{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-second-cohomology-of-regular-semisimple","title":"The second cohomology of regular semisimple Hessenberg varieties from GKM theory","arxiv_id":"2203.11580","date":"2022-03-22","proceeding":null,"authors":["Anton Ayzenberg","Mikiya Masuda","Takashi Sato"],"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.","url_abs":"https://arxiv.org/abs/2203.11580v1","url_pdf":"https://arxiv.org/pdf/2203.11580v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-second-cohomology-of-regular-semisimple","repo_url":"https://github.com/antonayzenberg/gkm-homology","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}