Frobenius Categories Versus Brauer Blocks: The Grothendieck Group of the Frobenius Category of a Brauer Block

This book contributes to important questions in modern representation theory of finite groups. 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 either of finite groups or of blocks, and for the purpose of an eventual classification. On the other hand, it gives the application of the abstract setting to the blocks. In particular, it develops a framework for a deeper understanding of one of the central open problems in representation theory, known as Alperin 's Weight Conjecture (AWC). One of the main results of the book is a reduction theorem of the author 's own form of AWC to quasi-simple groups. Although it is a research monograph rather than a textbook, all the arguments are widely developed to make it accessible to interested graduate students. A long introduction gives a motivating insight to each chapter and provides a basic guideline.

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 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)