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.