Reflection Groups and Coxeter Groups

A self-contained graduate textbook introducing the basic theory of Coxeter groups.

James E. Humphreys (Author)

9780521436137, Cambridge University Press

Paperback, published 1 October 1992

220 pages
22.9 x 15.6 x 1.5 cm, 0.35 kg

"...a useful book. The style is informal and the arguments are clear." Louis Solomon, Mathematical Reviews

This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.

Part I. Finite and Affine Reflection Groups: 1. Finite reflection groups
2. Classification of finite reflection groups
3. Polynomial invariants of finite reflection groups
4. Affine reflection groups
Part II. General Theory of Coxeter Groups: 5. Coxeter groups
6. Special case
7. Hecke algebras and Kazhdan–Lusztig polynomials
8. Complements
Bibliography.

Subject Areas: Groups & group theory [PBG]