Quantum Product and Parabolic Orbits in Homogeneous Spaces

Chaput, Manivel, and Perrin proved in [3] 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 relate 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. For all classical Grassmannians, we give a complete description of parabolic orbits associated to cominuscule weights, and we make the decomposition of the Hasse diagram explicit.