$$\label{eqn:H.gamma.polynomials}
    {\mathscr{H}}_\gamma {\cong}{\mathrm{H}}^\bullet(B{\boldsymbol{{\mathrm{GL}}}}_\gamma) 
        {\cong}{\bigotimes}_{i\in Q_0} {\mathbb{Q}}[\omega_{i,1},\ldots,\omega_{i,\gamma(i)}]^{{\mathfrak{S}}_{\gamma(i)}}$$ where $B$ denotes the [Borel construction]{} for equivariant cohomology and ${\mathfrak{S}}_p$ is the symmetric group on $p$ letters. The action of each ${\mathfrak{S}}_{\gamma(i)}$ is by permuting the variables $\{\omega_{i,j}:j\in[\gamma(i)]\}$. Hence, in the sequel we realize elements of ${\mathscr{H}}_\gamma$ as polynomials $f(\omega_{i,j})$ which are separately symmetric in the variables $\omega_{i,j}$ for each $i\in Q_0$.

Subquiver partitions {#ss:quivers.subquiver.partitions}
--------------------

A quiver $Q$ is *nonempty* if $Q_0 \neq \emptyset$, and $Q$ is *connected* if its underlying non-oriented graph is connected. A *subquiver* $Q'$ of $Q$ is a quiver with $Q'_0 \subseteq Q_0$ and $Q'_1 \subseteq Q_1$.

\[defn:dsqp\] Let ${{\mathcal}{Q}^\bullet}= \{Q^1,\ldots,Q^\ell\}$ be a set of subquivers of $Q$.

1.  We say that ${{\mathcal}{Q}^\bullet}$ is a *subquiver partition* if each $Q^j$ is a non-empty connected subquiver, and $Q_0$ is the disjoint union of the $Q^j_0$ vertex sets.

2.  We say that ${{\mathcal}{Q}^\bullet}$ is a *Dynkin subquiver partition* if it is a subquiver partition and each $Q^j$ is a Dynkin quiver.

3.  We say that ${{\mathcal}{Q}^\bullet}$ is an *admissible subquiver partition* if it is a subquiver partition and the quiver obtained from $Q$ by contracting each subquiver $Q^j$ to a single vertex is also acyclic.

4.  In the contracted quiver 