism w → πw from W to S(R).
(ii) πt (t, ε) = (t, −ε), for all t ∈ T .
Proof. We verify the assertions in several steps.
(1) It was already shown that πs2 = idR .
(2) Let s, s ∈ S and m(s, s ) = p = ∞. We claim that
(πs πs )p = idR .
To prove this, let
si =

s , if i is odd,
s, if i is even

and let s denote the word s1 s2 . . . s2p = s ss ss . . . s s. Let T (s) =
(t1 , t2 , . . . , t2p ); that is,
ti = s1 s2 . . . si . . . s2 s1 = (s s)i−1 s ,

1 ≤ i ≤ 2p.

Since (s s)p = e, we have that
tp+i = ti ,

1 ≤ i ≤ p,

which implies that n(s; t) is even for all t ∈ T . Let
(t , ε ) = (πs πs )p (t, ε) = πs2p πs2p−1 . . . πs1 (t, ε).
Then, t = s2p . . . s1 t s1 . . . s2p = t, since s1 s2 . . . s2p = (s s)p = e.
Furthermore, using (1.15), we get
2p

η(si ; si−1 . . . s1 t s1 . . . si−1 ) = ε (−1)n(s;t) = ε.

ε =ε
i=1

So, the claim is proved.
(3) By the universality property and what has just been shown, the
mapping s → πs extends to a homomorphism w → πw of W . If w =


14

1. The basics

sk sk−1 . . . s1 , we compute
πw (t, ε) = πsk πsk−1 . . . πs1 (t, ε)
k

=

sk . . . s1 ts1 . . . sk , ε

η(si ; si−1 . . . s1 t s1 . . . si−1 )

i=1
n(s1 s2 ...sk ;t)

= (w t w−1 , ε (−1)

).

(1.16)

In particular, the parity of n(s1 s2 . . . sk ; t) only depends on w and t.
(4) Suppose that w = e. Choose an expression w = sk sk−1 . . . s1 with k
minimal, and let T (s1 s2 . . . sk ) = (t1 , t2 , . . . , tk ). By Lemma 1.3.1, all ti ’s
are distinct, so n(s1 s2 . . . sk ; ti ) = 1. Therefore, πw (ti , ε) = (wti w−1 , −ε)
for 1 ≤ i ≤ k by equation (1.16), which shows that πw = idR . Hence, the
homomorphism is injective.
(5) We show part (ii) of the theorem by induction on the size of a symmetric
expression for t. Let
t = s1 s2 . . . sp . . . s2 s1 ,

si ∈ S.

The case p = 1 is clear by definition. Then, by induction,
πs1 ...sp ...s1 (s1 . . . sp . . . s1 , ε)
= πs1 πs2 ...sp ...s2 (s2 . . . sp . . . s2 , ε η(s1 ; s1 . . . sp . . . s1 ))
= πs1 (s2 . . . sp . . . s2 , −ε η(s1 ; s2 . . . sp . . . s2 ))
= (s1 . . . sp . . . s1 , −ε η 2 (s1 ; s2 . . . sp . . . s2 ))
= (t, −ε).
✷
For w ∈ W and t ∈ T , let
def

η(w ; t) = (−1)n(s1 s2 ...sk ;t) ,

(1.17)

where w = s1 s2 . . . sk is an arbitrary expression, si ∈ S. Step (3) of the
proof shows that this is well defined. This definition extends that of η(s ; t)
for s ∈ S given in equation (1.14) and makes it possible to rewrite equation
(1.16) as follows:
πw (t, ε) = (w t w−1 , ε η(w−1 ; t)).

(1.18)

1.4 Reduced words and the exchange property
In this section, we prove some fundamental combinatorial properties of the
system of words representing any given element of a Coxeter group.

1.4. Reduced words and the exchange property

15

Let (W, S) be a Coxeter system. Each element w ∈ W can be written as
a product of generators:
w = s1 s2 . . . sk ,

si ∈ S.

If k is minimal among all such expressions for w, then k is called the length
of w (written (w) = k) and the word s1 s2 . . . sk is called a reduced word
(or reduced decomposition or reduced expression) for w. As discussed in
Section 1.1, we let “s1 s2 . . . sk ” denote both the product of these generators
(an element of W ) and the word formed by listing them in this order (an
element of the free monoid S ∗ ).
The following is an immediate consequence of the Universality Property.
Lemma 1.4.1 The map ε : s → −1, for all s ∈ S, extends to a group
homomorphism ε : W → {+1, −1}. ✷
Here are some basic properties of the length function.
Proposition 1.4.2 For all u, w ∈ W :
(i) ε(w) = (−1)

(w)

,

(ii) (uw) ≡ (u) + (w) (mod 2),
(iii) (sw) = (w) ± 1, for all s ∈ S,
(iv) (w−1 ) = (w),

(v) | (u) − (w)| ≤ (uw) ≤ (u) + (w),
(vi) (uw−1 ) is a metric on W .
Proof. Parts (i) – (iii) follow from Lemma 1.4.1. We leave the rest as
exercises. ✷
It is a consequence of Lemma 1.4.1 that the elements of even length form
a subgroup of W of index 2. This is called the alternating subgroup (following the terminology of the symmetric group) or the rotation subgroup
(following the terminology of finite reflection groups) of W .
We now come to the so-called “exchange property,” which is a fundamental combinatorial property of a Coxeter group. In its basic version,
appearing in the following section, the condition t ∈ T in the statement
below is weakened to t ∈ S, hence the adjective “strong” for the version
given here.
Theorem 1.4.3 (Strong Exchange Property) Suppose w = s1 s2 . . . sk
(si ∈ S) and t ∈ T . If (tw) < (w), then tw = s1 . . . si . . . sk for some
i ∈ [k].
Proof. Recall the number η(w ; t) ∈ {+1, −1} defined in definition (1.17).
We prove the equivalence of these two conditions:
(a) (tw) < (w),