Rings of invariants for modular representations of elementary abelian p-groups

TL;DR

This paper studies the structure of invariant rings for modular representations of elementary abelian p-groups, providing explicit computations for low-dimensional cases and conjecturing a general property for three-dimensional representations.

Contribution

It offers explicit descriptions of invariant rings for low-dimensional modular representations and proposes a conjecture for all three-dimensional cases.

Findings

Invariant rings for 2D representations are generated by two algebraically independent elements.

Invariant rings of the symmetric square of a 2D representation are hypersurfaces.

Invariant rings for 3D representations of rank ≤3 are complete intersections.

Abstract

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three dimensional representation of an elementary abelian p-group is a complete intersection.