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On a theorem of R. Steinberg on rings of coinvariants

Authors:	Larry Smith

Institution:	AG-Invariantentheorie, Mathematisches Institut der Universität, Bunsenstraße 3-5, D37073 Göttingen, Federal Republic of Germany

Abstract:	Let $\rho: G\hookrightarrow\mathrm{GL}(n,\mathbb{F})$ be a representation of a finite group 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 acts on 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 -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 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 and give a counterexample in the modular case when .

Keywords:	Invariant theory pseudoreflection groups

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