Rings of invariants for three dimensional modular representations

TL;DR

This paper computes the rings of invariants for 3-dimensional modular representations of elementary abelian p-groups, confirming a conjecture about their structure as complete intersection rings with specific embedding dimensions.

Contribution

It extends previous results by proving the conjecture for all r, describing the invariant rings as complete intersections with a precise embedding dimension.

Findings

Rings of invariants are complete intersections.

Embedding dimension is loor;r/2 + 3.

Conjecture by Campbell, Shank, and Wehlau is proven for all r.

Abstract

Let $p>3$ be a prime number. We compute the rings of invariants of the elementary abelian $p$-group $(\mathbb Z/p\mathbb Z)^r$ for $3$-dimensional generic representations. Furthermore we show that these rings of invariants are complete intersections rings with embedding dimension $\lceil r/2\rceil +3$. This proves a conjecture of Campbell, Shank and Wehlau in [CSW], which they proved for $r=3$, and later Pierron and Shank proved it for $r=4$.