{
  "id": "2109.02021",
  "version": "v1",
  "published": "2021-09-05T08:16:05.000Z",
  "updated": "2021-09-05T08:16:05.000Z",
  "title": "The Terwilliger algebra of the halved cube",
  "authors": [
    "Chia-Yi Wen",
    "Hau-Wen Huang"
  ],
  "categories": [
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let $D\\geq 3$ denote an integer. For any $x\\in \\mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $\\mathbb F_2^D$ consisting of all $x\\in \\mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved cube $\\frac{1}{2}H(D,2)$ is a finite simple connected graph with vertex set $X$ and $x,y\\in X$ are adjacent if and only if $w(x-y)=2$. Fix a vertex $x\\in X$. The Terwilliger algebra $\\mathcal T=\\mathcal T(x)$ of $\\frac{1}{2}H(D,2)$ with respect to $x$ is the subalgebra of ${\\rm Mat}_X(\\mathbb C)$ generated by the adjacency matrix $A$ and the dual adjacency matrix $A^*=A^*(x)$ where $A^*$ is a diagonal matrix with $$ A^*_{yy}=D-2w(x-y) \\qquad \\hbox{for all $y\\in X$}. $$ In this paper we decompose the standard $\\mathcal T$-module into a direct sum of irreducible $\\mathcal T$-modules.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-05T08:16:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "16D70",
      "33D45"
    ],
    "keywords": [
      "terwilliger algebra",
      "dual adjacency matrix",
      "finite simple connected graph",
      "dimensional halved cube",
      "direct sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}