Introduction to Complex Reflection Groups and Their Braid Groups

ISBN-13: 9783642111747 / Angielski / Miękka / 2010 / 144 str.

Michel Broue

Opis

toComplexRe ectionGroups and Their Braid Groups 123 Michel Broue Universite Paris Diderot Paris 7 UFR de Mathematiques 175 Rue du Chevaleret 75013 Paris France broue@math. jussieu. fr ISBN: 978-3-642-11174-7 e-ISBN: 978-3-642-11175-4 DOI: 10. 1007/978-3-642-11175-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009943837 Mathematics Subject Classi cation (2000): 20, 13, 16, 55 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speci cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on micro lm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of aspeci c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover illustration: c Anouk Grinberg Cover design: SPi Publisher Services Printed on acid-free paper springer. com Preface Weyl groups are ?nite groups acting as re?ection groups on rational vector spaces. It iswellknownthat theserationalre?ectiongroupsappearas ske- tons of many important mathematical objects: algebraic groups, Hecke algebras, Artin Tits braid groups, etc."

Recenzje

From the reviews:

"It is the aim of the present notes to give an introduction to complex reflection groups in such a way as to lead the reader to the very forefront of research in this area. ... it is a useful addition to the growing literature on complex reflection groups." (Stephen P. Humphries, Mathematical Reviews, Issue 2011 d)

Spis treści

Preliminaries.- Prerequisites and Complements in Commutative Algebra.- Polynomial Invariants of Finite Linear Groups.- Finite Reflection Groups in Characteristic Zero.- Eigenspaces and Regular Elements.

Blurb

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.