Vinberg representations and related geometry

In the 70’s Vinberg explained that there are many representations that behave like the adjoint representation of a simple complex Lie group from the point of view of invariant theory. Implicitly, many of these representations have been used to describe various finite covers of moduli spaces of curves of low genus. In some cases there are also toric and tropical varieties naturally associated with the moduli space. I’ll explain one example $\wedge^4 C^8$ concerning curves of genus 3 and some results in recent work with Ren, Schrader, and Sturmfels.