Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties

TL;DR

This paper proves the Shareshian–Wachs conjecture linking the symmetric group action on Hessenberg variety cohomology to chromatic quasisymmetric functions, using geometric and combinatorial methods.

Contribution

It confirms the conjecture by connecting geometric cohomology with combinatorial symmetric functions through a novel proof.

Findings

Proves the Shareshian–Wachs conjecture.

Establishes a bijective proof linking cohomology and chromatic quasisymmetric functions.

Uses local invariant cycle theorem to relate cohomology to combinatorial structures.

Abstract

Motivated by a 1993 conjecture of Stanley and Stembridge, Shareshian and Wachs conjectured that the characteristic map takes the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety to $\omega X_G(t)$, where $X_G(t)$ is the chromatic quasisymmetric function of the incomparability graph $G$ of the corresponding natural unit interval order, and $\omega$ is the usual involution on symmetric functions. We prove the Shareshian--Wachs conjecture. Our proof uses the local invariant cycle theorem of Beilinson-Bernstein-Deligne to obtain a surjection from the cohomology of a regular Hessenberg variety of Jordan type $\lambda$ to a space of local invariant cycles; as $\lambda$ ranges over all partitions, these spaces collectively contain all the information about the dot action on a regular semisimple Hessenberg variety. Using a palindromicity argument, we show that in our case the surjections are actually isomorphisms, thus reducing the Shareshian-Wachs conjecture to computing the cohomology of a regular Hessenberg variety. But this cohomology has already been described combinatorially by Tymoczko; we give a bijective proof (using a generalization of a combinatorial reciprocity theorem of Chow) that Tymoczko's combinatorial description coincides with the combinatorics of the chromatic quasisymmetric function.