Séminaire Lotharingien de Combinatoire, 80B.49 (2018), 12 pp.

Megumi Harada and Martha Precup

The Cohomology of Abelian Hessenberg Varieties and the Stanley-Stembridge Conjecture

Abstract. We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We prove that the cohomology of an abelian regular semisimple Hessenberg variety, with respect to the symmetric group action defined by Tymoczko, is a non-negative combination of tabloid representations. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case. As part of our arguments, we obtain inductive formulas for the Betti numbers of regular Hessenberg varieties.


Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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