{"title":"On denominator conjecture for cluster algebras of finite type","authors":"Changjian Fu, Shengfei Geng","doi":"arxiv-2409.10914","DOIUrl":null,"url":null,"abstract":"We continue our investigation on denominator conjecture of Fomin and\nZelevinsky for cluster algebras via geometric models initialed in \\cite{FG22}.\nIn this paper, we confirm the denominator conjecture for cluster algebras of\nfinite type. The new contribution is a proof of this conjecture for cluster\nalgebras of type $\\mathbb{D}$ and an algorithm for the exceptional types. For\nthe type $\\mathbb{D}$ cases, our approach involves geometric model provided by\ndiscs with a puncture. By removing the puncture or changing the puncture to an\nunmarked boundary component, this also yields an alternative proof for the\ndenominator conjecture of cluster algebras of type $\\mathbb{A}$ and\n$\\mathbb{C}$ respectively.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"4 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10914","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We continue our investigation on denominator conjecture of Fomin and
Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}.
In this paper, we confirm the denominator conjecture for cluster algebras of
finite type. The new contribution is a proof of this conjecture for cluster
algebras of type $\mathbb{D}$ and an algorithm for the exceptional types. For
the type $\mathbb{D}$ cases, our approach involves geometric model provided by
discs with a puncture. By removing the puncture or changing the puncture to an
unmarked boundary component, this also yields an alternative proof for the
denominator conjecture of cluster algebras of type $\mathbb{A}$ and
$\mathbb{C}$ respectively.