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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 ω*X*_{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.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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