{
  "id": "1909.12896",
  "version": "v1",
  "published": "2019-09-27T20:22:19.000Z",
  "updated": "2019-09-27T20:22:19.000Z",
  "title": "Cluster modular groups of dimer models and networks",
  "authors": [
    "Terrence George"
  ],
  "categories": [
    "math.CO",
    "math.AG"
  ],
  "abstract": "Associated to a convex integral polygon $P$ is a dimer cluster integrable system $\\mathcal X_P$. We compute the group of symmetries of $\\mathcal X_P$, called the cluster modular group, showing that it matches a conjecture of Fock and Marshakov. Probabilistically, non-torsion elements of $G_P$ are ways of shuffling the underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrically, $G_P$ is the degree zero subgroup of the Picard group of a certain algebraic surface associated to $P$. We also prove analogous results for resistor networks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-27T20:22:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cluster modular group",
      "dimer models",
      "degree zero subgroup",
      "dimer cluster integrable system",
      "convex integral polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}