Schubert calculus of Richardson varieties stable under spherical Levi subgroups
arXiv:1209.0739 · doi:10.1007/s10801-013-0427-z
Abstract
We observe that the expansion in the basis of Schubert cycles for 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 . Namely, we describe when and are inverse to Grassmannian permutations with unique descents at and , respectively. We offer some conjectures for similar rules in types and , associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\C)$ and $SO(2n,\C)$, respectively.
Section 4 significantly shortened, and other minor changes made as suggested by referees. Final version, to appear in Journal of Algebraic Combinatorics