{
  "id": "0810.3638",
  "version": "v1",
  "published": "2008-10-20T19:10:57.000Z",
  "updated": "2008-10-20T19:10:57.000Z",
  "title": "Cluster expansion formulas and perfect matchings",
  "authors": [
    "Gregg Musiker",
    "Ralf Schiffler"
  ],
  "comment": "19 pages, 8 figures",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph $G_{T,\\gamma}$ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph $G_{T,\\gamma}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-20T19:10:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S99",
      "05E99",
      "05C70"
    ],
    "keywords": [
      "cluster expansion formulas",
      "perfect matchings",
      "laurent polynomial expansion",
      "study cluster algebras",
      "principal coefficient systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.3638M"
    }
  }
}