Let $G=\mathrm{GL}_n(\mathbb{C})$ and $X_{\omega}=\overline{B_-wB/B}\subset G/B$ be a Schubert variety. Denote by $C(X_\omega)$ the conormal variety inside $T^*(G/B)$ , $\mu:T^*(G/B)\to \mathcal{N}$ the Springer resolution. It is known that $\mu(C(X_\omega))$ is an orbital variety inside $\mathcal{N}$. (An orbital variety is defined as a component of $(G\cdot x) \cap \mathfrak{u}$, $\mathfrak{u}$ is lower triangular matrix with zeros on diagonal.) $G$-orbits of $\mathcal{N}$ are indexed by partition of $n$. Then we can get a map from $\mathrm{Permutation}(n)\to \mathrm{Partition}(n)$. Do we know anything about this map?
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