\leq GL$, and subspace $Z_m \leq {\mathrm{Hom}}$.

Then $[\Omega_m]$ may be computed by Theorem \[thm:multidegree\] where $[Z_m]$ is the product of the weights of all blocks in ${\mathrm{Hom}}$ that are strictly above the “block diagonal". For types $A$ and $D$ the $K$-polynomial $[\Omega_m]^K_{{\mathrm{Hom}}}$ may be computed by Theorem \[thm:Kpoly\], in which $[Z_m]^K_{\mathrm{Hom}}$ is the product of terms of the form $1-e^{-\gamma}$ where $\gamma$ runs over those same weights as in $[Z_m]$.

Let $Q$ be the equioriented $A_n$ quiver: $$\begin{aligned}
    \pspicture(-.5,-.5)(4.5,.5)    \psset{arrows=->,arrowsize=3pt 3}    \psdots(0,0)(1,0)(2,0)(3,0)    \psline[arrows=->](0,0)(1,0)    \psline[arrows=->](1,0)(2,0)    \psline[arrows=->,linestyle=dashed](2,0)(3,0)    \rput(0,-.4){$1$}    \rput(1,-.4){$2$}    \rput(2,-.4){$3$}    \rput(3,-.4){$n$}    \endpspicture
  \end{aligned}$$ The simple roots of $A_n$ are given by $\alpha_{ij}=\alpha_i+\alpha_{i+1}+\dotsm+\alpha_j$ for $1\le i\le
  j\le n$. Write $I_{ij}$ for the indecomposable representation of $A_n$ corresponding to $\alpha_{ij}$. A suitable ordering on the indecomposables is given by $I_{11},I_{12},I_{22},I_{13},I_{23},I_{33},\dotsc,I_{nn}$. Let us consider the specific example $n=3$, $d=(2,3,2)$, and $\Omega$ given by the $GL$-orbit closure of $I_{12}^{\oplus 2} \oplus I_{23}\oplus I_{33}$. Geometrically, $\Omega$ is defined by requiring the map $V(2) \to V(3)$ to have rank $\leq 1$, and the composite map $V(1)\to V(3)$ to vanish.

The decompositions $d(i) = \sum_{j=1}^N d_j(i)$ are $$d(1) = 2+0+0, \quad d(2) = 2+1+0, \quad d(3) = 0+1+1,$$ so the parabolic $P_m\subset GL$ and the linear subspace $Z_m\subset{\mathrm{Hom}}$ 