Frobenius Categories versus Brauer Blocks The Grothendieck Group of
This book contributes to important questions in the representation theory of finite groups over fields of positive characteristic — an area of research initiated by Richard Brauer sixty years ago with the introduction of the blocks of characters. On the o
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Series Editors H. Bass J. Oesterlé A. Weinstein
Lluís Puig
Frobenius Categories versus Brauer Blocks The Grothendieck Group of the Frobenius Category of a Brauer Block
Birkhäuser Basel · Boston · Berlin
Author: Lluís Puig CNRS, Institut de Mathématiques de Jussieu Université Denis Diderot (Paris VII) 175, Rue du Chevaleret 75013 Paris France e-mail: [email protected]
2000 Mathematics Subject Classification 20C11 Library of Congress Control Number: 2009921943 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN 978-3-7643-9997-9 Birkhäuser Verlag AG, Basel ∙ Boston ∙ Berlin 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, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-9997-9
e-ISBN 978-3-7643-9998-6
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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 General notation and quoted results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Frobenius P -categories: the first definition . . . . . . . . . . . . . . . . . . . . . . . . 27 3 The Frobenius P -category of a block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Nilcentralized, selfcentralizing and intersected objects in Frobenius P -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Alperin fusions in Frobenius P -categories . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exterior quotient of a Frobenius P -category over the selfcentralizing objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
39 47 57 73
Nilcentralized and selfcentralizing Brauer pairs in blocks . . . . . . . . . . . 93
8 Decompositions for Dade P -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9 Polarizations for Dade P -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 10 A gluing theorem for Dade P -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11 The nilcentralized chain k*-functor of a block . . . . . . . . . . . . . . . . . . . . . . 151 12 Quotients and normal subcategories in Frobenius P -categories . . . . . 179 13 The hyperfocal subcategory of a Frobenius P -category . . . . . . . . . . . . .195 14 The Grothendieck groups of a Frobenius P -category . . . . . . . . . . . . . . . 211 15 Reduction results for Grothendieck grou
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