{
  "id": "1212.6487",
  "version": "v1",
  "published": "2012-12-28T08:55:19.000Z",
  "updated": "2012-12-28T08:55:19.000Z",
  "title": "Hall-Littlewood polynomials and vector bundles on the Hilbert scheme",
  "authors": [
    "Erik Carlsson"
  ],
  "comment": "12 pages, 0 figures",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "Let $E$ be the bundle defined by applying a polynomial representation of $GL_n$ to the tautological bundle on the Hilbert scheme of $n$ points in the complex plane. By a result of Haiman, the Cech cohomology groups $H^i(E)$ vanish for all $i>0$. It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative coefficients in the torus variables $z_1,z_2$, because they count the dimensions of the weight spaces of $H^0(E)$. We derive a very explicit asymmetric formula for this Euler characteristic which has this property, by expanding known contour integral formulas for the Euler characteristic stemming from the quiver description in $z_2$, and calculating the coefficients using Jing's Hall-Littlewood vertex operator with parameter $z_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-28T08:55:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "05E05",
      "05E10"
    ],
    "keywords": [
      "hilbert scheme",
      "hall-littlewood polynomials",
      "vector bundles",
      "jings hall-littlewood vertex operator",
      "standard two-dimensional torus action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.6487C"
    }
  }
}