{
  "id": "2409.10914",
  "version": "v1",
  "published": "2024-09-17T06:15:04.000Z",
  "updated": "2024-09-17T06:15:04.000Z",
  "title": "On denominator conjecture for cluster algebras of finite type",
  "authors": [
    "Changjian Fu",
    "Shengfei Geng"
  ],
  "comment": "27pages, comments welcome",
  "categories": [
    "math.RT",
    "math.CO",
    "math.RA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-17T06:15:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cluster algebras",
      "denominator conjecture",
      "finite type",
      "geometric model",
      "exceptional types"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}