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

Abstract

Shareshian-Wachs, Brosnan-Chow, and Guay-Pacquet [Adv. Math. $295$ (2016), $329$ (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 ${\mathcal{X}}_{\Psi }$ 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.

16pp, v2 typographical corrections and arXiv number inserted in a reference