= (1,1,0)$, $\gamma_2 = (0,3,3)$, $\gamma_3 = (0,0,1)$.

Only (i) is consistent with ${{\mathcal}{Q}^\bullet}$. While (ii) satisfies the first bullet point of Definition \[defn:consistent.gamma.lists\], it fails the second with $u=1$ and $v=2$. The list (iii) fails the first bullet point since $\gamma_1 = (1,1,0)$ is not supported on either of the subquivers $Q^1$ or $Q^2$.

For the remainder of the subsection, we require that ${{\mathcal}{Q}^\bullet}$ is an admissible, ordered, Dynkin subquiver partition of $Q$. Hence each subquiver $Q^j$ has an associated set of positive roots $\Phi_+(Q^j)$ which are also naturally dimension vectors of $Q$ with entries of zero at each $i \notin Q^j_0$. Let $\Phi_+({{\mathcal}{Q}^\bullet}) = {\bigcup}_{j\in[\ell]} \Phi_+(Q^j)$, and we extend the notion of a Reineke order on each $\Phi_+(Q^j)$ to the union $\Phi_+({{\mathcal}{Q}^\bullet})$ as follows. Write $r_j = \#\Phi_+(Q^j)$ and $\Phi_+(Q^j) = \{\beta^j_1,\ldots,\beta^j_{r_j}\}$ in a Reineke order for the Dynkin subquiver $Q^j$. Then a *Reineke order* on $\Phi_+({{\mathcal}{Q}^\bullet})$ has the form $$(\beta^1_1 \prec \cdots \prec \beta^1_{r_1}) \prec (\beta^2_1 \prec \cdots \prec \beta^2_{r_2}) \prec \cdots \prec (\beta^\ell_{1} \prec \cdots \prec \beta^\ell_{r_\ell}).$$ Observe that the ordering condition on ${{\mathcal}{Q}^\bullet}$ and the ordering condition together imply that if $\beta'$ precedes $\beta''$ where $\beta',\beta''$ are

-   positive roots from *the same* subquiver, then 