{"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-cohomology-of-the-free-loop-spaces-of-su","title":"The cohomology of the free loop spaces of $SU(n+1)/T^n$","arxiv_id":"2106.03440","date":"2021-06-07","proceeding":null,"authors":["Matthew I. Burfitt","Jelena Grbić"],"abstract":"We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the cohomology algebra of the free loop spaces. We build new theory to allow for the computation of $H^*(\\Lambda(SU(n+1)/T^{n});\\mathbb{Z})$, a significantly more complicated structure than other known examples. In addition to our theoretical results, we explicitly implement a novel integral Gr\\\"obner basis procedure for computation. This procedure is applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. The power of this procedure is illustrated by the explicit calculation of $H^*( \\Lambda(SU(4)/T^3);\\mathbb{Z})$. We also provide a python library with examples of all procedures used in the paper.","url_abs":"https://arxiv.org/abs/2106.03440v2","url_pdf":"https://arxiv.org/pdf/2106.03440v2.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-cohomology-of-the-free-loop-spaces-of-su","repo_url":"https://github.com/MatthewBurfitt/Integral-Grobner-basis-for-spectral-sequence","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"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}