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

Patrick Brosnan ; Timothy Y. Chow

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

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 ωX_G(t), where X_G(t) is the chromatic quasisymmetric function of the incomparability graph G of the corresponding natural unit interval order, and ω 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 λ to a space of local invariant cycles; as λ 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.