Poincaré duality algebras and rings of coinvariants

Abstract:

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