A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra

TL;DR

This paper presents a new proof of the Shareshian--Wachs conjecture using a novel Hopf algebra approach, offering an alternative to the existing geometric deformation proof and aiming to deepen understanding of Hessenberg varieties and symmetric functions.

Contribution

It introduces a new Hopf algebra framework to recursively decompose Hessenberg varieties, providing a novel proof of the Shareshian--Wachs conjecture.

Findings

Provides a second proof of the conjecture

Introduces a new Hopf algebra for Hessenberg variety decomposition

Enhances understanding of symmetric functions and algebraic geometry

Abstract

This is a set of working notes which give a second proof of the Shareshian--Wachs conjecture, the first (and recent) proof being by Brosnan and Chow in November 2015. The conjecture relates some symmetric functions constructed combinatorially out of unit interval graphs (their $q$-chromatic quasisymmetric functions), and some symmetric functions constructed algebro-geometrically out of Tymoczko's representation of the symmetric group on the equivariant cohomology ring of a family of subvarieties of the complex flag variety, called regular semisimple Hessenberg varieties. Brosnan and Chow's proof is based in part on the idea of deforming the Hessenberg varieties. The proof given here, in contrast, is based on the idea of recursively decomposing Hessenberg varieties, using a new Hopf algebra as the organizing principle for this recursion. We hope that taken together, each approach will shed some light on the other, since there are still many outstanding questions regarding the objects under study.