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Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law

JiSun Huh, Byung-Hak Hwang, Donghyun Kim, Jang Soo Kim, Jaeseong Oh

TL;DR

This work refines the $e$-positivity phenomenon for chromatic quasisymmetric functions associated with natural unit interval orders by integrating Abreu–Nigro’s $g$-functions with a restricted modular law. It introduces three interrelated function families, $E_{f m}(x;q)$, $G_{f m}(x;q)$, and $S_{f m}(x;q)$, and proves they coincide for all Hessenberg inputs, along with a common refinement $E_{{f m},k}(x;q)=G_{{f m},k}(x;q)$. The authors further establish that a restricted modular law suffices to determine these functions from their values on disjoint unions of paths, enabling immediate $e$-positivity at $q=1$ and a Schur expansion of $ extstyle hspace{-0.5em}\sum_{k=1}^n e_k(x) g_{{f m},n-k}(x;q)$ via $P$-tableaux with $T(1,1)=1$. The path-case is worked out explicitly, yielding concrete formulas and illustrating the multiplicativity and a sink-type combinatorial interpretation, which together deepen the relationship between Hessenberg varieties, symmetric functions, and graph-theoretic decompositions.

Abstract

We study the symmetric functions \( g_{\mm,k}(x;q) \), introduced by Abreu and Nigro for a Hessenberg function \( \mm \) and a positive integer \( k \), which refine the chromatic symmetric function. Building on Hikita's recent breakthrough on the Stanley--Stembridge conjecture, we prove the \( e \)-positivity of \( g_{\mm,k}(x;1) \), refining Hikita's result. We also provide a Schur expansion of the sum \( \sum_{k=1}^n e_k(x) g_{\mm,n-k}(x;q) \) in terms of \( P \)-tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs.

Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law

TL;DR

This work refines the -positivity phenomenon for chromatic quasisymmetric functions associated with natural unit interval orders by integrating Abreu–Nigro’s -functions with a restricted modular law. It introduces three interrelated function families, , , and , and proves they coincide for all Hessenberg inputs, along with a common refinement . The authors further establish that a restricted modular law suffices to determine these functions from their values on disjoint unions of paths, enabling immediate -positivity at and a Schur expansion of via -tableaux with . The path-case is worked out explicitly, yielding concrete formulas and illustrating the multiplicativity and a sink-type combinatorial interpretation, which together deepen the relationship between Hessenberg varieties, symmetric functions, and graph-theoretic decompositions.

Abstract

We study the symmetric functions \( g_{\mm,k}(x;q) \), introduced by Abreu and Nigro for a Hessenberg function and a positive integer , which refine the chromatic symmetric function. Building on Hikita's recent breakthrough on the Stanley--Stembridge conjecture, we prove the -positivity of \( g_{\mm,k}(x;1) \), refining Hikita's result. We also provide a Schur expansion of the sum \( \sum_{k=1}^n e_k(x) g_{\mm,n-k}(x;q) \) in terms of -tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs.

Paper Structure

This paper contains 22 sections, 44 theorems, 179 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Organization
  4. Preliminaries
  5. Compositions, partitions, and symmetric functions
  6. Hessenberg functions and associated objects
  7. Chromatic symmetric functions
  8. Hikita's model
  9. Algorithm for restricted modular law
  10. Restricted modular law for $E$, $G$, and $S$
  11. Restricted modular law for $E$
  12. Restricted modular law for $G$
  13. Restricted modular law for $S$
  14. Multiplicativity for $E$, $G$ and $S$
  15. Multiplicativity for $E$
  16. ...and 7 more sections

Key Result

Theorem 1.2

The chromatic symmetric function $X_\mathbf{m}(x;1)$ of a natural unit interval order $\mathbf{m}$ is $e$-positive. In other words, $c_\lambda(1) \geq 0$ for all $\lambda$.

Figures (13)

  • Figure 1: The Dyck path, the Hasse diagram of the natural unit interval order, and the incomparability graph of the natural unit interval order associated with $\mathbf{m}=(2,3,5,5,5)\in \mathbb{H}_5$.
  • Figure 2: The transition probabilities for $\mathbf{m}=(2,3,5,5,5)$. Here, $p_\mathbf{m}(S;q)=q[2]_q/[3]_q$, $p_\mathbf{m}(R;q)=q/[4]_q$, and $p_\mathbf{m}(T;q)=1/[3]_q[4]_q$. When computing $\psi^{(r)}_k$, the cells containing an integer greater than $r$ are colored red.
  • Figure 3: A modular triple $(\mathbf{m},\mathbf{m}',\mathbf{m}")$ of type I. Note that $\mathbf{m}$, $\mathbf{m}'$, and $\mathbf{m}"$ differ only in the highlighted region.
  • Figure 4: A restricted modular triple $(\mathbf{m},\mathbf{m}',\mathbf{m}")$ of type II. Note that $\mathbf{m}$, $\mathbf{m}'$, and $\mathbf{m}"$ differ only in the highlighted region.
  • Figure 5: The Hessenberg function $\mathbf{m}_1$ is flat, while $\mathbf{m}_2$ is non-flat. The highlighted regions indicate the $\beta(\mathbf{m}_1)$-th and $\beta(\mathbf{m}_2)$-th columns of the grids.
  • ...and 8 more figures

Theorems & Definitions (82)

  • Conjecture 1.1: Stanley1993Stanley1995Shareshian2016
  • Theorem 1.2: Hikita2024
  • Theorem 1.3: Brosnan2018Guay-Paquet2016
  • Theorem 1.4: Abreu2023
  • Conjecture 1.5: Abreu2023
  • Theorem 1.6: Gasharov1996Shareshian2016
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Definition 2.1
  • ...and 72 more