A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
By Mathieu Guay-Paquet. 2016
94 citations recorded by the source corpus.
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.
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Citations in the corpus
- Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties (2015, unknown): “, m − ) will always denote a Hessenberg function , by which we mean a sequence of positive integers satisfying We also define 5 Given m , let P ( m ) denote the poset on the vertex set [ n ] whose order relation ≺ is given Such a poset is called a natural unit interval order .”
- Recent Trends in Quasisymmetric Functions (2018, methodology): “Guay-Paquet [48] proved that the chromatic symmetric function of a (3+1)-free poset is a convex combination of chromatic symmetric functions of posets which are both (3+1)-free and (2+2)-free.”