Flagged LLT polynomials, nonsymmetric plethysm, and nonsymmetric Macdonald polynomials

TL;DR

This paper introduces nonsymmetric flagged LLT polynomials, explores their properties, and formulates a nonsymmetric Macdonald positivity conjecture, providing new algebraic and combinatorial tools for Macdonald polynomial theory.

Contribution

It defines nonsymmetric flagged LLT polynomials, constructs a plethysm operator, and proposes a new nonsymmetric Macdonald positivity conjecture with positive sum expansions.

Findings

Flagged LLT polynomials admit algebraic and combinatorial descriptions.

The plethysm operator maps flagged LLT polynomials over signed to unsigned alphabets.

A conjecture that modified nonsymmetric Macdonald polynomials are atom positive.

Abstract

The plethystic transformation $f[X] \mapsto f[X/(1-t)]$ and LLT polynomials are central to the theory of symmetric Macdonald polynomials. In this work, we introduce and study nonsymmetric flagged LLT polynomials. We show that these admit both an algebraic and a combinatorial description, that they Weyl symmetrize to the usual symmetric LLT polynomials, and we conjecture that they expand positively in terms of Demazure atoms. Additionally, we construct a nonsymmetric plethysm operator $\Pi_{t,x}$ on $\mathfrak{K}[x_1,\ldots,x_n]$, which serves as an analogue of $f[X] \mapsto f[X/(1-t)]$. We prove that $\Pi_{t,x}$ remarkably maps flagged LLT polynomials defined over a signed alphabet to ones over an unsigned alphabet. Our main application of this theory is to formulate a nonsymmetric version of Macdonald positivity, similar in spirit to conjectures of Knop and Lapointe, but with several new features. To do this, we recast the Haglund-Haiman-Loehr formula for nonsymmetric Macdonald polynomials $\mathcal{E}_{\mu }(x;q,t)$ as a positive sum of signed flagged LLT polynomials. Then, after applying a suitable stable limit of $\Pi_{t,x}$ to a stable version of $\mathcal{E}_{\mu }(x;q,t)$, we obtain modified nonsymmetric Macdonald polynomials which are positive sums of flagged LLT polynomials and thus are conjecturally atom positive, strengthening the Macdonald positivity conjecture.