Reflection Groups and Coxeter Groups / Edition 1

by James E. Humphreys

Overview

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.

Product Details

ISBN-13:	9780521436137
Publisher:	Cambridge University Press
Publication date:	10/01/1992
Series:	Cambridge Studies in Advanced Mathematics , #29
Edition description:	Reprint
Pages:	220
Product dimensions:	6.14(w) x 9.02(h) x 0.59(d)

About the Author

James E. Humphreys was born in Erie, Pennsylvania, and received his A.B. from Oberlin College, 1961, and his Ph.D. from Yale University, 1966. He has taught at the University of Oregon, Courant Institute (NYU), and the University of Massachusetts at Amherst (now retired). He visits IAS Princeton, Rutgers. He is the author of several graduate texts and monographs.

Table of Contents

Part I. Finite and Affine Reflection Groups: 1. Finite reflection groups; 2. Classification of finite reflection groups; 3. Polynomial invariants of finite reflection groups; 4. Affine reflection groups; Part II. General Theory of Coxeter Groups: 5. Coxeter groups; 6. Special case; 7. Hecke algebras and Kazhdan–Lusztig polynomials; 8. Complements; Bibliography.

Editorial Reviews

"This is a book which can be recommended to both beginners and more experienced workers with an interest in Coxeter groups. In common with all of Humphrey's books, it is written in a clear and helpful expository style, and so gives an excellent introduction to the subject. At the same time the material dealing with recent developments such as Kazhdan-Lusztig theory will be most useful to specialists in the area." Bulletin of the London Mathematical Society

"...a useful book. The style is informal and the arguments are clear." Louis Solomon, Mathematical Reviews

From the Publisher