A filtration of the modular representation functor

Abstract

Let F and K be algebraically closed fields of characteristics p > 0 and 0, respectively. For any finite group G we denote by K R F ( G ) = K ⊗ Z G 0 ( F G ) the modular representation algebra of G over K where G 0 ( F G ) is the Grothendieck group of finitely generated F G -modules with respect to exact sequences. The usual operations induction, inflation, restriction, and transport of structure with a group isomorphism between the finitely generated modules of group algebras over F induce maps between modular representation algebras making K R F an inflation functor. We show that the composition factors of K R F are precisely the simple inflation functors S C , V i where C ranges over all nonisomorphic cyclic p ′ -groups and V ranges over all nonisomorphic simple K Out ( C ) -modules. Moreover each composition factor has multiplicity 1. We also give a filtration of K R F .