{
  "id": "1907.09092",
  "version": "v1",
  "published": "2019-07-22T02:49:25.000Z",
  "updated": "2019-07-22T02:49:25.000Z",
  "title": "The counting matrix of a simplicial complex",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "15 pages, 5 figures and code",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x) is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The matrix K is always positive definite. The spectra of K and K^(-1) always agree so that the matrix Q=K-K^(-1) has the spectral symmetry spec(Q)=-spec(Q) and the zeta function z(s) summing l(k)^(-s) with eigenvalues l(k) of K satisfies the functional equation z(a+ib)=z(-a+ib). The energy theorem in this case tells that the sum of the matrix elements of K^(-1)(x,y) is equal to the number sets in G. In comparison, we had in the connection matrix case the identity that the sum of the matrix elements of L^(-1) is the Euler characteristic of G.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-22T02:49:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M15",
      "68R10"
    ],
    "keywords": [
      "counting matrix",
      "finite abstract simplicial complex",
      "matrix elements",
      "green function entries",
      "connection matrix case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}