A t-generalization for Schubert Representatives of the Affine Grassmannian

TL;DR

This paper introduces two new families of symmetric functions with a parameter t, generalizing Schubert representatives for the affine Grassmannian's cohomology and homology, with positive transition properties.

Contribution

It defines these families via combinatorial statistics and conjectures one family corresponds to k-atoms, extending the understanding of affine Grassmannian symmetric functions.

Findings

Families specialize to Schubert representatives at t=1

Transition matrix with Hall-Littlewood polynomials is t-positive

Conjecture that one family equals k-atoms

Abstract

We introduce two families of symmetric functions with an extra parameter t that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when t = 1. The families are defined by a statistic on combinatorial objects associated to the type-A affine Weyl group and their transition matrix with Hall-Littlewood polynomials is t-positive. We conjecture that one family is the set of k-atoms.