Mirrors and Reflections - (Universitext) by Alexandre V Borovik & Anna Borovik (Paperback)

About the Book

Starting 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.

Book Synopsis

"Mirrors and Reflections" is a systematic and elementary treatment of finite groups generated by reflections. The approach is based on fundamental geometric considerations in Coxeter complexes, and emphasizes the intuitive geometric aspects of the theory of reflection groups. The exposition is directed at advanced undergraduates and first-year graduate students, and features a large number of exercises at various levels of difficulty.

From the Back Cover

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. Oftenconsidered 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.

Review Quotes

From the reviews:

"In Mirrors and Reflections by Alexandre Borovik (Univ. of Manchester, UK) and Anna Borovik, readers get the whole stage ... . Mastering this book not only gives undergraduates a taste of the mathematics of special objects, but prepares the way to various more important abstract theories. Thoughtfully illustrated, compact, leisurely, and unique in its coverage, this work is the way to learn this critical material. Summing Up: Highly recommended. Academics students, all levels, and professionals." (D. V. Feldman, Choice, Vol. 48 (1), September, 2010)

"A different approach to the study of reflection groups: an intuitive geometric approach, suitable for undergraduate students. ... the book provides the reader with the necessary geometric background. ... The book ends with a very interesting appendix on the 'forgotten art of blackboard drawing', where the authors give advice on making usable mathematical drawings. ... the authors believe that pictures are indispensable tools which facilitate mathematical work. They also give hints and solutions to selected exercises." (Maria Chlouveraki, Mathematical Reviews, Issue 2011 b)

"This is a nice booklet! ... the authors present an almost purely geometric approach to the theory of reflection groups which can be followed even by undergraduates. ... The authors attach great value to geometric intuition ... which makes the theory easily accessible. A very recommendable booklet!" (G. Kowol, Monatshefte für Mathematik, Vol. 162 (2), February, 2011)