{
  "abstract": "In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden--Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the $(q,t)$-deformed class product (dual to the coproduct diagonal on the $J$-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov--Sklyanin operators $A^{(k)}$, derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D'Adderio.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30791",
  "author_total": 1,
  "authors": [
    {
      "name": "Jean-Yves Thibon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2m9nb2y7atzyf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30791",
  "fulltext_url": null,
  "id": "2m9nb2y7atzyf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30791",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204944487",
  "pdf_url": "https://arxiv.org/pdf/2608.30791",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2m9nb2y7atzyf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30791",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30791"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product",
  "title_text": "Shifted Macdonald Polynomials and the (q,t)-Deformed Goulden--Jackson Product",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2m9nb2y7atzyf",
  "venue": "arXiv",
  "versions": []
}