I'm asking this question as a continuation of discussion and answer given by Hugh Thomas at the following post: Why do people study semi-invariant ring (in general)?
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? For what kind of concept or calculation is it useful?
Even if the answer is not particularly in the context of Quiver representation, I would still be glad to know.