Schubert calculus of richardson varieties stable under spherical Levi subgroups

Wyser, Benjamin J.

Journal of Algebraic Combinatorics, Tome 38 (2013) no. 4, pp. 829-850.

Résumé

We observe that the expansion in the basis of Schubert cycles for $H ^{\ast }(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^{w}$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some roughly stated conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\mathbb C)$ and $SO(2n,\mathbb C)$, respectively.

Détail

Classification : 14M15, 05E99
Keywords: Schubert calculus, Richardson variety, Schubert variety, spherical subgroup, symmetric subgroup, flag variety