{
  "id": "1702.00385",
  "version": "v1",
  "published": "2017-02-01T18:25:52.000Z",
  "updated": "2017-02-01T18:25:52.000Z",
  "title": "Braid group symmetries of Grassmannian cluster algebras",
  "authors": [
    "Chris Fraser"
  ],
  "comment": "28 pages, 7 figures, comments welcome; includes SAGE code used in some proofs; overlaps with FPSAC 2016 extended abstract, but with updated results",
  "categories": [
    "math.CO",
    "math.GR",
    "math.RT"
  ],
  "abstract": "We define an action of the k-strand braid group on the set of cluster variables for the Grassmannian Gr(k,n), whenever k divides n. The action sends clusters to clusters, preserving the underlying quivers, defining a homomorphism from the braid group to the cluster modular group for Gr(k,n). The action is induced by certain rational maps on Gr(k,n). Each of these maps is a quasi-automorphism of the cluster structure, a notion we defined in previous work. We also describe a quasi-isomorphism between Gr(k,n) and a certain Fock-Goncharov space of SL_k-local systems in a disk. The quasi-isomorphism identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. Fomin and Pylyavskyy have proposed a description of the cluster combinatorics for Gr(3,n) in terms of Kuperberg's basis of non-elliptic webs. As our main application, we prove these conjectures for Gr(3,9). The proof relies on the fact that Gr(3,9) is of finite mutation type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-01T18:25:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60"
    ],
    "keywords": [
      "grassmannian cluster algebras",
      "braid group symmetries",
      "cluster modular group",
      "cluster structure",
      "cluster variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}