representations* $$\label{eqn:defn.Rep.gamma}
    {\boldsymbol{\mathrm{Rep}}}_\gamma(Q) = {\boldsymbol{\mathrm{Rep}}}_\gamma := {\bigoplus}_{a\in Q_1} \operatorname{Hom}\left( {\mathbb{C}}^{\gamma(ta)},{\mathbb{C}}^{\gamma(ha)} \right)$$ with action of the *base change group* ${\boldsymbol{{\mathrm{GL}}}}_\gamma := \prod_{i\in Q_0} {\mathrm{GL}}({\mathbb{C}}^{\gamma(i)})$ given by $$\label{eqn:defn.GGLL.action}
    (g_i)_{i\in Q_0} \cdot (\phi_a)_{a\in Q_1} = (g_{ha} \phi_a g_{ta}^{-1})_{a\in Q_1}.$$ We let $\chi:D \times D \to {\mathbb{Z}}$ denote the bilinear Euler form for $Q$, given by the formula $$\label{eqn:defn.Euler.form}
    \chi(\gamma_1,\gamma_2) = \sum_{i\in Q_0} \gamma_1(i)\gamma_2(i) - \sum_{a\in Q_1} \gamma_1(ta)\gamma_2(ha) \nonumber$$ for any $\gamma_1,\gamma_2 \in D$. We also consider the opposite antisymmetrization of $\chi$, $$\label{eqn:defn.lambda}
    \langle\gamma_1,\gamma_2\rangle := \chi(\gamma_2,\gamma_1) - \chi(\gamma_1,\gamma_2).\nonumber$$ Observe that for any $i,i'\in Q_0$ $$\label{eqn:lambda.simples}
    \langle e_i,e_{i'}\rangle = \#\{a\in Q_1: ta=i,\,ha=i'\} - \#\{a\in Q_1: ta=i',\,ha=i\},$$ that is, $\langle e_i,e_{i'} \rangle$ equals the number of arrows $i\to i'$ minus the number of arrows $i'\to i$.

A *path* in $Q$ is a concatenation of arrows $a_\ell\,\cdots\,a_2\,a_1$ such that $ha_j = ta_{j+1}$ for all $j\in[\ell-1]$. Such a path is an *oriented cycle* if $ha_\ell = ta_1$. A quiver is called *acyclic* if it contains no oriented cycles. In particular, acyclic quivers contain no loop arrows. Observe that the underlying non-directed graph associated to $Q$ may contain cycles, even if $Q$ is acyclic.

In the rest of the paper, we assume $Q$ is an acyclic quiver, and we 