{
  "id": "2405.09846",
  "version": "v1",
  "published": "2024-05-16T06:48:34.000Z",
  "updated": "2024-05-16T06:48:34.000Z",
  "title": "Delta Operators on Almost Symmetric Functions",
  "authors": [
    "Milo Bechtloff Weising"
  ],
  "comment": "20 pages. arXiv admin note: text overlap with arXiv:2307.05864",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We construct $\\Delta$-operators $F[\\Delta]$ on the space of almost symmetric functions $\\mathscr{P}_{as}^{+}$. These operators extend the usual $\\Delta$-operators on the space of symmetric functions $\\Lambda \\subset \\mathscr{P}_{as}^{+}$ central to Macdonald theory. The $F[\\Delta]$ operators are constructed as certain limits of symmetric functions in the Cherednik operators $Y_i$ and act diagonally on the stable-limit non-symmetric Macdonald functions $\\widetilde{E}_{(\\mu|\\lambda)}(x_1,x_2,\\ldots;q,t).$ Using properties of Ion-Wu limits, we are able to compute commutation relations for the $\\Delta$-operators $F[\\Delta]$ and many of the other operators on $\\mathscr{P}_{as}^{+}$ introduced by Ion-Wu. Using these relations we show that there is an action of $\\mathbb{B}_{q,t}^{\\text{ext}}$ on almost symmetric functions which we show is isomorphic to the polynomial representation of $\\mathbb{B}_{q,t}^{\\text{ext}}$ constructed by Gonz\\'{a}lez-Gorsky-Simental.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-16T06:48:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric functions",
      "delta operators",
      "stable-limit non-symmetric macdonald functions",
      "cherednik operators",
      "operators extend"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}