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

Let $n\ge 2$ 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 ${\mathrm{G}\mathrm{L}}_3({\mathcal{O}}_K)$-orbits. This generalizes a result of Birch–Merriman in the binary forms case. We also prove a similar finiteness result on the ${\mathrm{G}\mathrm{L}}_3({\mathcal{O}}_K)$-orbits of the 27-dimensional representation of ${\mathrm{G}\mathrm{L}}_3$ with highest weight $(4,2)$.