Mirrors and reflections : the geometry of finite reflection groups / Alexandre V. Borovik, Anna Borovik.

Format:

Book

Author/Creator:

Borovik, Alexandre.

Contributor:

Borovik, Anna.

Series:

Universitext

Language:

English

Subjects (All):

Reflection groups.

Coxeter complexes.

Physical Description:

xii, 171 pages. : illustrations ; 24 cm.

Place of Publication:

New York : Springer, [2010]

Summary:

Mirrors and Reflections presents an intuitive and elementary introduction to finite reflection groups. Starting with basic principles, this book provides a comprehensive classification of the various types of finite reflection groups and describes their underlying geometric properties.

Unique to this text is its emphasis on the intuitive geometric aspects of the theory of reflection groups, making the subject more accessible to the novice. Primarily self-contained, necessary geometric concepts are introduced and explained. Principally designed for coursework, this book is saturated with exercises and examples of varying degrees of difficulty. An appendix offers hints for solving the most difficult problems. Wherever possible, concepts are presented with pictures and diagrams intentionally drawn for easy reproduction.

Finite reflection groups is a topic of great interest to many pure and applied mathematicians. Often considered a cornerstone of modern algebra and geometry, an understanding of finite reflection groups is of great value to students of pure or applied mathematics. Requiring only a modest knowledge of linear algebra and group theory, this book is intended for teachers and students of mathematics at the advanced undergraduate and graduate levels.

Contents:

Part I Geometric Background

1 Affine Euclidean Space AR n 3

1.1 Euclidean Space R n 4

1.2 Affine Euclidean Space AR n 5

1.3 Affine Subspaces 5

1.3.1 Subspaces 6

1.3.2 Systems of Linear Equations 6

1.3.3 Points and Lines 7

1.3.4 Planes 7

1.3.5 Hyperplanes 7

1.3.6 Orthogonal Projection 7

1.4 Half-Spaces 8

1.5 Bases and Coordinates 8

1.6 Convex Sets 9

2 Isometries of AR n 11

2.1 Fixed Points of Groups of Isometries 11

2.2 Structure of Isom AR n 12

2.2.1 Translations 12

2.2.2 Orthogonal Transformations 13

3 Hyperplane Arrangements 17

3.1 Faces of a Hyperplane Arrangement 17

3.2 Chambers 18

3.3 Galleries 19

3.4 Polyhedra 20

4 Polyhedral Cones 25

4.1 Finitely Generated Cones 25

4.1.1 Cones 25

4.1.2 Extreme Vectors and Edges 26

4.2 Simple Systems of Generators 27

4.3 Duality 29

4.4 Duality for Simplicial Cones 30

4.5 Faces of a Simplicial Cone 31

Part II Mirrors, Reflections, Roots

5 Mirrors and Reflections 37

6 Systems of Mirrors 41

6.1 Systems of Mirrors 41

6.2 Finite Reflection Groups 44

7 Dihedral Groups 49

7.1 Groups Generated by Two Involutions 49

7.2 Proof of Theorem 7.1 50

7.3 Dihedral Groups: Geometric Interpretation 51

8 Root Systems 55

8.1 Mirrors and Their Normal Vectors 55

8.2 Root Systems 56

8.3 Planar Root Systems 57

8.4 Positive and Simple Systems 59

9 Root Systems A n-1 , BC n , D n 63

9.1 Root System A n-1 63

9.1.1 A Few Words about Permutations 63

9.1.2 Permutation Representation of Sym n 64

9.1.3 Regular Simplices 64

9.1.4 The Root System, A n-1 65

9.1.5 The Standard Simple System 66

9.1.6 Action of Sym n on the Set of all Simple Systems 66

9.2 Root Systems of Types C n and B n 68

9.2.1 Hyperoctahedral Group 68

9.2.2 Admissible Orderings 69

9.2.3 Root Systems C n and B n 70

9.2.4 Action of W on Φ 71

9.3 The Root System D n 72

Part III Coxeter Complexes

10 Chambers 79

11 Generation 83

11.1 Simple Reflections 83

11.2 Foldings 84

11.3 Galleries and Paths 85

11.4 Action of W on C 87

11.5 Paths and Foldings 87

11.6 Simple Transitivity of W on C: Proof of Theorem 11.6 89

12 Coxeter Complex 91

12.1 Labeling of the Coxeter Complex 91

12.2 Length of Elements in W 93

12.3 Opposite Chamber 93

12.4 Isotropy Groups 94

12.5 Parabolic Subgroups 95

13 Residues 99

13.1 Residues 99

13.2 Example 100

13.3 The Mirror System of a Residue 101

13.4 Residues are Convex 102

13.5 Residues: the Gate property 102

13.6 The Opposite Chamber 103

14 Generalized Permutahedra 105

Part IV Classification

15 Generators and Relations 113

15.1 Reflection Groups are Coxeter Groups 113

15.2 Proof of Theorem 15.1 115

16 Classification of Finite Reflection Groups 117

16.1 Coxeter Graph 117

16.2 Decomposable Reflection Groups 118

16.3 Labeled Graphs and Associated Bilinear Forms 118

16.4 Classification of Positive Definite Graphs 119

17 Construction of Root Systems 123

17.1 Root System A n 123

17.2 Root System B n , n ≥ 2 124

17.3 Root System C n , n ≥ 2 125

17.4 Root System D n , n ≥ 4 126

17.5 Root System E₈ 126

17.6 Root System E₇ 127

17.7 Root System E₆ 128

17.8 Root System F₄ 128

17.9 Root System G₂ 129

17.10 Crystallographic Condition 129

18 Orders of Reflection Groups 133

Part V Three-Dimensional Reflection Groups

19 Reflection Groups in Three Dimensions 139

19.1 Planar Mirror Systems 139

19.2 From Mirror Systems to Tessellations of the Sphere 139

19.3 The Area of a Spherical Triangle 141

19.4 Classification of Finite Reflection Groups in Three Dimensions 142

20 Icosahedron 147

20.1 Construction 147

20.2 Uniqueness and Rigidity 149

20.3 The Symmetry Group of the Icosahedron 151

Part VI Appendices.

Notes:

Includes bibliographical references and index.

ISBN:

9780387790657

0387790659

OCLC:

455831733