Frobenius Categories versus Brauer Blocks

The Grothendieck Group of the Frobenius Category of a Brauer Block

By: Lluís Puig

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Hardcover | 13 March 2009

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I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem a?rming that, for a primep and a ?nite groupG, if the quotient of the normalizer by the centralizer of anyp-subgroup ofG is a p-group then, up to a normal subgroup of order prime top,G is ap-group. Ofcourse,itwouldbeananachronismtopretendthatFrobenius,when doing this theorem, was thinking the category - notedF in the sequel - G where the objects are thep-subgroups ofG and the morphisms are the group homomorphisms between them which are induced by theG-conjugation. Yet Frobenius' hypothesis is truly meaningful in this category. I2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotencyoftheso-called Frobeniuskernelofa FrobeniusgroupGwithar- ments - at that time completely new - which might be rewritten in terms ofF ; indeed, some time later, following these kind of arguments, George G Glauberman [27] proved that, under some - rather strong - hypothesis onG, the normalizerNofasuitablenontrivial p-subgroup ofG controls fusion inG, which amounts to saying that the inclusionN?G induces an ? equivalence of categoriesF =F .

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From the reviews: "The concept of Frobenius category (also called saturated fusion system) has been developed by Lluis Puig many years ago. ! This volume is a research monograph containing many original results which appear for the first time in print ! . The book ends with Appendix developing the author's point of view on cohomology of small categories. ! the book is a necessary tool for any researcher in the field." (Andrei Marcus, Studia Universitatis Babes-Bolyai, Mathematica, Vol. LV (4), December, 2010) "In recent years a new concept--called Frobenius category by Puig and fusion system by others--has become very successful in the representation theory of finite groups. ! the book gives an account of the early developments and definitions presented by Puig, as they were given in his first manuscripts. This certainly is of great importance because it makes the first, original definitions of Puig accessible to a wide audience." (Alexander Zimmermann, Mathematical Reviews, Issue 2010 j)

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Introduction	p. 1
General notation and quoted results	p. 15
Frobenius P-categories: the first definition	p. 27
The Frobenius P-category of a block	p. 39
Nilcentralized, selfcentralizing and intersected objects in Frobenius P-categories	p. 47
Alperin fusions in Frobenius P-categories	p. 57
Exterior quotient of a Frobenius P-category over the selfcentralizing objects	p. 73
Nilcentralized and selfcentralizing Brauer pairs in blocks	p. 93
Decompositions for Dade P-algebras	p. 103
Polarizations for Dade P-algebras	p. 117
A gluing theorem for Dade P-algebras	p. 137
The nilcentralized chain k*-functor of a block	p. 151
Quotients and normal subcategories in Frobenius P-categories	p. 179
The hyperfocal subcategory of a Frobenius P-category	p. 195
The Grothendieck groups of a Frobenius P-category	p. 211
Reduction results for Grothendieck groups	p. 241
The local-global question: reduction to the simple groups	p. 287
Localities associated with a Frobenius P-category	p. 319
The localizers in a Frobenius P-category	p. 333
Solvability for Frobenius P-categories	p. 361
A perfect F-locality from a perfect Fsc-locality	p. 369
Frobenius P-categories: the second definition	p. 389
The basic F-locality	p. 397
Narrowing the basic F^sc-locality	p. 409
Looking for a perfect F^sc-locality	p. 437
Appendix	p. 449
References	p. 489
Index	p. 493
Table of Contents provided by Ingram. All Rights Reserved.

Frobenius Categories versus Brauer Blocks

The Grothendieck Group of the Frobenius Category of a Brauer Block

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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