These notes are based on a lecture given at the Workshop preceding the International Conference of Representations of Algebras 1990 at Tsukuba, Japan. The aim is to give a survey over recent progress in the study of blocks of tame representation type. These are the 2-blocks whose defect groups are dihedral or semidihedral or quaternion. Over the last few years, a range of new results on a class of algebras including such blocks have been obtained. The algebras are essentially defined in terms of their stable Auslander-Reiten quivers, and it has been proved that any such algebra is Morita equivalent to one of the algebras in a small list which is explicitly given by quivers and relations. In particular, this describes tame blocks; and it allows to extend classical results on the arithmetic properties of such blocks. We also show how the methods may be applied to the study of blocks of finite type. More generally, we give an outline of the classification of all symmetric algebras which are stably equivalent to Nakayama algebras. This is a different proof of the well-known theorem of Gabriel and Riedtmann.

Introduction

Let G be a finite group and K a field of characteristic p; we assume that K is algebraically closed. One approach to modular group representation theory is to study KG, as an algebra, and its module category. The group algebra KG is a direct sum of blocks; where a block is an indecomposable direct summand of KG, as an algebra.