{
  "id": "1712.07692",
  "version": "v1",
  "published": "2017-12-20T20:14:12.000Z",
  "updated": "2017-12-20T20:14:12.000Z",
  "title": "A diagram associated with the subconstituent algebra of a distance-regular graph",
  "authors": [
    "Supalak Sumalroj"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we consider a distance-regular graph $\\Gamma$. Fix a vertex $x$ of $\\Gamma$ and consider the corresponding subconstituent algebra $T$. The algebra $T$ is the $\\mathbb{C}$-algebra generated by the Bose-Mesner algebra $M$ of $\\Gamma$ and the dual Bose-Mesner algebra $M^*$ of $\\Gamma$ with respect to $x$. We consider the subspaces $M, M^*, MM^*, M^*M, MM^*M, M^*MM^*, \\dots$ along with their intersections and sums. In our notation, $MM^*$ means $Span\\{RS|R\\in M, S\\in M^*\\}$, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to $MM^*+M^*M$. For each subspace $U$ shown in this part of the diagram, we display an orthogonal basis for $U$ along with the dimension of $U$. For an edge $U\\subseteq W$ from this part of the diagram, we display an orthogonal basis for the orthogonal complement of $U$ in $W$ along with the dimension of this orthogonal complement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-20T20:14:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30"
    ],
    "keywords": [
      "distance-regular graph",
      "orthogonal complement",
      "orthogonal basis",
      "dual bose-mesner algebra",
      "corresponding subconstituent algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}