Abstract. This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and two-step flag varieties. As a consequence, we obtain a positive, geometric rule for expressing the structure constants 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 Grassmannians. We also obtain a positive, geometric rule for computing the classes of subvarieties of Grassmannians that arise as the projection of the intersection of two Schubert varieties in a partial flag variety. These rules have numerous applications to geometry, representation theory and the theory of symmetric functions. This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and twostep flag varieties. The first consequence of our study is a new positive, geometric rule for expressing the structure constants of the cohomology ring of Grassmannians in terms of their Schubert basis (Theorem 3.21). Such rules are known as Littlewood-Richardson rules. In fact, much more generally, Algorithm 3.12 provides a positive, geometric algorithm for computing the classes of subvarieties of the Grassmannian that arise as the projection of the intersection of two Schubert varieties in a partial flag variety. The second main application of our study is a positive, geometric rule for computing the structure constants of two-step flag varieties in terms of their Schubert basis (Theorem 4.21). Since the three-pointed Gromov-Witten invariants of a Grassmannian can be calculated as ordinary intersections of Schubert cycles in two-step flag varieties [BKT], we also obtain a quantum Littlewood-Richardson rule for Grassmannians (Theorem 5.1). 2000 Mathematics Subject Classification. Primary 14M15, 14N35, 32M10 . During the preparation of this article the author was partially supported by a Clay Mathematics Institute Liftoff Fellowship. Fix an ordered basis e 1 , . . . , e n of a vector space V . Let W 1 , . . . , W k be k distinct subspaces of V spanned by consecutive elements of the ordered basis. Consider the subvariety Σ of G(k, n) parameterizing k-planes that intersect each W i in a specified dimension (subject to the necessary compatibility conditions). The first problem we address in this paper is: Problem 1.1. Give a positive, geometric rule for computing the cohomlogy class of Σ. Note that when the vector spaces W i are totally ordered with respect to inclusion, the subvariety Σ is a Schubert variety. If the vector spaces W i have no inclusion relations among them, then Σ is the intersection of two Schubert varieties in G(k, n). More generally, the projection of the intersection of two Schubert varieties in F (k 1 , . . . , k r ; n) to G(k i , n) can be described as a variety of the form Σ. Our first main result is a positive, geometric rule for expressing the class of Σ in terms of Schubert cycles. The second main result of this paper is a positive geometric rule for computing the structure constants of two-step flag varieties in terms of their Schubert basis. The strategy that leads to the rule works to compute the classes of subvarieties of other partial flag varieties. However, to keep this paper at a reasonable length, we restrict our discussion to two-step flag varieties. The reader may consult [CV] for an example for three-step flag varieties. In addition to being a step in understanding the geometry of arbitrary partial flag varieties, the study of two-step flag varieties is interesting in its own right. For instance, the work of Buch, Kresch and Tamvakis ([BKT]) has highlighted a beautiful relation between the small quantum cohomology of Grassmannians and the cohomology of two-step flag varieties. The final result of this paper is a quantum Littlewood-Richardson rule for Grassmannians.
Coskun REF gave a positive geometric rule to compute the structure constants of the cohomology ring of two-step flag varieties in terms of Mondrian tableaux.
A Littlewood–Richardson rule for two-step flag varieties
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