about the original flag $F$, and moreover, the action of $GL(V) \times GL(Q)$ on $Flags(V) \times Flags(Q)$ is transitive. Thus we shall stop thinking about $F_V$ and $F_Q$ as the induced flags of $F$, and instead think of them as an independent pair of generic flags on $V$ and $Q$ respectively.

\[rmk:genericinducedflags\] We have in fact just observed that the induced flags on $V$ and ${\mathbb{C}}^n/V$ are generic, if $V \in Gr(r,n)$ is a generic point of intersection of Schubert varieties in general position.

Let $S$ be a subspace of $V$ whose Schubert position relative to the flag $F_V$ is $\rho$. Put $d = \dim S$. If $\phi \in {\mathrm{Hom}}(V,Q)$ and $\ker \phi
\supset S$, then $\phi$ descends to a map $\tilde{\phi} \in {\mathrm{Hom}}(V/S,Q)$. We now consider the space $$X_\lambda{\big /}S = \{\tilde\phi \in {\mathrm{Hom}}(V/S,Q)\ |\ \phi \in X_\lambda,
\ \ker\phi \supset S\} = X_\lambda \cap {\mathrm{Hom}}(V/S,Q).$$

$X_\lambda{\big /}S = X_{\lambda'}$, where $\lambda' \in \Lambda(r-d, n-r)$ is the subpartition $$\lambda' = \mathbf{0} \leq \lambda_{k_1} \leq \cdots \leq 
\lambda_{k_{r-d}} \leq \mathbf{n-r}$$ where $k_j$ are the positions of the [[‘$0$’]{}s ]{}of $\rho$. The flag on $V/S$ associated to $X_{\lambda'}$ is the induced flag $(F_V)_{V/S}=:F_{V/S}$.

The action of $P$ simply gives a change 