The recursive nature of cominuscule Schubert calculus

Abstract

The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood–Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood–Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.