Non-symmetric Macdonald polynomials and Demazure-Lusztig operators

TL;DR

This paper introduces generalized non-symmetric Macdonald polynomials, explores their properties under Demazure-Lusztig operators, and connects them to classical symmetric functions, providing new positive expansion formulas and unifying existing results.

Contribution

It extends non-symmetric Macdonald polynomials to a broader family, defines general-basement variants, and establishes their behavior under Demazure-Lusztig operators, linking them to classical symmetric functions.

Findings

General-basement Macdonald polynomials satisfy triangularity properties.

Hall-Littlewood polynomials expand positively into t-deformations of Demazure atoms.

Schur polynomials decompose into t-deformations of Demazure atoms with non-negative coefficients.

Abstract

We extend the family non-symmetric Macdonald polynomials and define general-basement Macdonald polynomials. We show that these also satisfy a triangularity property with respect to the monomials bases and behave well under the Demazure-Lusztig operators. The symmetric Macdonald polynomials $J_\lambda$ are expressed as a sum of general-basement Macdonald polynomials via an explicit formula. By letting $q=0$, we obtain $t$-deformations of key polynomials and Demazure atoms and we show that the Hall--Littlewood polynomials expand positively into these. This generalizes a result by Haglund, Luoto, Mason and van Willigenburg. As a corollary, we prove that Schur polynomials decompose with non-negative coefficients into $t$-deformations of general Demazure atoms and thus generalizing the $t=0$ case which was previously known. This gives a unified formula for the classical expansion of Schur polynomials in Hall-Littlewood polynomials and the expansion of Schur polynomials into Demazure atoms.