A geometric realization of the chromatic symmetric function of a unit interval graph

TL;DR

This paper presents a new geometric realization of the chromatic symmetric function of unit interval graphs using Betti cohomology, providing a novel combinatorial expression and a refined conjecture related to Stanley-Stembridge.

Contribution

It introduces a geometric realization via Betti cohomology and proposes a refined conjecture, advancing understanding of chromatic symmetric functions of unit interval graphs.

Findings

New geometric realization in terms of Betti cohomology

Inductive combinatorial expression of chromatic symmetric functions

Proposed geometric refinement of the Stanley-Stembridge conjecture

Abstract

Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. ${\bf 295}$ (2016), ${\bf 329}$ (2018), arXiv:1601.05498] realized the chromatic (quasi-)symmetric function of a unit interval graph in terms of Hessenberg varieties. Here we exhibit another realization of these chromatic (quasi-)symmetric functions in terms of the Betti cohomology of the variety $\mathscr X_\Psi$ defined in [arXiv:2301.00862]. This yields a new inductive combinatorial expression of these chromatic symmetric functions. Based on this, we propose a geometric refinement of the Stanley-Stembridge conjecture, whose validity would imply the Shareshian-Wachs conjecture.