Geodesic
The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory
Informatics and Automation, 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 = {Informatics and Automation},
     pages = {5--26},
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     volume = {317},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_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. Informatics and Automation, Toric Topology, Group Actions, Geometry, and Combinatorics. Part 1, Tome 317 (2022), pp. 5-26. http://geodesic.mathdoc.fr/item/TRSPY_2022_317_a0/