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Introduction to Complex Reflection Groups and Their Braid Groups

  • Michel Broué

Part of the Lecture Notes in Mathematics book series (LNM, volume 1988)

Table of contents

  1. Front Matter
    Pages I-XI
  2. Michel Broué
    Pages 1-9
  3. Michel Broué
    Pages 97-118
  4. Back Matter
    Pages 119-138

About this book

Introduction

Weyl groups are particular cases of complex reflection groups, i.e. finite subgroups of GLr(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.

Keywords

Braids Groups Invariants Reflections Representation theory

Authors and affiliations

  • Michel Broué
    • 1
  1. 1.Inst. Mathématiques JussieuUniversité Paris VIIParisFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-11175-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 2010
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-11174-7
  • Online ISBN 978-3-642-11175-4
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site