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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 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 D. Kazhdan and 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.

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20G05 Representation theory for linear algebraic groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
51F15 Reflection groups, reflection geometries
20H15 Other geometric groups, including crystallographic groups