Mirrors and Reflections

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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.

Author(s): Alexandre V. Borovik, Anna Borovik (auth.)
Series: Universitext
Edition: 1
Publisher: Springer-Verlag New York
Year: 2010

Language: English
Pages: 172
Tags: Group Theory and Generalizations;Geometry;Topological Groups, Lie Groups;Linear and Multilinear Algebras, Matrix Theory;Mathematical Methods in Physics

Front Matter....Pages i-xii
Affine Euclidean Space $$\mathbb{A}\mathbb{R}^{n}$$ ....Pages 3-10
Isometries of $$\mathbb{A}\mathbb{R}^{n}$$ ....Pages 11-16
Hyperplane Arrangements....Pages 17-23
Polyhedral Cones....Pages 25-34
Mirrors and Reflections....Pages 37-40
Systems of Mirrors....Pages 41-47
Dihedral Groups....Pages 49-53
Root Systems....Pages 55-62
Root Systems A n−1 , BC n , D n ....Pages 63-75
Chambers....Pages 79-82
Generation....Pages 83-90
Coxeter Complex....Pages 91-97
Residues....Pages 99-104
Generalized Permutahedra....Pages 105-109
Generators and Relations....Pages 113-116
Classification of Finite Reflection Groups....Pages 117-122
Construction of Root Systems....Pages 123-131
Orders of Reflection Groups....Pages 133-135
Reflection Groups in Three Dimensions....Pages 139-146
Icosahedron....Pages 147-153
Back Matter....Pages 155-171