A uniform model for Kirillov-Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at $t=0$ and Demazure characters

TL;DR

This paper proves the equality between specialized nonsymmetric Macdonald polynomials at t=0 and Demazure characters of certain modules, extending previous symmetric cases and providing combinatorial formulas for these specializations.

Contribution

It establishes a new connection between nonsymmetric Macdonald polynomials at t=0 and Demazure module characters, generalizing prior symmetric polynomial results.

Findings

Equality of specialized nonsymmetric Macdonald polynomials and Demazure characters

Two combinatorial formulas for the polynomial specialization

Extension of previous symmetric polynomial results

Abstract

We establish the equality of the specialization $E_{w\lambda}(x;q,0)$ of the nonsymmetric Macdonald polynomial $E_{w\lambda}(x;q,t)$ at $t=0$ with the graded character $\mathop{\rm gch} U_{w}^{+}(\lambda)$ of a certain Demazure-type submodule $U_{w}^{+}(\lambda)$ of a tensor product of "single-column" Kirillov--Reshetikhin modules for an untwisted affine Lie algebra, where $\lambda$ is a dominant integral weight and $w$ is a (finite) Weyl group element, this generalizes our previous result, that is, the equality between the specialization $P_{\lambda}(x;q,0)$ of the symmetric Macdonald polynomial $P_{\lambda}(x;q,t)$ at $t=0$ and the graded character of a tensor product of single-column Kirillov--Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of a nonsymmetric Macdonald polynomial: one in terms of quantum Lakshmibai-Seshadri paths and the other in terms of the quantum alcove model.