Endotrivial modules and the Auslander-Reiten quiver

Abstract

As Dade stated it in [8]: “There are just too many modules over p-groups!” More precisely, if P is a p-group and R a suitable commutative valuation ring, then almost always the group algebra RP is of wild representation type and there is no classification of all its indecomposable modules. Searching for a useful family of modules that could still be classified Dade was led to study endopermutation RP-modules, i.e. RP-lattices whose R-endomorphisms form a permutation RP-module. These modules play an important rôle for example in the study of sources of simple modules. The isomorphism classes of indecomposable endopermutation RP-modules with vertex P form an abelian group under a multiplication induced by tensor product. For abelian P, Dade determined the structure of this group [8]; for non-abelian P Puig [11] proved at least that this group is finitely generated.

This work has been supported by the Deutsche Forschungsgemeinschaft