{
  "id": "1707.03314",
  "version": "v1",
  "published": "2017-07-11T15:10:37.000Z",
  "updated": "2017-07-11T15:10:37.000Z",
  "title": "Combinatorics of generalized exponents",
  "authors": [
    "Cedric Lecouvey",
    "Cristian Lenart"
  ],
  "comment": "28 pages",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also prove a related conjecture of the first author as an application, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-11T15:10:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "17B10"
    ],
    "keywords": [
      "generalized exponents",
      "combinatorics",
      "finite type",
      "symplectic lie algebras",
      "zero weight spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}