Title: On R. Steinberg's Theorem on Algebras of Coinvariants
Author Larry Smith (AG-Invariantentheorie}
Name of the .PDF file: steintri.pdf

Steinberg's Theorem on the coinvariant algebra $\C[V]_G$ of a complex
representation $\rho : G \hra \GL(n, \C)$ of a finite group $G$ says
that $\C[V]_G$ is a Poincar\'e duality algebra if and only if the
invariant algebra $\C[V]^G$ is a polynomial algebra. The extension of
this to the nonmodular case has been achieved in stages, the final
result being obtained by W.G. Dwyer and C.W.  Wilkerson. We show that
the main module theoretic tool they use extends to the following
characteristic free result: If $\F[V]_G$ is a Poincar\'e duality algebra
of formal dimension $d$\/, then $\F[V]^G$ is a polynomial algebra if and
only if $\Hom_{\F[V]^G} (\F[V], \F[V])$ contains a nonzero element of
degree $-d$\/.  In the nonmodular case an easy transfer argument then
recovers their extension of Steinberg's Theorem by means of some
representation theory.  Combined with some new results concerning the
$\Delta$ operators of Demazure, our characteristic free result yields
the following for reflection groups: A reflection group $G$ for which
$\F[V]_G$ is a Poincar\'e duality algebra in which the trivial
$G$-representation $1_G$ occurs only once as a subrepresentation has a
polynomial algebra for its invariant algebra $\F[V]^G$\/.