Birkhauser Frobenius Categories versus Brauer Blocks: The Grothendieck Group of the Frobenius Category of a Brauer Block: 274 (Progress in Mathematics, 274)

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 . Review 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) From the Back Cover 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 the 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 on the open question about the existence and the uniquenes of a perfect locality ― also called centric linking system in the literature. We have developed a long introductio