A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

Abstract

Let G be a finite group and k a field of characteristic p > 0. Maschke’s theorem says that the group algebra KG is semisimple if and only if p does not divide the group order, ) G I. We refer to RG as a modular group algebra when p divides / G I. In this case the representation theory of KG presents many unsolved problems. Of special interest is the structure of the indecomposable modules for kG, even when G is an abelian non-cyclic p-group [ 1, p. 431; 7, Introduction]. In this paper we begin a study of modular group algebras and their representations using the deformation theory of Gerstenhaber [4] and NijenhuisRichardson [6]. A major intuition in the program derives from the correlation for k-algebras between deformability and the failure of semi-simplicity. In Section 1 we show how, by using deformations, Maschke’s theorem may be recovered for abelian modular group algebras. One might conjecture furthermore that any modular group algebra deforms into a predictable semisimple algebra. Some non-abelian examples in Section 2 support this conjecture. There we exhibit deformations of certain group algebras of the dihedral and symmetric groups into the “expected” semisimple algebras. In Section 3 we propose further questions and avenues for investigation.