Mirrors and Reflections

This graduate/advanced undergraduate textbook contains 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. Key features include: many important concepts in the proofs are illustrated in simple drawings, which give easy access to the theory; a large number of exercises at various levels of difficulty; some Euclidean geometry is included along with the theory of convex polyhedra; no prerequisites are necessary beyond the basic concepts of linear algebra and group theory; and a good index and bibliography The exposition is directed at advanced undergraduates and first-year graduate students.

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)