K-orbits on G/B and Schubert constants for pairs of signed shuffles in types C and D

TL;DR

This paper provides positive formulas for Schubert structure constants in types C and D by relating certain orbit closures to Richardson varieties and using Brion's theorem on their decomposition.

Contribution

It introduces a new geometric approach to compute Schubert constants in types C and D via orbit closures and weak order paths.

Findings

Positive descriptions for Schubert structure constants in types C and D.

Identification of certain orbit closures with Richardson varieties.

Application of Brion's theorem to decompose orbit closures in the Schubert basis.

Abstract

We give positive descriptions for certain Schubert structure constants $c_{u,v}^w$ for the full flag variety in Lie types $C$ and $D$. This is accomplished by first observing that a number of the $K=GL(n,\C)$-orbit closures on these flag varieties coincide with Richardson varieties, and then applying a theorem of M. Brion on the decomposition of such an orbit closure in the Schubert basis in terms of paths in the weak order graph.