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 representations 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 representations, I would still be glad to know.

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 representations 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 calculation or concept is it useful? Or, in general, what information does it give?

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