The varieties for some Specht modules
by Kay Jin Lim


\begin{abstract} J. Carlson introduced the cohomological and rank variety
for a module over a finite group algebra. We give a general form for
the largest component of the variety for the Specht module for the
partition $(p^p)$ of $p^2$ restricted to a maximal elementary
abelian $p$-subgroup of rank $p$. We determine the varieties of a
large class of Specht modules corresponding to $p$-regular
partitions. To any partition $\mu$ of $np$ of not more than $p$
parts with empty $p$-core we associate a unique partition
$\Phi(\mu)$ of $np$, where the rank variety of the restricted Specht
module $S^\mu{\downarrow_{E_n}}$ to a maximal elementary abelian
$p$-subgroup $E_n$ of rank $n$ is $V_{E_n}^\sharp(k)$ if and only if
$V_{E_n}^\sharp(S^{\Phi(\mu)})=V_{E_n}^\sharp(k)$. In some cases
where $\Phi(\mu)$ is a 2-part partition, we show that the rank
variety $V_{E_n}^\sharp(S^\mu)$ is $V^\sharp_{E_n}(k)$. In
particular, the complexity of the Specht module $S^\mu$ is $n$.
\end{abstract}