Rings of invariants for the three dimensional modular representations of elementary abelian $p$-groups of rank four

TL;DR

This paper proves that the rings of invariants for three-dimensional modular representations of elementary abelian p-groups of rank four are complete intersections with small embedding dimension, confirming previous conjectures.

Contribution

It establishes that these invariant rings are complete intersections with embedding dimension at most five, validating conjectures by Campbell, Shank, and Wehlau.

Findings

Rings of invariants are complete intersections

Embedding dimension is at most five

Conjectures of Campbell, Shank, and Wehlau are confirmed

Abstract

We show that the rings of invariants for the three dimensional modular representations of an elementary abelian $p$-group of rank four are complete intersections with embedding dimension at most five. Our results confirm the conjectures of Campbell, Shank and Wehlau.