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

By P. Brosnan, T. Chow. 2015

140 citations recorded by the source corpus.

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

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38d5d24d27653d9ad7f233ceb379088889a72dfd

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In Discrete Mathematics and Combinatorics

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. Sarah K. Mason
6. Martha Precup
7. James Haglund
8. T. Panov

In Geometry and Topology

1. M. Masuda
2. T. Horiguchi
3. Hiraku Abe
4. Megumi Harada
5. Sarah K. Mason
6. Martha Precup
7. T. Panov
8. D. Suh

In Mathematical Physics

1. Megumi Harada
2. Sarah K. Mason
3. Hiraku Abe
4. M. Masuda
5. T. Horiguchi
6. Martha Precup

Citations in the corpus

• The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture (2017, result): “For any strictly negative Hessenberg function of the form h = (m1,m2, n, n, . . . , n), Shareshian and Wachs proved results on the associated chromatic quasisymmetric polynomial which, when paired with Brosnan and Chow’s Theorem 2.10, proves Conjecture 2.8 for that case.”
• Recent Trends in Quasisymmetric Functions (2018, methodology): “This conjecture was proved by Brosnan and Chow [22] and, through a different approach, by Guay-Paquet [49], providing an alternate proof of Schur positivity.”
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, methodology): “mmediately yield further information about the product structure on the rings H∗(Hess(N,h)) and H∗(Hess(S,h))Sn. Thus, for our special case, our Theorem B is stronger than the corresponding result in [9]. Moreover, while Brosnan and Chow’s arguments utilize deep and powerful results in the theory of”