Title: Root system combinatorics and Schubert calculus

 

Abstract: We discuss some results in Schubert calculus obtained using the combinatorial model of root-theoretic Young diagrams (RYDs). In joint work with A. Yong, we give nonnegative rules for the Schubert calculus of the (co)adjoint varieties of classical type, and use these rules to suggest a connection between planarity of the root poset and polytopality of the nonzero Schubert structure constants. In joint work with O. Pechenik, we introduce a deformation of the cohomology of generalized flag varieties. A special case is the Belkale-Kumar deformation, introduced in 2006 by P. Belkale-S. Kumar. This construction yields a new, short proof that the Belkale-Kumar product is well-defined. Another special case preserves the Schubert structure constants corresponding to triples of Schubert varieties that behave nicely under projections. We also present an RYD rule for the Belkale-Kumar product for flag varieties of type A (after the puzzle rule of A. Knutson-K. Purbhoo).