On a theorem of R. Steinberg on rings of coinvariants

Abstract:

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 $n=4$.