I'm asking this question as a continuation of [discussion](https://math.stackexchange.com/questions/4218687/why-do-people-study-semi-invariant-ring-in-general/4223408#comment8771354_4223408) and [answer](https://math.stackexchange.com/a/4223408) given by Hugh Thomas at the following post: [Why do people study semi-invariant ring (in general)?](https://math.stackexchange.com/q/4218687/884739)

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.