Root-theoretic Young diagrams and Schubert Calculus

Abstract

A longstanding problem in algebraic combinatorics is to find nonnegative combinatorial rules for the Schubert calculus of generalized flag varieties; that is, for the structure constants of their cohomology rings with respect to the Schubert basis. There are several natural choices of combinatorial indexing sets for the Schubert basis classes. This thesis examines a number of Schubert calculus problems from the common lens of root-theoretic Young diagrams (RYDs). In terms of RYDs, we present nonnegative Schubert calculus rules for the (co)adjoint varieties of classical Lie type. Using these we give polytopal descriptions of the set of nonzero Schubert structure constants for the (co)adjoint varieties where the RYDs are all planar, and suggest a connection between planarity of the RYDs and polytopality of the nonzero Schubert structure constants. This is joint work with A. Yong. For the family of (nonmaximal) isotropic Grassmannians, we characterize the RYDs and give a bijection between RYDs and the k-strict partitions of A. Buch, A. Kresch and H. Tamvakis. We apply this bijection to show that the (co)adjoint Schubert calculus rules agree with the Pieri rules of A. Buch, A. Kresch and H. Tamvakis, which is needed for the proofs of the (co)adjoint rules. We also use RYDs to study the Belkale-Kumar deformation of the ordinary cup product on cohomology of generalized flag varieties. This product structure was introduced by P. Belkale and S. Kumar and used to study a generalization of the Horn problem. A structure constant of the Belkale-Kumar product is either zero or equal to the corresponding Schubert structure constant, hence the Belkale-Kumar product captures a certain subset of the Schubert structure constants. We give a new formula (after that of A. Knutson and K. Purbhoo) in terms of RYDs for the Belkale-Kumar product on flag varieties of type A. We also extend this formula outside of type A to the (co)adjoint varieties of classical type. With O. Pechenik, we introduce a new deformed product structure on the cohomology of generalized flag varieties, whose nonzero structure constants can be understood in terms of projections to smaller flag varieties. We draw comparisons with the ordinary cup product and the Belkale-Kumar product