{
  "id": "2306.08071",
  "version": "v1",
  "published": "2023-06-13T18:36:44.000Z",
  "updated": "2023-06-13T18:36:44.000Z",
  "title": "Some combinatorial interpretations of the Macdonald identities for affine root systems and Nekrasov--Okounkov type formulas",
  "authors": [
    "David Wahiche"
  ],
  "comment": "All comments are welcome",
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP",
    "math.RT"
  ],
  "abstract": "We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This yields on the other hand a combinatorial interpretation of the Macdonald identities for affine root systems of the $7$ infinite families in terms of Schur functions, symplectic and special orthogonal Schur functions. From these results, we are able to derive $q$-Nekrasov--Okounkov formulas associated to each type. The latter for limit cases of $q$ yield Nekrasov--Okounkov type formulas corresponding to all the specializations given by Macdonald.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-13T18:36:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E10",
      "05A15",
      "05A17"
    ],
    "keywords": [
      "affine root systems",
      "macdonald identities",
      "combinatorial interpretation",
      "special orthogonal schur functions",
      "yield nekrasov-okounkov type formulas corresponding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}