{
  "id": "0809.2593",
  "version": "v2",
  "published": "2008-09-15T19:35:41.000Z",
  "updated": "2008-09-18T17:44:31.000Z",
  "title": "On cluster algebras arising from unpunctured surfaces II",
  "authors": [
    "Ralf Schiffler"
  ],
  "comment": "36 pages, 9 figures",
  "categories": [
    "math.RT",
    "math.RA"
  ],
  "abstract": "We study cluster algebras with principal and arbitrary 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 certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincar\\'e characteristic of quiver Grassmannians in Dynkin type $A$ and affine Dynkin type $\\tilde A$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-09-18T17:44:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16S99",
      "05E99",
      "16G20"
    ],
    "keywords": [
      "cluster algebras arising",
      "unpunctured surfaces",
      "direct formula",
      "study cluster algebras",
      "affine dynkin type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0809.2593S"
    }
  }
}