{
  "id": "1112.5188",
  "version": "v1",
  "published": "2011-12-21T21:56:13.000Z",
  "updated": "2011-12-21T21:56:13.000Z",
  "title": "Macdonald polynomials in superspace: conjectural definition and positivity conjectures",
  "authors": [
    "O. Blondeau-Fournier",
    "P. Desrosiers",
    "L. Lapointe",
    "P. Mathieu"
  ],
  "comment": "18 pages",
  "journal": "Letters in Mathematical Physics 101 (2012) 27-47",
  "doi": "10.1007/s11005-011-0542-5",
  "categories": [
    "math-ph",
    "hep-th",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\\infty seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q=t=0 family) are polynomials in q and t with nonnegative integer coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-21T21:56:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "macdonald polynomials",
      "superspace",
      "conjectural definition",
      "nonnegative integer coefficients",
      "expansion coefficients"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Letters in Mathematical Physics",
      "year": 2012,
      "month": "Jul",
      "volume": 101,
      "number": 1,
      "pages": 27
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1082755,
      "adsabs": "2012LMaPh.101...27B"
    }
  }
}