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A proof of the Stanley--Stembridge conjecture

Tatsuyuki Hikita

TL;DR

This work proves the Stanley--Stembridge conjecture by constructing a probabilistic representation for the $e$-expansion coefficients of the chromatic quasisymmetric function of unit interval graphs. Central to the approach is a probabilistic formula that expresses ${\mathbf{X}}_{\Gamma}(x;q)$ as a weighted sum over partitions, with weights interpreted as probabilities arising from a time-inhomogeneous Markov process on standard Young tableaux guided by conjugate Hessenberg data. The proof reduces to verifying an operator-level modular law (via Abreu–Nigro’s framework) and hinges on three key lemmas proved through residue calculus and a generalized skyline-detour construction. The result not only establishes $e$-positivity for unit interval graphs (and hence the Stanley--Stembridge conjecture in this setting) but also connects combinatorics with geometry and representation theory through Kato’s formula and related structures. The probabilistic interpretation provides a flexible framework for further refinements and potential extensions to broader classes of graphs and symmetric-function positivity problems.

Abstract

We give a probabilistic interpretation of the coefficients of the elementary symmetric function expansion of the chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley--Stembridge conjecture.

A proof of the Stanley--Stembridge conjecture

TL;DR

This work proves the Stanley--Stembridge conjecture by constructing a probabilistic representation for the -expansion coefficients of the chromatic quasisymmetric function of unit interval graphs. Central to the approach is a probabilistic formula that expresses as a weighted sum over partitions, with weights interpreted as probabilities arising from a time-inhomogeneous Markov process on standard Young tableaux guided by conjugate Hessenberg data. The proof reduces to verifying an operator-level modular law (via Abreu–Nigro’s framework) and hinges on three key lemmas proved through residue calculus and a generalized skyline-detour construction. The result not only establishes -positivity for unit interval graphs (and hence the Stanley--Stembridge conjecture in this setting) but also connects combinatorics with geometry and representation theory through Kato’s formula and related structures. The probabilistic interpretation provides a flexible framework for further refinements and potential extensions to broader classes of graphs and symmetric-function positivity problems.

Abstract

We give a probabilistic interpretation of the coefficients of the elementary symmetric function expansion of the chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley--Stembridge conjecture.

Paper Structure

This paper contains 51 sections, 24 theorems, 126 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Conjugate Hessenberg functions
  4. Probabilistic ansatz
  5. Maya diagrams
  6. Probabilistic models
  7. Probabilistic formula
  8. Organization of the paper
  9. Recent developments
  10. Survey
  11. Notation
  12. Symmetric functions
  13. Young tableaux
  14. Graphs and posets
  15. Graphs and posets
  16. ...and 36 more sections

Key Result

Theorem 1.6

Let ${\mathsf{e}}:[n]\rightarrow{\mathbb{Z}}$ be a conjugate Hessenberg function and $\Gamma$ the corresponding unit interval graph. Then we have where the probabilities are viewed as rational functions in $q$.

Figures (6)

  • Figure 1: Maya diagram $\delta=(1^{\infty},0,1,0,1,0^{\infty})$ associated with $T$ and $r=4$. In this case, we have $W(\delta)=\{1,3,5\}$ and $R(\delta)=\{2,4\}$.
  • Figure 2: An example of the transition probabilities at step 7 for ${\mathsf{e}}(7)=4$.
  • Figure 3: Dyck paths and corresponding unit interval graphs
  • Figure 4: Bijections between Catalan objects: Dyck path $\pi$, Hessenberg function ${\mathsf{h}}=(3,4,4,5,5)$, conjugate Hessenberg function ${\mathsf{e}}=(0,0,0,1,3)$, and unit interval graph $\Gamma=\Gamma_{\pi}$. These objects also correspond to the area sequence ${\mathsf{a}}=(0,1,2,2,1)$ and the 312-avoiding permutation $w=34251$.
  • Figure 5: Modular triples $({\mathsf{e}},{\mathsf{e}}',{\mathsf{e}}")$ of type I and II. In both cases, ${\mathsf{e}}=(0,0,1,1,2,3,5)$ corresponds to the black path, ${\mathsf{e}}'$ corresponds to the red path, and ${\mathsf{e}}"$ corresponds to the blue path. The last conditions in Definition \ref{['Def:Modular_Triple']} mean that the green dotted path intersects the Dyck path corresponding to ${\mathsf{e}}$ vertically.
  • ...and 1 more figures

Theorems & Definitions (70)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Conjecture 2.1: Stanley--Stembridge SS93
  • Conjecture \ref{Conj_SS}': Sta95
  • ...and 60 more