A Littlewood-Richardson rule for two-step flag varieties

摘要

This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and two-step flag varieties. As a conse_quence, we obtain a positive, geometric rule for expressing the structure con_stants of the cohomology of two-step flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for computing the structure constants of the small quantum cohomology of Gras smannians. We also ob_tain a positive, geometric rule for computing the classes of subvarieties of Grassmannians that arise as the projection of the intersection of two Schu_bert varieties in a partial flag variety. These rules have numerous applications to geometry, representation theory and the theory of symmetric functions.