Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$

TL;DR

This paper studies non-symmetric Macdonald polynomials at special parameters, revealing symmetry properties, factorization, and expansion positivity, with implications for Demazure characters and Schur polynomial products.

Contribution

It provides new insights into the structure and positivity properties of non-symmetric Macdonald polynomials at q=1 and q=0, including their symmetry, factorization, and expansion into permuted-basement atoms.

Findings

At q=1, E_λ(x;1,t) is symmetric and t-independent for partitions.

At q=0, permuted-basement t-atoms expand positively into Demazure characters.

Product of a permuted-basement atom and a Schur polynomial remains positive in the same basis.

Abstract

We examine the non-symmetric Macdonald polynomials $E_\lambda(x;q,t)$ at $q=1$, as well as the more general permuted-basement Macdonald polynomials. When $q=1$, we show that $E_\lambda(x;1,t)$ is symmetric and independent of $t$ whenever $\lambda$ is a partition. Furthermore, we show that for general $\lambda$, this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of $t$, while the non-symmetric part only depends on the relative order of the entries in $\lambda$. We also examine the case $q=0$, which give rise to so called permuted-basement $t$-atoms. We prove expansion-properties of these, and as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that a product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis, and thus interpolates between two results by Haglund, Luoto, Mason and van Willigenburg. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by the first author.