{
  "id": "2407.14964",
  "version": "v1",
  "published": "2024-07-20T19:01:02.000Z",
  "updated": "2024-07-20T19:01:02.000Z",
  "title": "Projective geometries, $Q$-polynomial structures, and quantum groups",
  "authors": [
    "Paul Terwilliger"
  ],
  "comment": "30 pages",
  "categories": [
    "math.CO",
    "math.QA"
  ],
  "abstract": "In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $\\varphi$ that takes any positive real value. For $\\varphi=1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}(\\mathfrak{sl}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-20T19:01:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30"
    ],
    "keywords": [
      "polynomial structure",
      "quantum group",
      "projective geometry",
      "polynomial distance-regular graphs",
      "positive real value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}