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On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley-Stembridge Conjecture

Sean T. Griffin, Anton Mellit, Marino Romero, Kevin Weigl, Joshua Jeishing Wen

TL;DR

This work advances the study of the Stanley–Stembridge conjecture by embedding the Shareshian–Wachs $q$-chromatic symmetric functions for unit interval graphs into the Macdonald framework via the $\mathbb{A}_{q,t}$ algebra. The authors derive an explicit expansion of $\chi_\mathbf{e}[X;q]$ in terms of modified Macdonald polynomials $\widetilde{\mathrm{H}}_\mu[X;q,t]$, and show that plethystic substitution yields an $e$-basis expansion at $t=1$, reproducing Hikita's $e$-positive formula; at $t=0$, the expansion specializes to Hall–Littlewood functions. A central contribution is the tableau-driven coefficient formula, which ties the positivity at $q=1$ to the original Stanley–Stembridge conjecture and provides a combinatorial interpretation via admissible colorings. The Hall–Littlewood expansion and the connections to LLT/Hessenberg geometry underscore the broader impact of the $\mathbb{A}_{q,t}$ action in understanding chromatic-like symmetric functions and their positivity properties.

Abstract

The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a $(3+1)$-free graph is $e$-positive. Recently, Hikita proved this conjecture by giving an explicit $e$-expansion of the Shareshian-Wachs $q$-chromatic refinement for unit interval graphs. Using the $\mathbb{A}_{q,t}$ algebra, we give an expansion of these $q$-chromatic symmetric functions into Macdonald polynomials. Upon setting $t=1$, we obtain another proof of the Stanley-Stembridge conjecture and rederive Hikita's formula. Upon setting $t=0$, we obtain an expansion into Hall-Littlewood symmetric functions.

On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley-Stembridge Conjecture

TL;DR

This work advances the study of the Stanley–Stembridge conjecture by embedding the Shareshian–Wachs -chromatic symmetric functions for unit interval graphs into the Macdonald framework via the algebra. The authors derive an explicit expansion of in terms of modified Macdonald polynomials , and show that plethystic substitution yields an -basis expansion at , reproducing Hikita's -positive formula; at , the expansion specializes to Hall–Littlewood functions. A central contribution is the tableau-driven coefficient formula, which ties the positivity at to the original Stanley–Stembridge conjecture and provides a combinatorial interpretation via admissible colorings. The Hall–Littlewood expansion and the connections to LLT/Hessenberg geometry underscore the broader impact of the action in understanding chromatic-like symmetric functions and their positivity properties.

Abstract

The Stanley-Stembridge conjecture asserts that the chromatic symmetric function of a -free graph is -positive. Recently, Hikita proved this conjecture by giving an explicit -expansion of the Shareshian-Wachs -chromatic refinement for unit interval graphs. Using the algebra, we give an expansion of these -chromatic symmetric functions into Macdonald polynomials. Upon setting , we obtain another proof of the Stanley-Stembridge conjecture and rederive Hikita's formula. Upon setting , we obtain an expansion into Hall-Littlewood symmetric functions.

Paper Structure

This paper contains 12 sections, 10 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Symmetric functions
  4. The $\mathbb{A}_{q,t}$ algebra
  5. $q$-chromatic symmetric functions
  6. Macdonald symmetric functions
  7. The action of $d_{\pm}$ on $I_{\lambda,w}$
  8. Main proof
  9. The relation to Hikita's formula
  10. Proof of the main theorem
  11. A Hall--Littlewood expansion
  12. Acknowledgments

Key Result

Theorem 1.1

We have explicit tableau formulas for the coefficients $C_{\mathbf{e},\mu}(q,t)$ appearing in the expansion expanding $\chi_\mathbf{e}(q)$ in terms of plethystically evaluated modified Macdonald polynomials. In particular, when $t=1$, we get Upon setting $q=1$, we find that the coefficient of $e_\mu$ is nonnegative, so the Stanley--Stembridge conjecture holds. Furthermore, the coefficient of $e_

Figures (1)

  • Figure 1: The Dyck path $D$ and unit interval graph $\Gamma_\mathbf{e}$ associated to the reverse hessenberg function $\mathbf{e} = (0,1,1,2,2,3,4)$. Cells are labeled by the edges, $ij$, present in the graph.

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2: Carlsson-Mellit
  • Proposition 2.3
  • Proposition 2.4: Carlsson-Gorsky-Mellit
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Remark 3.5
  • Example 3.6
  • ...and 10 more