{
  "id": "2405.14915",
  "version": "v1",
  "published": "2024-05-23T16:04:04.000Z",
  "updated": "2024-05-23T16:04:04.000Z",
  "title": "Cluster expansion formulas and perfect matchings for type B and C",
  "authors": [
    "Azzurra Ciliberti"
  ],
  "comment": "22 pages. arXiv admin note: substantial text overlap with arXiv:2403.11308",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "Let $\\mathbf{P}_{2n+2}$ be the regular polygon with $2n+2$ vertices, and let $\\theta$ be the rotation of 180$^\\circ$. Fomin and Zelevinsky showed that $\\theta$-invariant triangulations of $\\mathbf{P}_{2n+2}$ are in bijection with the clusters of cluster algebras of type $B_n$ or $C_n$. Moreover, cluster variables correspond to the orbits of the action of $\\theta$ on the diagonals of $\\mathbf{P}_{2n+2}$. In this paper, we associate a labeled modified snake graph $\\mathcal{G}_{ab}$ to each $\\theta$-orbit $[a,b]$, and we get the cluster variables of type $B_n$ and $C_n$ which correspond to $[a,b]$ as perfect matching Laurent polynomials of $\\mathcal{G}_{ab}$. This extends the work of Musiker for cluster algebras of type B and C to every seed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-23T16:04:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cluster expansion formulas",
      "cluster algebras",
      "perfect matching laurent polynomials",
      "cluster variables correspond",
      "regular polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}