Finiteness theorems for some representations of $\mathrm{GL}_3$

TL;DR

This paper establishes finiteness results for the classification of certain algebraic forms and representations of ng over number fields, extending known results from binary to ternary forms.

Contribution

It generalizes finiteness theorems for ng representations from binary forms to ternary forms and specific higher-dimensional representations.

Findings

Finiteness of ng-orbits of ternary n-ic forms with fixed discriminant.

Finiteness of ng-orbits of a 27-dimensional representation with highest weight (4, 2).

Extension of Birch--Merriman's result from binary to ternary forms.

Abstract

Let $n \geq 2$ be an integer and let $K$ be a number field with ring of integers $\mathcal{O}_K$. We prove that the set of ternary $n$-ic forms with coefficients in $\mathcal{O}_K$ and fixed nonzero discriminant, breaks up into finitely many $\mathrm{GL}_3(\mathcal{O}_K)$-orbits. This generalizes a result of Birch--Merriman in the binary forms case. We also prove a similar finiteness result on the $\mathrm{GL}_3(\mathcal{O}_K)$-orbits of the 27-dimensional representation of $\mathrm{GL}_3$ with highest weight $(4, 2)$.