Geodesic
The Cohomology of Abelian Hessenberg Varieties and the Stanley-Stembridge Conjecture
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@article{SLC_2018_80B_a48,
     author = {Megumi Harada and Martha Precup},
     title = {The {Cohomology} of {Abelian} {Hessenberg} {Varieties} and the {Stanley-Stembridge} {Conjecture}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a48/}
}
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VL  - 80B
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Megumi Harada; Martha Precup. The Cohomology of Abelian Hessenberg Varieties and the Stanley-Stembridge Conjecture. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a48/