Schubert calculus of Richardson varieties stable under spherical Levi subgroups

Abstract

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 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.

Section 4 significantly shortened, and other minor changes made as suggested by referees. Final version, to appear in Journal of Algebraic Combinatorics

References in corpus (5)

• K-orbits on G/B and Schubert constants for pairs of signed shuffles in types C and D
• Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci
• RC-graphs and a generalized Littlewood-Richardson rule
• Closures of K-orbits in the flag variety for U(p,q)
• Some conjectures regarding certain Schubert structure constants in Lie types B and D

Cited by in corpus (8)

• Singularities of Richardson varieties
• On pattern avoidance in matchings and involutions
• K-orbit closures and Barbasch-Evens-Magyar varieties
• Some conjectures regarding certain Schubert structure constants in Lie types B and D
• Polynomials for GL_p x GL_q orbit closures in the flag variety
• Combinatorial results on (1,2,1,2)-avoiding $GL(p,\mathbb{C})\times GL(q,\mathbb{C})$-orbit closures on $GL(p+q,\mathbb{C})/B$
• $DIII$ clan combinatorics for the orthogonal Grassmannian
• Chains in Weak Order Posets Associated to Involutions