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Mirrors and Re¬‚ections:
The Geometry of Finite Re¬‚ection Groups

Incomplete Draft Version 01


Alexandre V. Borovik Anna S. Borovik
alexandre.borovik@umist.ac.uk anna.borovik@freenet.co.uk


25 February 2000
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 i

Introduction

This expository text contains an elementary treatment of ¬nite groups gen-
erated by re¬‚ections. There are many splendid books on this subject, par-
ticularly [H] provides an excellent introduction into the theory. The only
reason why we decided to write another text is that some of the applications
of the theory of re¬‚ection groups and Coxeter groups are almost entirely
based on very elementary geometric considerations in Coxeter complexes.
The underlying ideas of these proofs can be presented by simple drawings
much better than by a dry verbal exposition. Probably for the reason of
their extreme simplicity these elementary arguments are mentioned in most
books only brie¬‚y and tangently.

We wish to emphasize the intuitive elementary geometric aspects of
the theory of re¬‚ection groups. We hope that our approach allows an
easy access of a novice mathematician to the theory of re¬‚ection groups.
This aspect of the book makes it close to [GB]. We realise, however,
that, since classical Geometry has almost completely disappeared from
the schools™ and Universities™ curricula, we need to smugle it back and
provide the student reader with a modicum of Euclidean geometry and
theory of convex polyhedra. We do not wish to appeal to the reader™s
geometric intuition without trying ¬rst to help him or her to develope
it. In particular, we decided to saturate the book with visual material.
Our sketches and diagrams are very unsophisticated; one reason for this
is that we lack skills and time to make the pictures more intricate and
aesthetically pleasing, another is that the book was tested in a M. Sc.
lecture course at UMIST in Spring 1997, and most pictures, in their even
less sophisticated versions, were ¬rst drawn on the blackboard. There was
no point in drawing pictures which could not be reproduced by students
and reused in their homework. Pictures are not for decoration, they are
indispensable (though maybe greasy and soiled) tools of the trade.

The reader will easily notice that we prefer to work with the mirrors
of re¬‚ections rather than roots. This approach is well known and fully
exploited in Chapter 5, §3 of Bourbaki™s classical text [Bou]. We have
combined it with Tits™ theory of chamber complexes [T] and thus made
the exposition of the theory entirely geometrical.

The book contains a lot of exercises of di¬erent level of di¬culty. Some
of them may look irrelevant to the subject of the book and are included for
the sole purpose of developing the geometric intuition of a student. The
more experienced reader may skip most or all exercises.
ii

Prerequisites
Formal prerequisites for reading this book are very modest. We assume
the reader™s solid knowledge of Linear Algebra, especially the theory of
orthogonal transformations in real Euclidean spaces. We also assume that
they are familiar with the following basic notions of Group Theory:

groups; the order of a ¬nite group; subgroups; normal sub-
groups and factorgroups; homomorphisms and isomorphisms;
permutations, standard notations for them and rules of their
multiplication; cyclic groups; action of a group on a set.

You can ¬nd this material in any introductory text on the subject. We
highly recommend a splendid book by M. A. Armstrong [A] for the ¬rst
reading.
A. & A. Borovik • Mirrors and Re¬‚ections • Version 01 • 25.02.00 iii

Acknowledgements
The early versions of the text were carefully read by Robert Sandling and
Richard Booth who suggested many corrections and improvements.
Our special thanks are due to the students in the lecture course at
UMIST in 1997 where the ¬rst author tested this book:

Bo Ahn, Ay¸e Berkman, Richard Booth, Nazia Kalsoom, Vaddna
s
Nuth.
iv
Contents

1 Hyperplane arrangements 1
1.1 A¬ne Euclidean space ARn . . . . . . . . . . . . . . . . . 1
1.1.1 How to read this section . . . . . . . . . . . . . . . 1
1.1.2 Euclidean space Rn . . . . . . . . . . . . . . . . . . 2
1.1.3 A¬ne Euclidean space ARn . . . . . . . . . . . . . 2
1.1.4 A¬ne subspaces . . . . . . . . . . . . . . . . . . . . 3
1.1.5 Half spaces . . . . . . . . . . . . . . . . . . . . . . 5
1.1.6 Bases and coordinates . . . . . . . . . . . . . . . . 6
1.1.7 Convex sets . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Hyperplane arrangements . . . . . . . . . . . . . . . . . . 8
1.2.1 Chambers of a hyperplane arrangement . . . . . . . 8
1.2.2 Galleries . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Isometries of ARn . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1 Fixed points of groups of isometries . . . . . . . . . 14
1.4.2 Structure of Isom ARn . . . . . . . . . . . . . . . . 15
1.5 Simplicial cones . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 Finitely generated cones . . . . . . . . . . . . . . . 20
1.5.3 Simple systems of generators . . . . . . . . . . . . . 22
1.5.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5.5 Duality for simplicial cones . . . . . . . . . . . . . . 25
1.5.6 Faces of a simplicial cone . . . . . . . . . . . . . . . 27

2 Mirrors, Re¬‚ections, Roots 31
2.1 Mirrors and re¬‚ections . . . . . . . . . . . . . . . . . . . . 31
2.2 Systems of mirrors . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Dihedral groups . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Planar root systems . . . . . . . . . . . . . . . . . . . . . . 46
2.6 Positive and simple systems . . . . . . . . . . . . . . . . . 49
2.7 Root system An’1 . . . . . . . . . . . . . . . . . . . . . . . 51

v
vi

2.8 Root systems of type Cn and Bn . . . . . . . . . . . . . . . 56
2.9 The root system Dn . . . . . . . . . . . . . . . . . . . . . 60

3 Coxeter Complex 63
3.1 Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Generation by simple re¬‚ections . . . . . . . . . . . . . . . 65
3.3 Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4 Galleries and paths . . . . . . . . . . . . . . . . . . . . . . 67
3.5 Action of W on C . . . . . . . . . . . . . . . . . . . . . . . 69
3.6 Labelling of the Coxeter complex . . . . . . . . . . . . . . 73
3.7 Isotropy groups . . . . . . . . . . . . . . . . . . . . . . . . 74
3.8 Parabolic subgroups . . . . . . . . . . . . . . . . . . . . . 77
3.9 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.10 Generalised permutahedra . . . . . . . . . . . . . . . . . . 79

4 Classi¬cation 83
4.1 Generators and relations . . . . . . . . . . . . . . . . . . . 83
4.2 Decomposable re¬‚ection groups . . . . . . . . . . . . . . . 83
4.3 Classi¬cation of ¬nite re¬‚ection groups . . . . . . . . . . . 85
4.4 Construction of root systems . . . . . . . . . . . . . . . . . 85
4.5 Orders of re¬‚ection groups . . . . . . . . . . . . . . . . . . 91
List of Figures

1.1 Convex and non-convex sets. . . . . . . . . . . . . . . . . . 7
Line arrangement in AR2 . . . . . . . . . .
1.2 . . . . . . . . . 8
1.3 Polyhedra and polytopes . . . . . . . . . . . . . . . . . . . 12
1.4 A polyhedron is the union of its faces . . . . . . . . . . . . 12
1.5 The regular 2-simplex . . . . . . . . . . . . . . . . . . . . . 13
1.6 For the proof of Theorem 1.4.1 . . . . . . . . . . . . . . . . 14
1.7 Convex and non-convex sets. . . . . . . . . . . . . . . . . . 20
1.8 Pointed and non-pointed cones . . . . . . . . . . . . . . . . 22
1.9 Extreme and non-extreme vectors. . . . . . . . . . . . . . . 22
1.10 The cone generated by two simple vectors . . . . . . . . . 24
1.11 Dual simplicial cones. . . . . . . . . . . . . . . . . . . . . . 26

2.1 Isometries and mirrors (Lemma 2.1.3). . . . . . . . . . . . 32
2.2 A closed system of mirrors. . . . . . . . . . . . . . . . . . . 35
2.3 In¬nite planar mirror systems . . . . . . . . . . . . . . . . 36
2.4 Billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 For Exercise 2.2.3. . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Angular re¬‚ector . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 The symmetries of the regular n-gon . . . . . . . . . . . . 42
2.8 Lengths of roots in a root system. . . . . . . . . . . . . . . 45
2.9 A planar root system (Lemma 2.5.1). . . . . . . . . . . . . 47
2.10 A planar mirror system (for the proof of Lemma 2.5.1). . . 47
2.11 The root system G2 . . . . . . . . . . . . . . . . . . . . . . 48
2.12 The system generated by two simple roots . . . . . . . . . 50
2.13 Simple systems are obtuse (Lemma 2.6.1). . . . . . . . . . 51
2.14 Symn is the group of symmetries of the regular simplex. . . 53
2.15 Root system of type A2 . . . . . . . . . . . . . . . . . . . . 53
2.16 Hyperoctahedron and cube. . . . . . . . . . . . . . . . . . 57
2.17 Root systems B2 and C2 . . . . . . . . . . . . . . . . . . . . 58
2.18 Root system D3 . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1 The fundamental chamber. . . . . . . . . . . . . . . . . . . 64
3.2 The Coxeter complex BC3 . . . . . . . . . . . . . . . . . . . 64

vii
viii

3.3 Chambers and the baricentric subdivision. . . . . ..... 65
3.4 Generation by simple re¬‚ections (Theorem 3.2.1). ..... 65
3.5 Folding . . . . . . . . . . . . . . . . . . . . . . . . ..... 67
3.6 Folding a path (Lemma 3.5.4) . . . . . . . . . . . ..... 71
3.7 Labelling of panels in the Coxeter complex BC3 . . . ..... 73
3.8 Permutahedron for Sym4 . . . . . . . . . . . . . . ..... 79
3.9 Edges and mirrors (Theorem 3.10.1). . . . . . . . ..... 80
3.10 A convex polytope and polyhedral cone (Theorem 3.10.1). 81
3.11 A permutahedron for BC3 . . . . . . . . . . . . . . ..... 82

4.1 For the proof of Theorem 4.1.1. . . . . . . . . . . . . . . . . 84
Chapter 1

Hyperplane arrangements

A¬ne Euclidean space ARn
1.1

1.1.1 How to read this section

This section provides only a very sketchy description of the a¬ne geometry
and can be skipped if the reader is familiar with this standard chapter of
Linear Algebra; otherwise it would make a good exercise to restore the
proofs which are only indicated in our text1 . Notice that the section con-
tains nothing new in comparision with most standard courses of Analytic
Geometry. We simply transfer to n dimensions familiar concepts of three
dimensional geometry.
The reader who wishes to understand the rest of the course can rely on
his or her three dimensional geometric intuition. The theory of re¬‚ection
groups and associated geometric objects, root systems, has the most for-
tunate property that almost all computations and considerations can be
reduced to two and three dimensional con¬gurations. We shall make every
e¬ort to emphasise this intuitive geometric aspect of the theory. But, as a
warning to students, we wish to remind you that our intuition would work
only when supported by our ability to prove rigorously ˜intuitively evident™
facts.



1
To attention of students: the material of this section will not be included in the
examination.


1
2

Euclidean space Rn
1.1.2
Let Rn be the Euclidean n-dimensional real vector space with canonical
scalar product ( , ). We identify Rn with the set of all column vectors
« 
a1
¬.·
±= .  .
an

of length n over R, with componentwise addition and multiplication by
scalars, and the scalar product
« 
b1
¬.·
(±, β) = ±t β = (a1 , . . . , an )  . 
.
an


= a1 b1 + · · · + an bn ;

here t denotes taking the transposed matrix.
This means that we ¬x the canonical orthonormal basis 1, . . . , n in
Rn , where
«
0
¬.·
¬.·.
¬·
i = ¬ 1 · ( the entry 1 is in the ith row) .
¬.·
..
0

The length |±| of a vector ± is de¬ned as |±| = (a, a). The angle A
between two vectors ± and β is de¬ned by the formula

(±, β)

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