The ABC's of affine Grassmannians and Hall-Littlewood polynomials

TL;DR

This paper introduces new combinatorial formulas for affine Grassmannian Schubert classes and k-Schur functions, linking them to Hall-Littlewood polynomials and Kostka-Foulkes polynomials, advancing algebraic combinatorics.

Contribution

It provides a novel combinatorial description of the Pieri rule for k-Schur functions and new formulas for Schubert class representatives in affine Grassmannians.

Findings

New combinatorial formulas for Schubert classes

Connection between k-Schur functions and Hall-Littlewood polynomials

Insights into Kostka-Foulkes polynomials

Abstract

We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and k-Schur functions.