  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.
