From my understanding, a quiver is basically a finite directed graph (a multidigraph). And a representation assigns a vector space to each vertex, and a linear map to each arrow. Formally, $Q = (Q_0, Q_1, s, t)$ where $Q_0$ are vertices, $Q_1$ are arrows, $s$ assigns arrows to the source, $t$ assigns arrows to the target. And a representation $V$ of $Q$ is the set of vector spaces $\{V(x) \mid x \in Q_0\}$ and the set of linear maps $\{V(a): t(a) \to s(a) \mid a \in Q\}$.
But I am wondering if there is a visualization or geometric realization that this representation is talking about. The questions this brings up are:
- If there is a corresponding geometric realization. Or if not, wondering how I am supposed to interpret the representation. It seems that since it is mapped to a vector space with these "morphisms" between spaces that it should be somehow visual, but I am probably missing something.
- What kinds of vector spaces typically get assigned. Wondering what the purpose is of the representation, what it is typically used for.