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On a theorem of R. Steinberg on rings of coinvariants

Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 131 (2003), 1043-1048
MSC (1991): Primary 13A50; Secondary 20F55
Published electronically: July 26, 2002
MathSciNet review: 1948093
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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 $n=4$.

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

Keywords: Invariant theory, pseudoreflection groups
Received by editor(s): April 30, 2001
Received by editor(s) in revised form: November 5, 2001
Published electronically: July 26, 2002
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2002 American Mathematical Society