The recursive nature of cominuscule Schubert calculus

Details The recursive nature of cominuscule Schubert calculus The recursive nature of cominuscule Schubert calculus

2006-07-26

arxiv.org/abs/math/0607669v2

Advances in Mathematics, 217 (2008), pp. 1962--2004.

Mathematics

Algebraic Geometry

41 pages, revisions to improve clarity of exposition

Scientific paper

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.

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