On a theorem of R. Steinberg on rings of coinvariants

Abstract

Let p: G → GL (n, F) be a representation of a finite group G over the field F. Denote by F[V] the algebra of polynomial functions on the vector space V = F n . The group G acts on V and hence also on F[V]. The algebra of coinvariants is F[V] G = F[V] /h(G), where h(G) C F[V] is the ideal generated by all the homogeneous G-invariant forms of strictly positive degree. If the field F has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that F[V] G is a Poincare duality algebra if and only if G is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case n = 2 and give a counterexample in the modular case when n = 4.