Poincaré duality algebras and rings of coinvariants

Let ϱ : G ↪ G L ( n , F ) \varrho :G\hookrightarrow GL(n, \mathbb {F}) be a faithful representation of a finite group G G over the field F \mathbb {F} . Via ϱ \varrho the group G G acts on V = F n V=\mathbb {F} ^n and hence on the algebra F [ V ] {\mathbb {F}}[V] of homogenous polynomial functions on the vector space V V . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field F \mathbb {F} has characteristic 0 0 , then F [ V ] G {\mathbb {F}}[V] _G is a Poincaré duality algebra if and only if G G is a pseudoreflection group. The purpose of this note is to extend this result to the case | G | ∈ F × |G|\in \mathbb {F} ^{\times } (i.e. the order | G | |G| of G G is relatively prime to the characteristic of F \mathbb {F} ).