Some formulae relating modular representations of elementary abelian $p$-groups

Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that $V_2 \otimes V_i \cong V_{i+1} \oplus V_{i-1}$ provided $i$ is not divisible by $p$, and that $V_2 \otimes V_p$ is indecomposable if $n>1$. Our results generalise results of Almkvist and Fossum for representations of cyclic groups of order $p$. We show how our results give formulae for the direct sum decomposition of $V_i \otimes V_j$ for $i<p$ and $j<j$ modulo summands projective to $\bigoplus_{r=0}^{p-1}V_{rp}$ and conjecture that these formulae extend to the case $i<q$ and $j<q$. We provide some evidence for our conjecture.