further assume that the set of vertices $Q_0 = [n]$ is ordered so that for every arrow $a\in Q_1$, we have $ha < ta$. Such an ordering is always possible in an acyclic quiver, but is not unique in general; see e.g., [@hdjw2017 Exercise 1.5.2]. Moreover, when $Q$ is acyclic note that exactly counts (with a sign, but no cancellation) the number of arrows between the vertices $i$ and $i'$. Therefore, our assumption on the ordering of $Q_0$ implies that $\langle e_i,e_{i'} \rangle \leq 0$ whenever $i<i'$.

Dynkin quivers {#ss:dynkins}
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A quiver $Q$ is a *Dynkin quiver* if it is an orientation of a simply-laced Dynkin diagram (i.e., of type $A$, $D$, or $E$). We identify the *simple roots* of the associated root system with the set of simple dimension vectors $e_i$ for each $i\in Q_0$. Hence each element, $\beta$, of the set of *positive roots* $\Phi_+(Q) = \Phi_+$ has the form $$\beta = \sum_{i\in Q_0} d^i_{\beta} e_i$$ for some non-negative integers $d^i_\beta$. In this way, we realize positive roots as dimension vectors. As is our convention, we will suppress $Q$ in the notation when it is clear from context.

We have already described an ordering on the vertices 