A multiplicative formula for structure constants in the cohomology of flag varieties

TL;DR

This paper introduces a multiplicative formula for structure constants in the cohomology of flag varieties, enabling simplified calculations and generalizations in Schubert calculus related to complex semi-simple Lie groups.

Contribution

It presents a natural product formula for structure constants in cohomology of flag varieties, extending to Levi-movable constants and branching Schubert calculus.

Findings

Structure constants can be expressed as products from smaller flag varieties.

The formula simplifies computation of Levi-movable structure constants.

Generalization to branching Schubert calculus is established.

Abstract

Let G be a complex semi-simple Lie group and let P,Q be a pair of parabolic subgroups of G such that Q contains P. Consider the flag varieties G/P, G/Q and Q/P. We show that certain structure constants in H^*(G/P) with respect to the Schubert basis can be written as a product of structure constants coming from H^*(G/Q) and H^*(Q/P) in a very natural way. The primary application is to compute Levi-movable structure constants defined by Belkale and Kumar. We also give a generalization of this product formula in the branching Schubert calculus setting.