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

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.

36 pages. Draft version, but includes all elements of the proof

References in corpus (3)

• Permutation actions on equivariant cohomology
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
• Power sum expansion of chromatic quasisymmetric functions

Cited by in corpus (19)

• LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
• Geometry of Hessenberg varieties with applications to Newton-Okounkov bodies
• The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture
• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$
• Chromatic symmetric functions of Dyck paths and q-rook theory
• Geometry of regular Hessenberg varieties
• Cancelation free formula for the antipode of linearized Hopf monoid
• A survey of recent developments on Hessenberg varieties
• Permutation module decomposition of the second cohomology of a regular semisimple Hessenberg variety
• Upper-triangular linear relations on multiplicities and the Stanley-Stembridge conjecture
• Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties
• Torus actions, localization and induced representations on cohomology
• Hessenberg varieties, Slodowy slices, and integrable systems
• The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials
• The volume polynomial of regular semisimple Hessenberg varieties and the Gelfand-Zetlin polytope
• Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell
• Splines on Cayley Graphs of the Symmetric Group
• Chromatic Signed-Symmetric Functions of Signed Graphs