The parametrization of interior algebras

For indecomposable representations of a finite group G in characteristic p , the theory of vertices and sources introduced by J.A. Green in 1959 [G1] is a fundamental tool in modular representation theory. The vertex and source of an indecomposable module are (up to conjugation) two invariants of the module and only finitely many modules (up to isomorphism) have the same invariants. Thus a natural question is the existence of a third invariant which would distinguish further the modules and lead to a bijective parametrization of indecomposable modules using three invariants. When the base field of characteristic p is algebraically closed, the answer lies in the concept of multiplicity module introduced by Puig [P3], although the result is not explicitly stated in Puig’s work. It turns out that this third invariant is an indecomposable projective module over a twisted group algebra of the group N/P , where P is a vertex and N is the inertial subgroup of a source. This invariant has been used for the solution of some problems concerning almost split squences of group representations [P5], [T2]. More generally the same question arises for interior G-algebras but further complications appear (essentially because of the existence of outer automorphisms). The purpose of this paper is to give a complete description of the parametrization of primitive interior G-algebras with three invariants, including a description of the special case of indecomposable modules. Let O be a complete local commutative ring with residue field k of non-zero characteristic p (allowing the possibility O = k ). We assume that k is algebraically closed. By an O-algebra (or simply an algebra), we always mean an O-algebra which is finitely generated as an O-module. Let G be a finite group. Recall that an algebra A is called a G-algebra if it is endowed with an action of G by algebra automorphisms,