{
  "id": "2206.05177",
  "version": "v1",
  "published": "2022-06-10T15:13:54.000Z",
  "updated": "2022-06-10T15:13:54.000Z",
  "title": "$m$-Symmetric functions, non-symmetric Macdonald polynomials and positivity conjectures",
  "authors": [
    "Luc Lapointe"
  ],
  "comment": "33 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\\dots$ but have no special symmetry in the variables $x_1,\\dots,x_m$. We obtain $m$-symmetric Macdonald polynomials by $t$-symmetrizing non-symmetric Macdonald polynomials, and show that they form a basis of $R_m$. We define $m$-symmetric Schur functions through a somewhat complicated process involving their dual basis, tableaux combinatorics, and the Hecke algebra generators, and then prove some of their most elementary properties. We conjecture that the $m$-symmetric Macdonald polynomials (suitably normalized and plethystically modified) expand positively in terms of $m$-symmetric Schur functions. We obtain relations on the $(q,t)$-Koska coefficients $K_{\\Omega \\Lambda}(q,t)$ in the $m$-symmetric world, and show in particular that the usual $(q,t)$-Koska coefficients are special cases of the $K_{\\Omega \\Lambda}(q,t)$'s. Finally, we show that when $m$ is large, the positivity conjecture, modulo a certain subspace, becomes a positivity conjecture on the expansion of non-symmetric Macdonald polynomials in terms of non-symmetric Hall-Littlewood polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-10T15:13:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E05"
    ],
    "keywords": [
      "positivity conjecture",
      "symmetric functions",
      "symmetric schur functions",
      "koska coefficients",
      "hecke algebra generators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}