Reflection Groups and Coxeter Groups

Cambridge Studies in Advanced Mathematics #29

by James E. Humphreys

In this graduate textbook Professor Humphreys presents a concrete and up-to-date introduction to the theory of Coxeter groups. He assumes that the reader has a good knowledge of algebra, but otherwise the book is self contained. The first part is devoted to establishing concrete examples; the author begins by developing the most important facts about finite reflection groups and related geometry, and showing that such groups have a Coxeter representation. In the next chapter these groups are classified by Coxeter diagrams, and actual realizations of these groups are discussed. Chapter 3 discusses the polynomial invariants of finite reflection groups, and the first part ends with a description of the affine Weyl groups and the way they arise in Lie theory. The second part (which is logically independent of, but motivated by, the first) starts by developing the properties of the Coxeter groups. Chapter 6 shows how earlier examples and others fit into the general classification of Coxeter diagrams. Chapter 7 is based on the very important work of Kazhdan and Lusztig and the last chapter presents a number of miscellaneous topics of a combinatorial nature.

Genres: Mathematics

220 pages, Paperback

First published June 1, 1990

Reviews

[Mikhail Ignatev · Rating 4 out of 5]

Я прочёл первые три главы - и это действительно замечательно! Теорема Шевалле-Шепарда-Тодда изложена чрезвычайно понятно, и вообще всё очень просто и чётко написано. Так пока из прочитанных книг Хамфриса счёт 2:1 в его пользу (алгебры Ли и эта vs алгебраические группы).