{"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/nilpotent-matrices-having-a-given-jordan-type","title":"Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit","arxiv_id":"1409.2192","date":"2014-09-08","proceeding":null,"authors":["Anthony Iarrobino","Leila Khatami","Bart Van Steirteghem","Rui Zhao"],"abstract":"The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions $(P,Q)$, where $Q={\\mathcal Q}(P)$ is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type $ P$. T. Ko\\v{s}ir and P. Oblak have shown that $Q$ has parts that differ pairwise by at least two. Such partitions, which are also known as \"super distinct\" or \"Rogers-Ramanujan\", are exactly those that are stable or \"self-large\" in the sense that ${\\mathcal Q}(Q)=Q$. In 2012 P. Oblak formulated a conjecture concerning the cardinality of the set of partitions $P$ such that ${\\mathcal Q}(P)$ is a given stable partition $ Q$ with two parts, and proved some special cases. R. Zhao refined this to posit that those partitions $P$ such that ${\\mathcal Q}(P)= Q=(u,u-r)$ with $u>r\\ge 2$ could be arranged in an $(r-1)$ by $(u-r)$ table ${\\mathcal T}(Q)$ where the entry in the $k$-th row and $\\ell$-th column has $k+\\ell$ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions $P$ for which ${\\mathcal Q}(P)=Q$, for an arbitrary stable partition $Q$.","url_abs":"https://arxiv.org/abs/1409.2192v3","url_pdf":"https://arxiv.org/pdf/1409.2192v3.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":"nilpotent-matrices-having-a-given-jordan-type","repo_url":"https://github.com/FermenFlo/Oblak","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}