Rings of invariants of three-dimensional upper-triangular representations of finite groups

TL;DR

This paper characterizes when the ring of invariants for three-dimensional upper-triangular group representations over fields of odd characteristic is polynomial, based on properties of the unipotent subgroup.

Contribution

It provides a complete criterion linking the polynomiality of invariants to the generation by pseudoreflections or absence of transvections in the unipotent subgroup.

Findings

Ring of invariants is polynomial iff unipotent subgroup is generated by pseudoreflections or lacks transvections.

Establishes a clear criterion for polynomial invariants in three-dimensional upper-triangular groups.

Focuses on groups over fields of odd characteristic with abelian unipotent subgroups.

Abstract

The work proves that, for three-dimensional upper triangular groups over a field of odd characteristic with an abelian unipotent subgroup, the ring of invariants is polynomial if and only if the unipotent subgroup is generated by pseudoreflections or does not contain transvections.