Symmetric powers and modular invariants of elementary abelian p-groups

Let E be an 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 [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, S-d(V*) is projective relative to the set of subgroups of E with order <= m, and that the sequence S-d(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. (C) 2017 Elsevier Inc. All rights reserved.