A MODULAR ANALOG OF A THEOREM OF R. STEINBERG ON COINVARIANTS OF COMPLEX PSEUDOREFLECTION GROUPS

Abstract

Let $\rho:G\hookrightarrow {\rm GL}(n, {\bb F})$ be a representation of a finite group over the field ${\bb F}, V = {\bb F}^n$ the corresponding $G$-module, and ${\bb F}[V\hspace*{1.5pt}]$ the algebra of polynomial functions on $V$. The action of $G$ on $V$ extends to ${\bb F}[V\hspace*{1.5pt}]$, and ${\bb F}[V\hspace*{1.5pt}]^G$, respectively ${\bb F}[V\hspace*{1.5pt}]_G$, denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when ${\bb F} = {\bb C}$, dim$_{\bb C}$ (Tot$({\bb C}[V\hspace*{1.5pt}]_G))=|G|$ if and only if $G$ is a complex reflection group. Here Tot$({\bb F}[V\hspace*{1.5pt}]_G)$ denotes the direct sum of all the homogeneous components of the graded algebra ${\bb F}[V\hspace*{1.5pt}]_G$ and $|G|$ is the order of $G$. Chevalley's theorem tells us that the ring of invariants of a complex pseudoreflection representation $G\hookrightarrow {\rm GL}(n, {\bb C})$ is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim$_{\bb F}$(Tot$({\bb C}[V\hspace*{1.5pt}]_G) = |G|$ if and only if ${\bb C}[V\hspace*{1.5pt}]^G$ is a polynomial algebra. The purpose of this note is to show that the two conditions

(i) dim$_{\bb F}$(Tot(${\bb F}[V\hspace*{1.5pt}]_G))\,{=}\,|G|$,

(ii) ${\bb F}[V\hspace*{1.5pt}]^G$ is a polynomial algebra

are equivalent regardless of the ground field; in particular in the modular case.