{
  "id": "1207.3423",
  "version": "v1",
  "published": "2012-07-14T13:41:30.000Z",
  "updated": "2012-07-14T13:41:30.000Z",
  "title": "Classical and Signed Kazhdan-Lusztig Polynomials: Character Multiplicity Inversion by Induction",
  "authors": [
    "Wai Ling Yee"
  ],
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "The famous Kazhdan-Lusztig Conjecture of the 1970s states that the multiplicity of an irreducible composition factor of a Verma module can be computed by evaluating Kazhdan-Lusztig polynomials at 1. Thus the character of a Verma module is a linear combination of characters of irreducible highest weight modules where the coefficients in the linear combination are Kazhdan-Lusztig polynomials evaluated at 1. Kazhdan-Lusztig showed that inverting and writing the character of an irreducible highest weight module as a linear combination of characters of Verma modules, the coefficients in the linear combination are also Kazhdan-Lusztig polynomials evaluated at 1, up to a sign. In this paper, we show how to prove Kazhdan-Lusztig's character multiplicity inversion formula by induction using coherent continuation functors. Unitary representations may be identified by determining if characters and signature characters are the same. The signature character of a Verma module may be written as a linear combination of signature characters of irreducible highest weight modules where the coefficients in the linear combination are signed Kazhdan-Lusztig polynomials evaluated at 1. An analogous argument by induction using coherent continuation functors proves an analogous multiplicity inversion formula for signature characters: the signature character of an irreducible highest weight module is a linear combination of signature characters of Verma modules where the coefficients, up to a sign, are also signed Kazhdan-Lusztig polynomials evaluated at 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-14T13:41:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "05E10"
    ],
    "keywords": [
      "signed kazhdan-lusztig polynomials",
      "irreducible highest weight module",
      "linear combination",
      "signature character",
      "verma module"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.3423Y"
    }
  }
}