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Zbl 0725.20028
Humphreys, James E.
Reflection groups and Coxeter groups. (English)
[B] Cambridge Studies in Advanced Mathematics, 29. Cambridge etc.: Cambridge University Press. xii, 204 p. \sterling 25.00; \$ 39.50 (1990). ISBN 0-521-37510-X

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. \par The first part consists of 4 chapters: finite reflection groups, classification of finite reflection groups, polynomial invariants of finite reflection groups, affine reflection groups. \par 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.
[V.F.Molchanov (Tambov)]

MSC 2000:: *20F55 Coxeter groups
20G05 Representation theory of linear algebraic groups
20-02 Research monographs (group theory)
51F15 Reflection groups and geometries
20H15 Other geometric groups, including crystallographic groups

Keywords: textbook; Coxeter groups; finite reflection groups; polynomial invariants; affine reflection groups; representations; Hecke algebras; Bruhat ordering; Kazhdan-Lusztig polynomials; bibliography

Citations: Zbl 0186.330; Zbl 0499.20035

Cited in: Zbl 1191.05095 Zbl 0990.05139 Zbl 0933.20025 Zbl 0870.20011 Zbl 0768.20016 Zbl 0742.20040

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	Scientific prize winners of the ICM 2010
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	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

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