Geodesic
The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 1, Tome 317 (2022), pp. 5-26

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We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group $\mathfrak {S}_n$ by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an $\mathfrak {S}_n$-module. Our formula is not exactly the same as the known formula by Chow or Cho, Hong, and Lee, but they are equivalent. We also discuss its higher degree generalization.
Keywords: Hessenberg variety, GKM theory, equivariant cohomology
Mots-clés : torus action, permutation module. 
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     author = {Anton A. Ayzenberg and Mikiya Masuda and Takashi Sato},
     title = {The {Second} {Cohomology} of {Regular} {Semisimple} {Hessenberg} {Varieties} from {GKM} {Theory}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     year = {2022},
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     url = {http://geodesic.mathdoc.fr/item/TM_2022_317_a0/}
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Anton A. Ayzenberg; Mikiya Masuda; Takashi Sato. The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 1, Tome 317 (2022), pp. 5-26. http://geodesic.mathdoc.fr/item/TM_2022_317_a0/