{
  "id": "0804.1650",
  "version": "v1",
  "published": "2008-04-10T09:36:36.000Z",
  "updated": "2008-04-10T09:36:36.000Z",
  "title": "The Terwilliger Algebra of a Distance-Regular Graph of Negative Type",
  "authors": [
    "Stefko Miklavic"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Gamma$ denote a distance-regular graph with diameter $D \\ge 3$. Assume $\\Gamma$ has classical parameters $(D,b,\\alpha,\\beta)$ with $b < -1$. Let $X$ denote the vertex set of $\\Gamma$ and let $A \\in MX$ denote the adjacency matrix of $\\Gamma$. Fix $x \\in X$ and let $A^* \\in MX$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A, A^*$. We call $T$ the {\\em Terwilliger algebra} of $\\Gamma$ with respect to $x$. We show that up to isomorphism there exist exactly two irreducible $T$-modules with endpoint 1; their dimensions are $D$ and $2D-2$. For these $T$-modules we display a basis consisting of eigenvectors for $A^*$, and for each basis we give the action of $A$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-04-10T09:36:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30"
    ],
    "keywords": [
      "distance-regular graph",
      "terwilliger algebra",
      "negative type",
      "corresponding dual adjacency matrix",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.1650M"
    }
  }
}