{"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/factorizing-the-brauer-monoid-in-polynomial","title":"Factorizing the Brauer monoid in polynomial time","arxiv_id":"2402.07874","date":"2024-02-12","proceeding":null,"authors":["Daniele Marchei","Emanuela Merelli","Andrew Francis"],"abstract":"Finding a minimal factorization for a generic semigroup can be done by using the Froidure-Pin Algorithm, which is not feasible for semigroups of large sizes. On the other hand, if we restrict our attention to just a particular semigroup, we could leverage its structure to obtain a much faster algorithm. In particular, $\\mathcal{O}(N^2)$ algorithms are known for factorizing the Symmetric group $S_N$ and the Temperley-Lieb monoid $\\mathcal{T}\\mathcal{L}_N$, but none for their superset the Brauer monoid $\\mathcal{B}_{N}$. In this paper we hence propose a $\\mathcal{O}(N^4)$ factorization algorithm for $\\mathcal{B}_{N}$. At each iteration, the algorithm rewrites the input $X \\in \\mathcal{B}_{N}$ as $X = X' \\circ p_i$ such that $\\ell(X') = \\ell(X) - 1$, where $p_i$ is a factor for $X$ and $\\ell$ is a length function that returns the minimal number of factors needed to generate $X$.","url_abs":"https://arxiv.org/abs/2402.07874v2","url_pdf":"https://arxiv.org/pdf/2402.07874v2.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":"factorizing-the-brauer-monoid-in-polynomial","repo_url":"https://github.com/danielemarchei/brauermonoidfactorization","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}