Schubert calculus of Richardson varieties stable under spherical Levi subgroups

TL;DR

This paper provides combinatorial formulas for Schubert structure constants related to Richardson varieties stable under spherical Levi subgroups, extending known results in type A and proposing conjectures for types B and D.

Contribution

It introduces positive combinatorial descriptions of certain Schubert structure constants for Richardson varieties stable under spherical Levi subgroups, including new rules in type A and conjectures for types B and D.

Findings

Describes $c_{u,v}^w$ for inverse Grassmannian permutations in type A.

Provides combinatorial models for symmetric subgroup orbit closures.

Proposes conjectures for similar rules in types B and D.

Abstract

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ 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 conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\C)$ and $SO(2n,\C)$, respectively.