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Reflection groups and Coxeter groups. (English) Zbl 0725.20028
Cambridge Studies in Advanced Mathematics, 29. Cambridge etc.: Cambridge University Press. xii, 204 p. £ 25.00; $ 39.50 (1990).
This excellently written book is an advanced textbook on the theory of Coxeter groups. It pursues two objects. Firstly, it is an introduction to the book by {\it N. Bourbaki} on Lie groups and algebras (chapters 4-6, 1968; Zbl 0186.330). Secondly, it is an updating of the coverage. Correspondingly, the book is divided into two parts. The first part consists of 4 chapters: finite reflection groups, classification of finite reflection groups, polynomial invariants of finite reflection groups, affine reflection groups. The second part is inspired especially by the seminal work by {\it D. Kazhdan} and {\it G. Lusztig} [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)] on representations of Hecke algebras associated with Coxeter groups. This part consists of 4 chapters: Coxeter groups (here there is the Bruhat ordering), special cases, Hecke algebras and Kazhdan-Lusztig polynomials, complements (this chapter sketches a number of interesting complementary topics as well as connections with Lie theory). The book has an extensive bibliography on Coxeter groups and their applications.

20F55Reflection groups; Coxeter groups
20G05Representation theory of linear algebraic groups
20-02Research monographs (group theory)
51F15Reflection groups, reflection geometries
20H15Other geometric groups, including crystallographic groups