Quantum product and parabolic orbits in homogeneous spaces

TL;DR

This paper explores the relationship between quantum products, parabolic orbits, and the stratification of rational homogeneous spaces, providing new insights into their geometric and algebraic structures.

Contribution

It links quantum product formulas to P-orbit stratification in homogeneous spaces with Picard rank one, offering a new decomposition of the Hasse diagram.

Findings

Quantum product formula connected to P-orbit stratification

Decomposition of the Hasse diagram for homogeneous spaces

Enhanced understanding of geometric structures in rational homogeneous spaces

Abstract

Chaput, Manivel and Perrin proved a formula describing the quantum product by Schubert classes associated to cominuscule weights in a rational projective homogeneous space X. In the case where X has Picard rank one, we link this formula to the stratification of X by P-orbits, where P is the parabolic subgroup associated to the cominuscule weight. We deduce a decomposition of the Hasse diagram of X, i.e the diagram describing the cup-product with the hyperplane class.