Schubert polynomials for the affine Grassmannian

TL;DR

This paper confirms a conjecture by identifying polynomial representatives for Schubert classes in the affine Grassmannian as k-Schur and affine Schur functions, linking algebraic and geometric structures.

Contribution

It proves a conjecture by explicitly connecting Schubert classes with k-Schur and affine Schur functions using advanced algebraic tools.

Findings

Polynomial representatives for Schubert classes are identified as k-Schur and affine Schur functions.

The results establish a concrete link between algebraic functions and geometric Schubert classes.

The work confirms a longstanding conjecture in the field.

Abstract

Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar's nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on non-commutative k-Schur functions.