Borovik A.V., Borovik A. Mirrors and Reflections: The Geometry of Finite Reflection Groups

Springer Science+ Business Media, LLC, 2010. — 167 p. — (Universitext) — ISBN: 0387790659Starting with basic principles, this book provides a comprehensive classification of the various types of finite reflection groups and describes their underlying geometric properties. Numerous exercises at various levels of difficulty are included.
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.Geometric Background
Affine Euclidean Space ARn
Isometries of ARn
Hyperplane Arrangements
Polyhedral Cones
Mirrors, Reflections, Roots
Mirrors and Reflections
Systems of Mirrors
Dihedral Groups
Root Systems
Root Systems An−1, BCn, Dn
Coxeter Complexes
Chambers
Generation
Coxeter Complex
Residues
Generalized Permutahedra
Classification
Generators and Relations
Classification of Finite Reflection Groups
Construction of Root Systems
Orders of Reflection Groups
Three-Dimensional Reflection Groups
Reflection Groups in Three Dimensions
Icosahedron
Appendices
The ForgottenArt of Blackboard Drawing
Hints and Solutions to Selected Exercises