\title{Schur--Weyl duality over finite fields}
\author{David Benson}
\address{Department of Mathematics, University of Aberdeen, 
Aberdeen AB24 3EA, Scotland}
\author{Stephen Doty}
\address{Department of Mathematics, Loyola University Chicago, 
Chicago IL 60626, USA}
\thanks{Both authors would like to thank MSRI for its hospitality 
while this work was in progress}
\begin{document}

\begin{abstract}
We prove a version of Schur--Weyl duality over finite fields.  We
prove that for any field $k$, if $k$ has more than $r$ elements, then
Schur--Weyl duality holds for the $r$th tensor power of a finite
dimensional vector space $V$.  Moreover, if $\dim V$ is at least $r+1$
then the natural map $k\Sym_r \to \End_{GL(V)}(V^{\otimes r})$ is an
isomorphism; this isomorphism may fail if $\dim_k(V)$ is not strictly
larger than $r$.
\end{abstract}