{
  "id": "2212.11497",
  "version": "v1",
  "published": "2022-12-22T06:13:42.000Z",
  "updated": "2022-12-22T06:13:42.000Z",
  "title": "Intersection vectors over tilings with applications to gentle algebras and cluster algebras",
  "authors": [
    "Changjian Fu",
    "Shengfei Geng"
  ],
  "comment": "24 pages. Comments welcome!",
  "categories": [
    "math.RT",
    "math.CO",
    "math.RA"
  ],
  "abstract": "Under a mild condition, it is proved that a multiset of permissible arcs over a tiling is uniquely determined by its intersection vector. This generalizes a classical result over marked surfaces with triangulations. We apply this result to study $\\tau$-tilting theory of gentle algebras and denominator conjecture in cluster algebras. For gentle algebras, it is proved that different $\\tau$-rigid $A$-modules over a gentle algebra $A$ have different dimension vectors if and only if $A$ has no even oriented cycle with full relations. For cluster algebras, we establish the denominator conjecture for cluster algebras of type $\\mathbb{A}\\mathbb{B}\\mathbb{C}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-22T06:13:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cluster algebras",
      "gentle algebra",
      "intersection vector",
      "applications",
      "denominator conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}