and only if for $j \in \{n-r+1, \ldots, n\}$, $$\phi(x_j) \subset \langle x_{\alpha_1}, x_{\alpha_2}, \ldots, 
x_{\alpha_j} \rangle$$ But since ${\mathop{\mathrm{Image}}}(\phi) \subset \langle x_1, \ldots, x_{n-r} \rangle$, this is equivalent to

$$\phi(x_j) \subset V + \langle x_1, \ldots, x_l \rangle
= ({F^{\mathrm{std}}}_Q)_l$$ where $l$ is the number of [[‘$0$’]{}s ]{}before the $j^{\rm th}$ [[‘$1$’]{} ]{}. Thus $l = \lambda_j$.

In $Gr(4, {\mathbb{C}}^9)$, if $\lambda = \mathbf{0} \leq 0 \leq 1 \leq 3 \leq 3 \leq \mathbf{5}$, then with respect to the standard coordinate bases on $V$ and $Q$, $$\hat{X}_\lambda = \
\left (
\begin{matrix}
0 & * & * & * \\
0 & 0 & * & * \\
0 & 0 & * & * \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 
\end{matrix}
\right)$$ In general, if we write $\hat{X}_\lambda$ in this form, the number of $*$’s in column $j$ will be $\lambda_j$.

The action of $P$ on ${\mathrm{Hom}}(V,Q)$ is given by $$\left(
\begin{matrix}
A & 0 \\
C & B 
\end{matrix}
\right) 
\cdot \phi
= A \phi B^{-1}.$$ To compute the generic intersection of two subspaces $X_\lambda \cap X_\mu$, it suffices to take $X_\lambda = \hat{X}_\lambda$, and $X_\mu = 