Symmetric powers and modular invariants of elementary abelian p-groups

TL;DR

This paper investigates the structure of invariant rings of elementary abelian p-groups acting on symmetric powers of certain representations, establishing bounds on generators and revealing periodicity and projectivity properties.

Contribution

It extends known results on modular invariants from cyclic groups to elementary abelian p-groups using representation-theoretic methods.

Findings

Invariant rings are generated by elements of degree at most q and relative transfers.

For m < p, the invariant ring is generated by invariants of degree at most 2q-3.

The sequence of symmetric powers exhibits periodicity with period q, modulo certain projective summands.

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.