On a theorem of R. Steinberg on rings of coinvariants

Let $\rho: G\hookrightarrow\mathrm{GL}(n,\mathbb{F})$ be a representation of a finite group $G$ over the field $\mathbb{F}$. Denote by $\mathbb{F}[V]$ the algebra of polynomial functions on the vector space $V=\mathbb{F}^n$. The group $G$ acts on $V$ and hence also on $\mathbb{F}[V]$. The algebra of coinvariants is $\mathbb{F}[V]_G=\mathbb{F}[V]/\mathfrak{h}(G)$, where $\mathfrak{h}(G) \subset \mathbb{F}[V]$ is the ideal generated by all the homogeneous $G$-invariant forms of strictly positive degree. If the field $\mathbb{F}$ has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that $\mathbb{F}[V]_G$ is a Poincaré 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 $n=4$.