The second cohomology of regular semisimple Hessenberg varieties from GKM theory

Anton Ayzenberg; Mikiya Masuda; Takashi Sato

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arXiv · 2203.11580

The second cohomology of regular semisimple Hessenberg varieties from GKM theory

Anton Ayzenberg·Mikiya Masuda·Takashi Sato

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Abstract

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-Lee but they are equivalent. We also discuss its higher degree generalization.

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Anton Ayzenberg, Mikiya Masuda, Takashi Sato. 2022-03-22. The second cohomology of regular semisimple Hessenberg varieties from GKM theory. https://arxiv.org/abs/2203.11580

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