$1 \leq j \leq n-r$, $n-d < k \leq n$. The first two are naturally viewed as subspaces, while the last is more naturally viewed as a quotient space.

Since $S$ and $V$ are coordinate subspaces, $\hat{Y}_{\sigma}$ is spanned by some subset of the $E_{jk}$.

$\hat{Y}_{\sigma} = \{E_{jk}\ |\ \eta_j < \eta_k\}$.

$\eta = \eta_1 \ldots \eta_n$ is the element of the $S_n$ such that $\eta^{-1} \cdot {F^{\mathrm{std}}}= \hat{F}(\sigma)$. Thus ${\mathfrak{b}}(\hat{F}(\sigma)) = \eta^{-1} \cdot {\mathfrak{b}}({F^{\mathrm{std}}})$. One can easily check that the image in ${\mathfrak{g}}/{\mathfrak{p}}$ is $\{E_{jk}\ |\ \eta_j < \eta_k\}$.

\[ex:Y\] Let $d=2$, $r=5$, $n=9$. Then ${\mathfrak{g}}/{\mathfrak{p}}$ looks like $$\left(
\begin{matrix}
\cdot & \cdot & \cdot & \cdot & * & * & * & * & * \\
\cdot & \cdot & \cdot & \cdot & * & * & * & * & * \\
\cdot & \cdot & \cdot & \cdot & * & * & * & * & * \\
\cdot & \cdot & \cdot & \cdot & * & * & * & * & * \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & * & * \\
\cdot & \cdot & \cdot & \cdot & \cdot 