# Nonsymmetric Macdonald polynomials and a refinement of Kostka-Foulkes polynomials

## TL;DR

This paper analyzes the specialization of nonsymmetric Macdonald polynomials at t=0, demonstrating their expansion into fundamental slide polynomials and establishing a positive, stable, graded sum of Demazure characters that refines Kostka-Foulkes polynomials.

## Contribution

It provides a combinatorial proof that the specialization expands nonnegatively into fundamental slide polynomials and refines Kostka-Foulkes polynomials through Demazure characters.

## Key findings

- Specialization expands nonnegatively into fundamental slide polynomials.
- The specialization is a positive graded sum of Demazure characters.
- This provides a refinement of Kostka-Foulkes polynomials.

## Abstract

We study the specialization of the type A nonsymmetric Macdonald polynomials at $t=0$ based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of the Kostka--Foulkes polynomials.