{
  "id": "2303.09398",
  "version": "v1",
  "published": "2023-03-15T09:54:51.000Z",
  "updated": "2023-03-15T09:54:51.000Z",
  "title": "Cycle matrices: A combinatorial approach to the set-theoretic solutions of the Quantum Yang-Baxter Equation",
  "authors": [
    "Arpan Kanrar",
    "Saikat Panja"
  ],
  "comment": "19 pages, preliminary version, comments are welcome",
  "categories": [
    "math.GR",
    "math.CO",
    "math.QA"
  ],
  "abstract": "An $n\\times n$ matrix $M=[m_{ij}]$ with $m_{ij}\\in U_n=\\{1,2,\\ldots,n\\}$ will be called a cycle matrix if $(U_n,\\cdot)$ is a cycle set, where $i\\cdot j=m_{ij}$. We study these matrices in this article. Using these matrices, we give some recipes to construct solutions, which include the multipermutation level $2$ solutions. As an application of these, we construct a multi-permutation solution of level $r$ for all $r\\geq 1$. Our method gives alternate proof that the class of permutation groups of solutions contains all finite abelian groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-15T09:54:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum yang-baxter equation",
      "cycle matrix",
      "combinatorial approach",
      "set-theoretic solutions",
      "finite abelian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}