Blocks with Cyclic Defect Groups

Abstract

This chapter gives the theory of blocks with cyclic defect groups. We start by describing the Brauer tree, a combinatorial object that encodes first the decomposition matrix of the block, then Ext1 between simple modules in the block, and indeed the Morita equivalence type of the block (but not the source algebra). We then construct Brauer tree algebras, which are basic algebras that are Morita equivalent to blocks with cyclic defect groups. After describing the indecomposable modules for such a block, we turn to the classification of the possible Brauer trees, using the classification of finite simple groups.