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

TL;DR

We address when the invariant ring $\mathbb{K}[W]^G$ is polynomial for a finite group $G$ acting on a 3-dimensional space $W$ over a field of odd characteristic with an upper-triangular representation and abelian unipotent subgroup $H$. The authors develop a linearization technique that reduces questions to a linear action of $G/H$ on $W/H$, using formal power series to obtain a linear action and applying the Chevalley–Shephard–Todd theorem in the non-modular setting; they further dissect the 3D case by decomposing $G$ into subgroups and constructing explicit invariants via norm maps. The main result shows $\mathbb{K}[W]^G$ is non-polynomial if and only if $H$ contains transvections and is not generated by them; equivalently, the ring is polynomial when $H$ is generated by pseudoreflections. This work yields counterexamples to the converse of CST in dimension 3 for reducible indecomposable representations and advances modular invariant theory in low dimensions.

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.

Theorems & Definitions (25)

• Lemma 1
• Definition 1
• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Theorem 4
• Proposition 1
• Definition 2
• Lemma 2