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The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory

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

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Notes

  1. A. Ayzenberg, M. Masuda, and T. Sato, “Regular semisimple Hessenberg varieties with cohomology generated in degree two” (in preparation).

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Funding

The work of the first and second authors was performed within the framework of the HSE University Basic Research Program.

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Authors and Affiliations

  1. Faculty of Computer Science, HSE University, Pokrovskii bul. 11, Moscow, 109028, Russia

    Anton A. Ayzenberg

  2. Osaka City University Advanced Mathematical Institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Mikiya Masuda & Takashi Sato

  3. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Takashi Sato

Authors
  1. Anton A. Ayzenberg
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  2. Mikiya Masuda
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  3. Takashi Sato
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Corresponding author

Correspondence to Anton A. Ayzenberg.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 317, pp. 5–26 https://doi.org/10.4213/tm4289.

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Ayzenberg, A.A., Masuda, M. & Sato, T. The Second Cohomology of Regular Semisimple Hessenberg Varieties from GKM Theory. Proc. Steklov Inst. Math. 317, 1–20 (2022). https://doi.org/10.1134/S0081543822020018

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  • DOI: https://doi.org/10.1134/S0081543822020018

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