{
  "id": "2308.07851",
  "version": "v1",
  "published": "2023-08-15T15:55:18.000Z",
  "updated": "2023-08-15T15:55:18.000Z",
  "title": "An imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs",
  "authors": [
    "Hau-Wen Huang"
  ],
  "comment": "65 pages",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "The Clebsch--Gordan coefficients of $U(\\mathfrak{sl}_2)$ are expressible in terms of Hahn polynomials. The phenomenon can be explained by an algebra homomorphism from the universal Hahn algebra $\\mathcal H$ into $U(\\mathfrak{sl}_2)\\otimes U(\\mathfrak{sl}_2)$. Let $\\Omega$ denote a finite set and $2^\\Omega$ denote the power set of $\\Omega$. It is generally known that $\\mathbb C^{2^\\Omega}$ supports a $U(\\mathfrak{sl}_2)$-module. Fix an element $x_0\\in 2^\\Omega$. By the linear isomorphism $\\mathbb C^{2^\\Omega}\\to \\mathbb C^{2^{\\Omega\\setminus x_0}}\\otimes \\mathbb C^{2^{x_0}}$ given by $x\\mapsto (x\\setminus x_0)\\otimes (x\\cap x_0)$ for all $x\\in 2^\\Omega$, this induces a $U(\\mathfrak{sl}_2)\\otimes U(\\mathfrak{sl}_2)$-module structure on $\\mathbb C^{2^\\Omega}$. Pulling back via the algebra homomorphism $\\mathcal H\\to U(\\mathfrak{sl}_2)\\otimes U(\\mathfrak{sl}_2)$, the $U(\\mathfrak{sl}_2)\\otimes U(\\mathfrak{sl}_2)$-module $\\mathbb C^{2^\\Omega}$ forms an $\\mathcal H$-module. The $\\mathcal H$-module $\\mathbb C^{2^\\Omega}$ enfolds the Terwilliger algebra of a Johnson graph. This result connects these two seemingly irrelevant topics: The Clebsch--Gordan coefficients of $U(\\mathfrak{sl}_2)$ and the Terwilliger algebras of Johnson graphs. Unfortunately some steps break down in the $q$-analog case. By making detours, the imperceptible connection between the Clebsch--Gordan coefficients of $U_q(\\mathfrak{sl}_2)$ and the Terwilliger algebras of Grassmann graphs is successfully disclosed in this paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-15T15:55:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "06A11",
      "16G30",
      "17B37",
      "33D45"
    ],
    "keywords": [
      "terwilliger algebra",
      "clebsch-gordan coefficients",
      "grassmann graphs",
      "imperceptible connection",
      "algebra homomorphism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}