Blocks and Families for Cyclotomic Hecke Algebras
The definition of Rouquier for the families of characters introduced by Lusztig for Weyl groups in terms of blocks of the Hecke algebras has made possible the generalization of this notion to the case of complex reflection groups. The aim of this book is
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1981
Maria Chlouveraki
Blocks and Families for Cyclotomic Hecke Algebras
123
Maria Chlouveraki École Polytechnique Fédérale de Lausanne SB IGAT CTG Bâtiment BCH 3105 1015 Lausanne Switzerland [email protected]
ISBN: 978-3-642-03063-5 DOI: 10.1007/978-3-642-03064-2
e-ISBN: 978-3-642-03064-2
Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009931326 Mathematics Subject Classification (2000): 20C08, 20F55, 20F36 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
This book contains a thorough study of symmetric algebras, covering topics such as block theory, representation theory and Clifford theory. It can also serve as an introduction to the Hecke algebras of complex reflection groups. Its aim is the study of the blocks and the determination of the families of characters of the cyclotomic Hecke algebras associated to complex reflection groups. I would like to thank my thesis advisor, Michel Brou´e, for his advice. These Springer Lecture Notes were, after all, his idea. I am grateful to Jean Michel for his help with the implementation and presentation of the programming part. I would like to thank Gunter Malle for his suggestion that I generalize my results on Hecke algebras, which led to the notion of “essential algebras”. I also express my thanks to C´edric Bonnaf´e, Meinolf Geck, Nicolas Jacon, Rapha¨el Rouquier and Jacques Th´evenaz for their useful comments. Finally, I thank Thanos Tsouanas for copy-editing this manuscript.
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Introduction
The finite groups of matrices with coefficients in Q generated by reflections, known as Weyl groups, are a fundamental building block in the classification of semisimple complex Lie algebras and Lie groups, as well as semisimple algebraic groups over arbitrary algebraically closed fields. They are also a foundation for many other significant mathematical theories, including braid groups and Hecke algebras. The Weyl groups are particular cases of complex reflection groups, finite groups of matrices with coefficients in a finite abelian extension of Q generated by “pseudo-reflectio
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