Tag - Schubert calculus

The theory of symmetric polynomials plays a key role in Representation Theory, Schubert Calculus, and Algebraic Combinatorics. Fundamental rules like the Pieri, Murnaghan-Nakayama, and Littlewood-Richardson rules describe the decomposition of products of Schubert classes into Schubert classes. We focus on the decomposition of polynomial representatives of Schubert classes in homology and K-homology of the affine Grassmannian of SL_n, as well as quantum Schubert classes in quantum cohomology and K-cohomology of the full flag manifold of type A. Specifically, we explore how to use the Peterson isomorphism to connect formulas between homology and quantum cohomology, and between K-homology and quantum K-cohomology, extending techniques from the work of Lam-Shimozono on Schubert classes.