The representation theory of fully group-graded algebras

Abstract

Let G be a finite group, k an algebraically closed field of characteristic p, and R a fully G-graded k-algebra of finite dimension. The most trivial example of such an algebra is kG. This paper attempts to generalize the p-modular representation theory of the group algebra kG to the groupgraded algebra R. It is shown that most of the theory is the same for R as for kG. Two areas in which the representation theory of R differs from that of kG are the dimensions of induced modules and the properties of the defect groups of the blocks of R. If H is a subgroup of G and V is an R,-module, then the dimension of VR is not necessarily divisible by the dimension of V. If B is a block of R and D is a defect group of B, then D is a p-group as would be expected. However, it is not necessarily true that D is a Sylow intersection, nor that D = O,(N,(D)). In the first section we define what it means for an algebra to have a G-graded structure. We show how to give such a structure to the algebra M,(k) of r by r matrices over the field k whenever IGl <r. A number of basic results about fully G-graded algebras are then given. The second section deals with R-modules. Proofs are given of the Mackey decomposition formula, the existence of vertices and sources, and the Green correspondence. These three results were already known to hold for R-for example, they are mentioned in [6]-but proofs have not previously been given in the literature. They are proved here to make this paper more or less self-contained. Many of the proofs of supporting results for these theorems will not appear, as they are easy generalizations of the known proofs for R = kG. The reader will often be referred to [ 1 ] for these proofs. The third and last section contains new results. The Burry-Carlson-Puig theorem is generalized, and a new definition of the defect groups of a block