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Positive expansions of permuted basement and quasisymmetric Macdonald polynomials at $t=0$

Olya Mandelshtam, Harper Niergarth, Kartik Singh

TL;DR

The paper proves that at the t=0 specialization, both quasisymmetric and nonsymmetric Macdonald polynomials admit positive expansions into the corresponding Demazure-atom and quasisymmetric-Schur bases, respectively. It develops a crystal-theoretic framework on multiline queues, using the Ferrari–Martin labeling, a collapsing map, and commuting crystal operators to track type data and show positivity via restricted tableau statistics (charge/maj). The authors derive explicit Kostka–Foulkes-type polynomials K_{αβ}(q) and K_{γτ}(q) governing the decompositions, and deduce that permuted-basement Macdonald polynomials E_α^σ(X;q,0) expand positively into Demazure atoms as a corollary. They further connect these results to tableau formulas (SSAF/SSCT) and ASEP combinatorics, highlighting a rich interplay between nonsymmetric Macdonald theory and quasisymmetric refinements. Overall, the work advances positivity phenomena in the t=0 limit, with potential extensions to broader nonsymmetric Macdonald landscapes and related combinatorial models.

Abstract

It is well known that the $q$-Whittaker polynomials, which are $t=0$ specializations of the Macdonald polynomials $P_λ(X;q,t)$, expand positively as the sum of Schur polynomials. Macdonald polynomials have a quasisymmetric refinement: the quasisymmetric Macdonald polynomials $G_γ(X;q,t)$, and a nonsymmetric refinement: the ASEP polynomials $f_α(X;q,t)$. We study the $t=0$ specializations of both these families of polynomials and show analogous properties: the quasisymmetric Macdonald polynomials expand positively as a sum of quasisymmetric Schur functions, $\text{QS}_γ(X)$, and the ASEP polynomials expand positively as a sum of Demazure atoms, $\mathcal{A}_α(X)$. As a corollary of the latter, we prove more generally that any permuted basement Macdonald polynomial has a positive expansion in the Demazure atoms at $t=0$. We give a description of the structure coefficients of $G_γ(X;q,0)$ and $f_α(X;q,0)$ in both cases in terms of the charge statistic on a restricted set of semistandard tableaux.

Positive expansions of permuted basement and quasisymmetric Macdonald polynomials at $t=0$

TL;DR

The paper proves that at the t=0 specialization, both quasisymmetric and nonsymmetric Macdonald polynomials admit positive expansions into the corresponding Demazure-atom and quasisymmetric-Schur bases, respectively. It develops a crystal-theoretic framework on multiline queues, using the Ferrari–Martin labeling, a collapsing map, and commuting crystal operators to track type data and show positivity via restricted tableau statistics (charge/maj). The authors derive explicit Kostka–Foulkes-type polynomials K_{αβ}(q) and K_{γτ}(q) governing the decompositions, and deduce that permuted-basement Macdonald polynomials E_α^σ(X;q,0) expand positively into Demazure atoms as a corollary. They further connect these results to tableau formulas (SSAF/SSCT) and ASEP combinatorics, highlighting a rich interplay between nonsymmetric Macdonald theory and quasisymmetric refinements. Overall, the work advances positivity phenomena in the t=0 limit, with potential extensions to broader nonsymmetric Macdonald landscapes and related combinatorial models.

Abstract

It is well known that the -Whittaker polynomials, which are specializations of the Macdonald polynomials , expand positively as the sum of Schur polynomials. Macdonald polynomials have a quasisymmetric refinement: the quasisymmetric Macdonald polynomials , and a nonsymmetric refinement: the ASEP polynomials . We study the specializations of both these families of polynomials and show analogous properties: the quasisymmetric Macdonald polynomials expand positively as a sum of quasisymmetric Schur functions, , and the ASEP polynomials expand positively as a sum of Demazure atoms, . As a corollary of the latter, we prove more generally that any permuted basement Macdonald polynomial has a positive expansion in the Demazure atoms at . We give a description of the structure coefficients of and in both cases in terms of the charge statistic on a restricted set of semistandard tableaux.
Paper Structure (17 sections, 33 theorems, 140 equations, 3 figures)

This paper contains 17 sections, 33 theorems, 140 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof sketch
  3. Preliminaries
  4. Symmetric and quasisymmetric functions
  5. Multiline queues
  6. Schur functions and their refinements
  7. $q$-Whittaker polynomials and their quasisymmetric and nonsymmetric refinements
  8. Crystal operators on multiline queues
  9. Schur positivity of $q$-Whittaker polynomials
  10. Crystal operators and type
  11. Column crystal operators and $\textup{type}$ change
  12. Properties of $\textup{type}$ under collapsing
  13. Nonsymmetric and quasisymmetric refinements of $\text{MLQ}_\lambda$
  14. Positive expansions of $f_\alpha(X;q,0)$ and $G_\gamma(X;q,0)$
  15. $E_\alpha^\sigma(X;q,0)$ is Demazure atom-positive
  16. ...and 2 more sections

Key Result

Lemma 2.10

Let $s_1, s_2$ be two strands in a non-wrapping multiline queue $M$, with row-1 balls $y_1$ and $y_2$ respectively, and suppose that $s_1$ is weakly longer than $s_2$. Then $s_1$ and $s_2$ can intersect each other at most once. Moreover, if they intersect, then $y_1$ is to the left of $y_2$.

Figures (3)

  • Figure 1: An example of a labeled multiline queue $M$ with its corresponding array $L_M$.
  • Figure 2: Starting with $M_0=M$ in \ref{['ex:collapsing']}, we show the multiline queues $M_r$ (in black) along with the recording tableaux $Q_r$ at each step of the collapsing procedure of \ref{['def:colmap']}.
  • Figure 3: We show the crystal $\text{NMLQ}_{(3,3,1,1)}$. The different nonsymmetric components $\text{NMLQ}$, for $\texttt{sort}(\alpha)=(3,3,1,1)$, are highlighted. In particular, note that the component $\text{NMLQ}((1,3,1,3))$, highlighted with dashed lines, is disconnected.

Theorems & Definitions (100)

  • Definition 1.1: CHMMW22
  • Example 2.1
  • Definition 2.2
  • Definition 2.3: FM algorithm
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6: Row word and Column word
  • Remark 2.7
  • Example 2.8
  • Definition 2.9: Major index of a multiline queue
  • ...and 90 more