I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: https://math.stackexchange.com/q/4218687/884739

I have been studying about semi-invariant rings in the context of Quiver representation but I don't really understand that if a semi-invariant ring turns out to be a polynomial ring or a hypersurface (or complete intersection), what "representation-theoretical" properties does it tell us about the quiver? Or, in general, what information does it give?

Even if the answer is not particularly in the context of Quiver representation, I would still be glad to know.