Higman´s criterion revisited

Abstract

Let V be a finite-dimensional k-vector space endowed with an action of a finite group G and hence endowed with the structure of a kG-module. According to Higman's criterion, that module is projective if and only if there exists a k-linear endomorphism of V such that gG g g -1 = Id V . This paper presents a generalization of that criterion to the more general context of symmetric algebras. Having in mind some functors used in the representation theory of finite reductive groups, we then generalize the appropriate version of Higman's criterion applied to relative projectivity to a situation where induction-restriction are replaced by functors induced by pairs of "exact bimodules".