I'm interested in studying about semi-invariant ring in the context of Quiver representations. I started reading about it in some books and few places on the internet. But, nowhere do they mention why they are important. Even on the following Wikipedia page https://en.wikipedia.org/wiki/Semi-invariant_of_a_quiver, it just says, "They form a ring whose structure reflects representation-theoretical properties of the quiver."
I would like to know why these semi-invariant rings are important and what "representation-theoretical properties" they reflect. Or, why did people start studying semi-invariant ring (not just in the context of quiver representations, but in general)?
Fix a quiver $Q$ with $n$ vertices numbered 1 to $n$, and a dimension vector $d=(d_1,\dots,d_n)$ of non-negative integers, and consider the representations of $Q$ of that dimension. This amounts to picking matrix entries for the linear map defined by each of the arrows.
We could consider the space of such choices. Let's call it $R(Q,d)$. It is just a big affine space, so it's not very interesting and it doesn't reflect the quiver in any deep way.
The next obvious thing to do is to say that I want to allow change of basis at the vertices of $Q$. This defines an action of $GL(d)=GL(d_1)\times\dots\times GL(d_n)$ on $R(Q,d)$. The orbits for this action are exactly the isomorphism classes of $d$-dimensional representations of $Q$.
It would be a natural guess to think "Oh, I should look for functions on $R(Q,d)$ which are invariant under this $GL(d)$." Unfortunately, for many quivers, only the constant functions on $R(Q,d)$ will be invariant under $GL(d)$. For this reason, we instead consider the weaker condition of invariance under $SL(d)$; the functions invariant under $SL(d)$ are the semi-invariants.
Another motivation is that semi-invariants are useful for understanding the geometric invariant theory of the quotient. Sections 2.2 and 2.8 of this paper by Derksen and Weyman cover this material; they also have a very nice book which is an elementary introduction to quiver representations. I assume this material can be found there too, though I don't have it in front of me to check.