{"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/homology-representations-of-compactified","title":"Homology representations of compactified configurations on graphs applied to $\\mathcal{M}_{2,n}$","arxiv_id":"2109.03302","date":"2021-09-07","proceeding":null,"authors":["Christin Bibby","Melody Chan","Nir Gadish","Claudia He Yun"],"abstract":"We obtain new calculations of the top weight rational cohomology of the moduli spaces $\\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\\Delta_{2,n}$, as a representation of $S_n$. These calculations are achieved fully for all $n\\leq 10$, and partially -- for specific irreducible representations of $S_n$ -- for $n\\le 22$. We also present conjectures, verified up to $n=22$, for the multiplicities of the irreducible representations $\\mathrm{std}_n$ and $\\mathrm{std}_n\\otimes \\mathrm{sgn}_n$. We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.","url_abs":"https://arxiv.org/abs/2109.03302v6","url_pdf":"https://arxiv.org/pdf/2109.03302v6.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":"homology-representations-of-compactified","repo_url":"https://github.com/ClaudiaHeYun/Homology-rep-of-compactified-conf-on-graphs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"homology-representations-of-compactified","repo_url":"https://github.com/claudiaheyun/bcgy","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"homology-representations-of-compactified","repo_url":"https://github.com/louishainaut/gh-confspace","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}