{
  "abstract": "We determine the bigraded Frobenius series of the superspace coinvariant ring $R^{(1,1)}_{\\mathbb{Z}_r\\wr\\mathfrak{S}_n}$ for the wreath products $\\mathbb{Z}_r\\wr\\mathfrak{S}_n$, which is the quotient of the ring $\\mathbb{C}[x_1,\\ldots,x_n]\\otimes\\bigwedge[θ_1,\\ldots,θ_n]$ of polynomial-valued differential forms by the ideal generated by diagonal $\\mathbb{Z}_r\\wr\\mathfrak{S}_n$-invariants with vanishing constant term. The bigraded multiplicity of each irreducible character is a sum over standard Young $r$-tableaux of an expression determined by the flag major index and flag descent number. This refines the Frobenius series graded by fermionic degree found by Bhattacharya--Rhoades (2026). It specializes at $r=2$ to the hyperoctahedral group $\\mathfrak{B}_n$ case, and it specializes at $r=1$ to recover the theorem of Murai--Rhoades--Wilson (2025) on the symmetric group $\\mathfrak{S}_n$. We give new formulas for the bigraded multiplicities of hook characters of the superspace coinvariant ring in types $A$ and $B$, yielding upper bounds on trigraded multiplicities of the sign character of $R^{(1,2)}_{\\mathfrak{S}_n}$ and the determinant character of $R^{(1,2)}_{\\mathfrak{B}_n}$. As a corollary, we deduce a conjecture of F. Bergeron (2024) on hook components of $R^{(1,1)}_{\\mathfrak{S}_n}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38875",
  "author_total": 5,
  "authors": [
    {
      "name": "Sutanay Bhattacharya"
    },
    {
      "name": "Sylvie Corteel"
    },
    {
      "name": "Yuhan Jiang"
    },
    {
      "name": "John Lentfer"
    },
    {
      "name": "Brendon Rhoades"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ew49b1xgv4zh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38875",
  "fulltext_url": null,
  "id": "8ew49b1xgv4zh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38875",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38875",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ew49b1xgv4zh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38875",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38875"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The bigraded Frobenius series of the wreath product superspace coinvariant rings",
  "title_text": "The bigraded Frobenius series of the wreath product superspace coinvariant rings",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8ew49b1xgv4zh",
  "venue": "arXiv",
  "versions": []
}