Quantum Product and Parabolic Orbits in Homogeneous Spaces

Abstract

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.

Key Words:

• Generalized flag varieties
• Hasse diagrams
• Parabolic orbits
• Quantum product

2010 Mathematics Subject Classification:

• 14M15
• 14N35

ACKNOWLEDGMENTS

This paper reports on work done during my thesis. I would like to thank my advisor, Laurent Manivel, for his support and for valuable discussions. I am also indebted to Nicolas Perrin for many useful remarks and comments.

Notes

Communicated by R. Piene.

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lagb.