{"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-kazhdan-lusztig-polynomials-of-uniform","title":"The Kazhdan-Lusztig polynomials of uniform matroids","arxiv_id":"1806.10852","date":"2018-06-28","proceeding":null,"authors":["Alice L. L. Gao","Linyuan Lu","Matthew H. Y. Xie","Arthur L. B. Yang","Philip B. Zhang"],"abstract":"The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [{\\it Adv. Math. 2016}]. Let $U_{m,d}$ denote the uniform matroid of rank $d$ on a set of $m+d$ elements. Gedeon, Proudfoot, and Young [{\\it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of $U_{m,d}$ using equivariant Kazhdan-Lusztig polynomials. In this paper we give two alternative explicit formulas, which allow us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of $U_{m,d}$ for $2\\leq m\\leq 15$ and all $d$'s. The case $m=1$ was previously proved by Gedeon, Proudfoot, and Young [{\\it S\\'{e}m. Lothar. Combin. 2017}]. We further determine the $Z$-polynomials of all $U_{m,d}$'s and prove the real-rootedness of the $Z$-polynomials of $U_{m,d}$ for $2\\leq m\\leq 15$ and all $d$'s. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of $U_{m,d}$'s without using the equivariant Kazhdan-Lusztig polynomials.","url_abs":"http://arxiv.org/abs/1806.10852v1","url_pdf":"http://arxiv.org/pdf/1806.10852v1.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-kazhdan-lusztig-polynomials-of-uniform","repo_url":"https://github.com/mathxie/kl_uniform_matroid","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}