Symmetric powers and modular invariants of elementary abelian p-groups

Let E be an elementary abelian p-group of order q=pn. Let W be a faithful indecomposable representation of E with dimension 2 over a field k of characteristic p, and let V=Sm(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 [18] 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 [6] 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, ⁎Sd(V⁎) is projective relative to the set of subgroups of E with order ≤m, and that the sequence ⁎Sd(V⁎)d≥0 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 [1] on cyclic groups of prime order.