{"title":"关于有限类型簇代数的分母猜想","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":"{\"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}","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}
On denominator conjecture for cluster algebras of finite type
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.