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 $d+m\ge q$, $S^d(V^{\ast })$ is projective relative to the set of subgroups of $E$ with order at most $m$, and that the sequence $S^d(V^{\ast }{)}_{d\in \mathbb{N}}$ 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.

21 pages. New version has more readable proposition 3.7