{
  "id": "2107.13069",
  "version": "v2",
  "published": "2021-07-27T20:06:40.000Z",
  "updated": "2022-11-10T01:52:28.000Z",
  "title": "Tensor diagrams and cluster combinatorics at punctures",
  "authors": [
    "Chris Fraser",
    "Pavlo Pylyavskyy"
  ],
  "comment": "64 pages, 11 figures, comments welcome; v2 incorporates referee feedback and corrects Lemma 3.7 and Proposition 5.6",
  "categories": [
    "math.CO"
  ],
  "abstract": "Fock and Goncharov introduced a family of cluster algebras associated with the moduli of SL(k)-local systems on a marked surface with extra decorations at marked points. We study this family from an algebraic and combinatorial perspective, emphasizing the structures which arise when the surface has punctures. When k is 2, these structures are the tagged arcs and tagged triangulations of Fomin, Shapiro, and Thurston. For higher k, the tagging of arcs is replaced by a Weyl group action at punctures discovered by Goncharov and Shen. We pursue a higher analogue of a tagged triangulation in the language of tensor diagrams, extending work of Fomin and the second author, and we formulate skein-algebraic tools for calculating in these cluster algebras. We analyze the finite mutation type examples in detail.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-11-10T01:52:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "05E99",
      "57M50"
    ],
    "keywords": [
      "tensor diagrams",
      "cluster combinatorics",
      "cluster algebras",
      "finite mutation type examples",
      "formulate skein-algebraic tools"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 64,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}