Symmetric powers and modular invariants of elementary abelian p-groups

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 ≤ 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 ≤ 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 ≥ q, S^d (V^∗) is projective relative to the set of subgroups of E with order ≤ m, and that the sequence S^d (V^∗) 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.