GL(p) x GL(q)-orbit closures on the flag variety and Schubert structure constants for (p,q)-pairs

TL;DR

This paper provides positive combinatorial formulas for Schubert structure constants in type A flag varieties for specific pairs called $(p,q)$-pairs, linking orbit closures and Richardson varieties.

Contribution

It introduces a new combinatorial approach to compute structure constants for $(p,q)$-pairs using orbit closures and Richardson varieties.

Findings

Positive combinatorial descriptions of structure constants for $(p,q)$-pairs

Identification of certain orbit closures as Richardson varieties

Connection between orbit closure intersections and Schubert calculus

Abstract

We give positive combinatorial descriptions of Schubert structure constants $c_{u,v}^w$ for the full flag variety in type $A_{n-1}$ when $u$ and $v$ form what we refer to as a "$(p,q)$-pair" ($p+q=n$). The key observation is that a certain subset of the $GL(p,\mathbb{C}) \times GL(q,\mathbb{C})$-orbit closures on the flag variety (those satisfying an easily stated pattern avoidance condition) are Richardson varieties. The result on structure constants follows when one combines this observation with a theorem of Brion concerning intersection numbers of spherical subgroup orbit closures and Schubert varieties.