{
  "id": "2008.00480",
  "version": "v1",
  "published": "2020-08-02T13:41:51.000Z",
  "updated": "2020-08-02T13:41:51.000Z",
  "title": "Cluster algebras from surfaces and extended affine Weyl groups",
  "authors": [
    "Anna Felikson",
    "John W. Lawson",
    "Michael Shapiro",
    "Pavel Tumarkin"
  ],
  "comment": "37 pages, many figures",
  "categories": [
    "math.CO",
    "math.GR",
    "math.RA"
  ],
  "abstract": "We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with every triangulation a basis in $V$, such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type $A$, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types $E$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-02T13:41:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "20F55",
      "51F15"
    ],
    "keywords": [
      "extended affine weyl group",
      "positive semi-definite quadratic",
      "exceptional skew-symmetric mutation-finite cluster algebras",
      "semi-definite quadratic space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}