{
  "id": "1811.08090",
  "version": "v2",
  "published": "2018-11-20T06:51:39.000Z",
  "updated": "2018-11-26T23:19:22.000Z",
  "title": "A Categorification of the Vandermonde Determinant",
  "authors": [
    "Alex Chandler"
  ],
  "comment": "19 pages, 15 figures",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "abstract": "In the spirit of Bar Natan's construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers $\\vec{x}=(x_1,...,x_n)$, we construct a commutative diagram in the shape of the Bruhat order on $S_n$ whose nodes are colored smoothings of the $2$-strand torus link $T_{2,n}$, and whose arrows are colored cobordisms. An application of a TQFT to this diagram yields a chain complex whose Euler characteristic is the Vandermonde determinant evaluated at $\\vec{x}$. A generalization to arbitrary link diagrams is given, producing categorifications of certain generalized Vandermonde determinants. We also address functoriality of this construction.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2018-11-26T23:19:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "categorification",
      "bar natans construction",
      "arbitrary link diagrams",
      "strand torus link",
      "address functoriality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}