{
  "id": "2408.03792",
  "version": "v1",
  "published": "2024-08-07T14:17:23.000Z",
  "updated": "2024-08-07T14:17:23.000Z",
  "title": "Log-concavity of cluster algebras of type $A_n$",
  "authors": [
    "Zhichao Chen",
    "Guanhua Huang",
    "Zhe Sun"
  ],
  "comment": "26 pages, 10 figures",
  "categories": [
    "math.RT",
    "math.CO",
    "math.GT"
  ],
  "abstract": "After Gross, Hacking, Keel, Kontsevich [GHKK18] introduced the theta basis which is shown to be indexed by its highest term exponent in cluster variables of any given seed, we are interested in all the non-vanishing exponents in these cluster variables. We prove that the coefficients of the exponents of any cluster variable of type $A_n$ are log-concave. We show that the cluster monomials of $A_2$ type are log-concave. As for larger generality, we conjecture that the log-concavity of cluster monomials is also true.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-07T14:17:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13F60",
      "05E99",
      "16S34"
    ],
    "keywords": [
      "cluster algebras",
      "log-concavity",
      "cluster variable",
      "cluster monomials",
      "highest term exponent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}