Symmetric powers and modular invariants of elementary abelian p-groups

Abstract

Let $E$ be a elementary abelian $p$-group of order $q=p^n$. Let $W$ be a faithful indecomposable representation of $E$ with dimension 2 over a field $k$ of characteristic $p$, and let $V= S^m(W)$ with $m<q$. We prove that the rings of invariants $k[V]^E$ are generated by elements of degree at most $q$ and relative transfers. This extends recent work of Wehlau on modular invariants of cyclic groups of order $p$. If $m<p$ we prove that $k[V]^E$ is generated by invariants of degree at most $2q-3$, extending a result of Fleischmann, Sezer, Shank and Woodcock for cyclic groups of order $p$. Our methods are primarily representation-theoretic, and along the way we prove that for any $d<q$ with $d+m \geq q$, $S^d(V^*)$ is projective relative to the set of subgroups of $E$ with order at most $m$, and that the sequence $S^d(V^*)_{d \in \mathbb{N}}$ is periodic with period $q$, modulo summands which are projective relative to the same set of subgroups. These results extend results of Almkvist and Fossum on cyclic groups of prime order.