{
  "abstract": "Frenkel and Wang obtained a representation of the Virasoro algebra by commuting Goulden's cut-and-join operator with the Heisenberg generators. A vertex-operator construction by Lascoux and the author extends this representation to $\\mathcal W_{1+\\infty}$ by means of differential operators whose eigenvalues are the power sums of the contents of a Young diagram. We develop a Jack deformation in the spherical degenerate double affine Hecke algebra and its stable limit. Starting from the Heckman--Polychronakos integrals, we isolate operators whose eigenvalues are the power sums of the $α$-contents. Because the Goulden--Jackson product is defined in the convention dual to the usual Calogero--Sutherland Hamiltonians, the multiplication operators $Δ_μ(α)$ are obtained by taking Hall adjoints. This gives conceptual derivations of the Jack cut-and-join operator and of the stable $3$-cycle operator. A normal-ordering construction due to Sergeev and Veselov makes the latter calculation explicit and suggests an integral form over $\\mathbb Z[α]$. The commutators of the cut-and-join operator contain one half of the usual Feigin--Fuchs realization, but this Virasoro completion is not the deformation of the Frenkel--Wang construction. The latter takes place in the deformed $\\mathcal W_{1+\\infty}$ algebra $\\mathbf{SH}^c$, equivalently in the affine Yangian of $\\mathfrak{gl}_1$: in our normalization its first nontrivial Cartan mode is $ψ_3=3Δ_2(α)+2(α-1)E$, and its commutators with the first raising and lowering modes recursively generate the remaining currents. At $α=1$ these relations specialize to the central-charge-one $\\mathcal W_{1+\\infty}$ representation used by Lascoux and the author.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10284",
  "author_total": 1,
  "authors": [
    {
      "name": "Jean-Yves Thibon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/12xfez249df6j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10284",
  "fulltext_url": null,
  "id": "12xfez249df6j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10284",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7212107252",
  "pdf_url": "https://arxiv.org/pdf/2609.10284",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/12xfez249df6j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10284",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10284"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Jack Content Operators and the Deformed ${\\mathcal W}_{1+\\infty}$ Algebra",
  "title_text": "Jack Content Operators and the Deformed 𝒲_1+∞ Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/12xfez249df6j",
  "venue": "arXiv",
  "versions": []
}