\author{David Benson}
\address{Department of Mathematics, University of Georgia}
\email{/b$\backslash$e/n$\backslash$s/o$\backslash$n/d%
$\backslash$j/ (without the slashes) at math dot uga dot edu}
\thanks{The research of the first author was partly supported by 
NSF grant DMS-0242909}
\author{Peter Webb}
\address{School of Mathematics, University of Minnesota, Minneapolis MN 55455}
\email{/w$\backslash$e/b$\backslash$b/
(without the slashes) at math dot umn dot edu}
\title{Unique factorization in invariant power series rings}
\subjclass[2000]{Primary 13A50; Secondary 20C20}
\keywords{invariant theory, symmetric powers, unique factorization,
modular representation}

\begin{abstract}
Let $G$ be a finite group, $k$ a perfect field of characteristic $p$,
and $V$ a finite dimensional $kG$-module.
We let $G$ act on the power series $k[[V]]$ by linear substitutions
and address the question of when the invariant
power series $k[[V]]^G$ form a unique factorization domain. We prove
that for a permutation module for a $p$-group, the answer is always
positive. On the other hand, if $G$ is a cyclic group of order $p$
and $V$ is an indecomposable $kG$-module of dimension $r$ with
$1\le r\le p$,  we show that the invariant
power series form a unique factorization domain if and only if 
$r$ is equal to $1$, $2$, $p-1$ or $p$. This contradicts a conjecture
of Peskin. We note also that
a theorem of Nakajima completely answers the question of when the
invariant polynomial functions $k[V]^G$ form a unique factorization domain. 
\end{abstract}