The second cohomology of regular semisimple Hessenberg varieties from GKM theory
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.11580
- arXiv:
- arXiv:2203.11580
- Bibcode:
- 2022arXiv220311580A
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Algebraic Topology;
- Mathematics - Representation Theory;
- 57S12 (Primary);
- 14M15 (Secondary)
- E-Print:
- 23 pages, 3 figures