Séminaire Lotharingien de Combinatoire, 78B.3 (2017), 11 pp.
Patrick Brosnan and Timothy Y. Chow
Unit Interval Orders and the Dot Action on the
Cohomology of Regular Semisimple Hessenberg
Varieties
Abstract.
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 ωXG(t),
where XG(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.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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