  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.
