.5) – (2.75, 1.25) – (3.25, 1.25) – (3.25,.25) – (4.25,.25) – (4.25, -.25) – (2.75,-.25) – (2.75, .5); (4,1) circle \[radius=.25\];

Given a dimension vector $\gamma$ for $Q$ we define the *support* of $\gamma$ to be the set $\{i\in Q_0: \gamma(i) \neq 0\}$. If $Q'$ is a subquiver of $Q$ then we say *$\gamma$ has its support in $Q'$* if the support of $\gamma$ is a subset of $Q'_0$.

\[defn:consistent.gamma.lists\] Let ${{\mathcal}{Q}^\bullet}= \{Q^1,\ldots,Q^\ell\}$ be an admissible, ordered subquiver partition of $Q$. We say that the list of dimension vectors $\gamma_1,\ldots,\gamma_r$ is *consistent with ${{\mathcal}{Q}^\bullet}$* if

-   for each $u\in[r]$, there exists $j(u)\in[\ell]$ such that $\gamma_u$ has its support in $Q^{j(u)}$, and

-   if $1\leq u<v \leq r$, then $1\leq j(u) \leq j(v) \leq \ell$.

For the equioriented $A_3$ quiver $1 \leftarrow 2 \leftarrow 3$, we have the admissible, ordered, Dynkin subquiver partition ${{\mathcal}{Q}^\bullet}= \{Q^1,Q^2\}$ where $Q^1$ is the $A_1$-subquiver consisting of the singleton vertex $1$ and $Q^2$ is the $A_2$-subquiver $2\leftarrow 3$. Consider the following lists of dimension vectors for $Q$:

1.  $\gamma_1 = (1,0,0)$, $\gamma_2 = (0,0,1)$, $\gamma_3 = (0,1,1)$, $\gamma_4 = (0,1,0)$;

2.  $\gamma_1 = (0,1,3)$, $\gamma_2 = (1,0,0)$, $\gamma_3 = (0,2,1)$, $\gamma_4 = (0,2,0)$;

3.  $\gamma_1 