$N$. This defines a standard parabolic subgroup $P_m\subset GL$ whose $i$th component (for $i\in Q_0$) is the block lower triangular subgroup of $GL({\mathbb{C}}^{d(i)})$ with the given diagonal block sizes.

The decompositions $d(i) = \sum_{j=1}^N d_j(i)$ also induce a block structure on each component ${\mathrm{Hom}}({\mathbb{C}}^{d(ta)},{\mathbb{C}}^{d(ha)})$ of ${\mathrm{Hom}}$, whose $(j,j')$ block is a $d_j(ta) \times d_{j'}(ha)$ rectangle. Define the linear subspace $Z_m\subset{\mathrm{Hom}}$ to be those elements with zeroes in all blocks strictly above the “block diagonal". This $Z_m$ is easily seen to be $P_m$-invariant. Reineke proves that [*for certain choices of ordering*]{} (built using reduced words for $w_0$ adapted to the quiver, as spelled out in Section \[sec:finitequivers\]) on $R_+$, the Kempf collapsing $GL\times^{P_m} Z_m \to {\mathrm{Hom}}$ is birational to $\Omega_m$.

Let $\{x_k^{(i)}\mid i\in Q_0,\, k \in \{1,\ldots,d(i)\} \}$ be a basis for the weight lattice $T^*$ of the standard maximal torus $T$ given by the tuples of diagonal matrices in $GL$. Then the $(k,k')$th matrix entry in the $a$th component of ${\mathrm{Hom}}$ has weight $x_k^{(ta)}-x_{k'}^{(ha)}$.

\[thm:application\] Let $Q$ be a Dynkin quiver, and $\{\beta_1,\beta_2,\dotsc,\beta_N\}$ a certain order on the set $R^+$ of positive roots (constructed explicitly in Section \[sec:finitequivers\]). Let $\Omega_m$ be a quiver locus, with associated multiplicities $m$, parabolic $P_m 