{
  "id": "1712.06968",
  "version": "v1",
  "published": "2017-12-19T15:06:53.000Z",
  "updated": "2017-12-19T15:06:53.000Z",
  "title": "Scattering diagrams and scattering fans",
  "authors": [
    "Nathan Reading"
  ],
  "comment": "58 pages, 7 figures, 5 tables",
  "categories": [
    "math.CO",
    "math.AG",
    "math.RT"
  ],
  "abstract": "Scattering diagrams arose in the context of mirror symmetry, but a special class of scattering diagrams (the cluster scattering diagrams) were recently developed to prove key structural results on cluster algebras. This paper studies scattering diagrams from a combinatorial and discrete-geometric point of view. We show that a consistent scattering diagram with minimal support cuts the ambient space into a complete fan. We give a simple derivation of the function attached to the limiting wall of a rank-2 cluster scattering diagram of affine type. In the skew-symmetric rank-2 affine case, this recovers a formula due to Reineke. In the same case, we point out that the generating function for signed Narayana numbers appears in a role analogous to a cluster variable. In acyclic finite type, cluster scattering fans are known to coincide with Cambrian fans because both coincide with the g-vector fan. Here, we construct scattering diagrams of acyclic finite type from Cambrian fans and sortable elements, with a simple direct proof. The paper includes two brief expositions of scattering diagrams, one largely following the conventions of Gross, Hacking, Keel, and Kontsevich, and the other (related by a global transpose) more compatible with the conventions of Fomin and Zelevinsky.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-19T15:06:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "14N35",
      "52C99",
      "05E10",
      "05A15"
    ],
    "keywords": [
      "scattering fans",
      "acyclic finite type",
      "cluster scattering diagram",
      "cambrian fans",
      "signed narayana numbers appears"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}