##
**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.” Paris: Hermann (1968; Zbl 0186.33001). 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.

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.

Reviewer: Vladimir F. Molchanov (Tambov)

### MathOverflow Questions:

How do I stop worrying about root systems and decomposition theorems (for reductive groups)?### MSC:

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 |

### Keywords:

textbook; Coxeter groups; finite reflection groups; polynomial invariants; affine reflection groups; representations; Hecke algebras; Bruhat ordering; Kazhdan-Lusztig polynomials; bibliography
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\textit{J. E. Humphreys}, Reflection groups and Coxeter groups. Cambridge etc.: Cambridge University Press (1990; Zbl 0725.20028)