Schubert Polynomials for the affine Grassmannian of the symplectic group

TL;DR

This paper explores the Schubert calculus of the affine Grassmannian for the symplectic group, linking homology and cohomology rings with symmetric functions and providing combinatorial descriptions and rules for Schubert classes.

Contribution

It introduces explicit combinatorial descriptions of the Schubert basis and defines affine type C Stanley symmetric functions, advancing the understanding of the affine Grassmannian's structure.

Findings

Identification of homology and cohomology rings with dual Hopf algebras of symmetric functions

Explicit combinatorial description of the Schubert basis

Pieri rule for products involving special Schubert classes

Abstract

We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.