{
  "id": "1210.1888",
  "version": "v3",
  "published": "2012-10-05T23:12:13.000Z",
  "updated": "2014-12-08T23:52:18.000Z",
  "title": "Tensor diagrams and cluster algebras",
  "authors": [
    "Sergey Fomin",
    "Pavlo Pylyavskyy"
  ],
  "comment": "73 pages, 66 figures; same results as in the earlier versions, numerous changes in the exposition",
  "categories": [
    "math.CO",
    "math.RA",
    "math.RT"
  ],
  "abstract": "The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. We show that when V is 3-dimensional, each of these rings carries a natural cluster algebra structure (typically, many of them) whose cluster variables include Weyl's generators. We describe and explore these cluster structures using the combinatorial machinery of tensor diagrams. A key role is played by the web bases introduced by G.Kuperberg.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-10-23T00:00:22.000Z",
      "comment": "71 pages, 64 figures; minor editorial changes",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-12-08T23:52:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "05E99",
      "13A50",
      "15A72"
    ],
    "keywords": [
      "tensor diagrams",
      "natural cluster algebra structure",
      "finite-dimensional complex vector space",
      "hermann weyl",
      "rings carries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 73,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.1888F"
    }
  }
}