Adjacency of Young tableaux and the Springer fibers
Let \(\mathcal F(V)\) be the full flag manifold of a vector space \(V\). For a nilpotent endomorphism \(x\) of \(V\) the Springer fiber is \[ \mathcal F_x= \{(\{0\}=V_0, V_1, \ldots, V_n) \in \mathcal F \mid x(V_i) \subseteq V_{i-1}, \forall i = 1, \ldots, n\}. \] To each flag \(\xi \in \mathcal F_x\), Spaltenstein has associated a standard Young tableau \(\sigma_{\xi}\). He showed that the set \(\mathcal F_{x,\sigma}\) of \(\xi\) such that \(\sigma_{\xi}=\sigma\) is a non-singular open irreducible subvariety of \(\mathfrak F_x\). Moreover, the irreducible components of \(\mathcal F_x\) are the closures \(\mathcal F_{\sigma}\) of the \({\mathcal F}_{x, \sigma}\). This paper achieves the following: (1) It shows that for any standard Young tableau \(\sigma\) of shape \(\lambda\) \[ \mathcal F_{\sigma} \subseteq \bigcup_{\gamma \preceq \sigma} \mathcal F_{x, \gamma}, \] where \(\preceq\) is an order previously introduced by the first author; (2) it shows that the Schubert cell \(\mathcal C_{w_{\sigma}}\) of \(\mathcal F\) intersecting \(\mathcal F_{\sigma}\) in an open dense subset is labelled by the permutation \[ w_{\sigma} = RS(\sigma_{\max}, \sigma^{\vee}), \] where \(\sigma_{\max}\) is the maximal with respect to \(\preceq\) standard Young tableau of shape \(\lambda\), \(^{\vee}\) is the Schützenberger involution, \(RS\) is the Robinson-Schensted correspondence, and the Schubert cell is taken with respect to basis compatible with the Jordan's form of \(x\); (3) it goes on to show that in the notation above \(\mathcal C_{w_{\sigma}} \cap \mathcal F_{x} \subseteq \mathcal F_{x, \sigma}\); (4) it gives a sufficient condition for smoothness of \(\mathcal F_{\sigma}\) in terms of \(k\)-adjacency to the so called Richardson components of \(\mathcal F_x\); (5) it shows that the map \(\sigma \mapsto w_\sigma\) gives an embedding of the adjacency graph of \(\lambda\) into Bruhat order, or as authors call it Bruhat graph of the symmetric group.
scientific article; zbMATH DE number 5189558