Reading about the semi-invariants of quivers, I see a fact which is frequently referred to in the literature, and is assumed to be trivial. However, I don't see that very easily. So, I was wondering if you could let me know if there is a crystal clear fact that I missing here.
For a given finite quiver $Q$ and a fixed dimension vector $\alpha$, it is claimed that every character of the algebraic group $\operatorname{GL}(Q,\alpha)$, namely every morphism of algebraic groups $\psi: \operatorname{GL}(Q,\alpha) \rightarrow K^*$, is of the form $\{A(x)|x \in Q_0\} \rightarrow \prod _{x\in Q_0}$ $(\det A(x))^{\sigma(x)}$, where $\sigma$ is a weight (one dimensional irreducible representation) of $\operatorname{Rep}(Q,\alpha)$.
It would be a great help if you could let me know whether this result is derived, as a corollary, from a general fact about the characters of algebraic groups of the given form, which I am not aware of.