Introduction to Complex Reflection Groups and Their Braid Groups

✍ Scribed by Michel Broué (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2010
• Tongue: English
• Leaves: 155
• Series: Lecture Notes in Mathematics 1988
• Edition: 1
• Category: Library

✦ Synopsis

Weyl groups are particular cases of complex reflection groups, i.e. finite subgroups of GL_r(C) generated by (pseudo)reflections. These are groups whose polynomial ring of invariants is a polynomial algebra.

It has recently been discovered that complex reflection groups play a key role in the theory of finite reductive groups, giving rise as they do to braid groups and generalized Hecke algebras which govern the representation theory of finite reductive groups. It is now also broadly agreed upon that many of the known properties of Weyl groups can be generalized to complex reflection groups. The purpose of this work is to present a fairly extensive treatment of many basic properties of complex reflection groups (characterization, Steinberg theorem, Gutkin-Opdam matrices, Solomon theorem and applications, etc.) including the basic findings of Springer theory on eigenspaces. In doing so, we also introduce basic definitions and properties of the associated braid groups, as well as a quick introduction to Bessis' lifting of Springer theory to braid groups.

✦ Table of Contents

Front Matter....Pages I-XI
Preliminaries....Pages 1-9
Prerequisites and Complements in Commutative Algebra....Pages 11-33
Polynomial Invariants of Finite Linear Groups....Pages 35-56
Finite Reflection Groups in Characteristic Zero....Pages 57-96
Eigenspaces and Regular Elements....Pages 97-118
Back Matter....Pages 119-138

✦ Subjects

Group Theory and Generalizations; Commutative Rings and Algebras; Associative Rings and Algebras; Algebraic Topology