Frobenius categories versus Brauer blocks : the Grothendieck group of the Frobenius category of a Brauer block / Lluís Puig.

Format:

Book

Author/Creator:

Puig, Luis.

Series:

Progress in mathematics (Boston, Mass.) ; v. 274.

Progress in mathematics ; 274

Language:

English

Subjects (All):

Frobenius algebras.

Frobenius groups.

Grothendieck groups.

Brauer groups.

Finite groups.

Representations of groups.

Physical Description:

498 pages : illustrations ; 24 cm.

Other Title:

Grothendieck group of the Frobenius category of a Brauer block

Place of Publication:

Basel, Switzerland ; Boston, Mass. : Birkhäuser, [2009]

Summary:

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 one hand, it introduces and develops the abstract setting of the Frobenius categories - also called the saturated fusion systems in the literature - created by the author fifteen years ago for a better understanding of what was loosely called the local theory of a finite group around a prime number p or, later, around a Brauer block, and for the purpose of an eventual classification - a reasonable concept of simple Frobenius category arises.

On the other hand, the book develops this abstract setting in parallel with its application to the Brauer blocks, giving the detailed translation of any abstract concept in the particular context of the blocks. One of the new features in this direction is a framework for a deeper understanding of one of the central open problems in modular representation theory, known as Alperin's Weight Conjecture (AWC). Actually, this new framework suggests a more general form of AWC, and a significant result of the book is a reduction theorem of this form of AWC to quasi-simple groups.

Although this book is a research monograph, all the arguments are widely developed to make it accessible to interested graduate students and, at the same time, to put them on the verge of the research on this new subject: the third part of the book on the localities associated to a Frobenius category gives some insight into the open question about the existence and the uniquenes of a perfect locality - also called centric linking system in the literature. The introduction provides a comprehensive guideline for the reader. A systematic appendix on the cohomology of categories completes the book.

Contents:

1 General notation and quoted results 15

2 Frobenius P-categories: the first definition 27

3 The Frobenius P-category of a block 39

4 Nilcentralized, selfcentralizing and intersected objects in Frobenius P-categories 47

5 Alperin fusions in Frobenius P-categories 57

6 Exterior quotient of a Frobenius P-category over the selfcentralizing objects 73

7 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 groups 241

16 The local-global question: reduction to the simple groups 287

17 Localities associated with a Frobenius P-category 319

18 The localizers in a Frobenius P-category 333

19 Solvability for Frobenius P-categories 361

20 A perfect F-locality from a perfect F<sc>sc</sc>-locality 369

21 Frobenius P-categories: the second definition 389

22 The basic F-locality 397

23 Narrowing the basic F sc -locality 409

24 Looking for a perfect F sc -locality 437.

Notes:

Includes bibliographical references (pages [489]-492) and index.

ISBN:

9783764399979

376439997X

OCLC:

297148429