{
  "abstract": "We consider a notion of $ω$-circular $*$-distribution $μ_ω$, defined in terms of a parameter $ω\\in \\mathbb{C}$ with $| ω|=1$. This $*$-distribution appears as limit law in an algebraic central limit theorem (CLT) for $ω$-commuting unitaries, and interpolates between the Fermionic setting (at $ω= -1$) and the Bosonic one (at $ω= 1$). On a combinatorial level, $μ_ω$ is described in terms of pair-partitions of sets $\\{1, ... ,2n\\}$, where one has a total order among the $n$ pairs of the partition and where every pair is assigned a direction, to the effect that one can define the sign of a crossing between two pairs. The main goal of the paper is to study a significant subspace, with \"Fock space\" flavour, of the $L^2$-space associated to a natural multi-variate version of $μ_ω$. The inner product on this subspace is defined by a sesquilinear form $\\langle \\cdot \\, , \\, \\cdot \\rangle_ω$ on non-commutative polynomials. We explicitly describe $\\langle \\cdot \\, , \\, \\cdot \\rangle_ω$ in terms of some tractable non-negative definite functions on symmetric groups, and we prove that it is non-degenerate if and only if $ω$ is not a root of unity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24443",
  "author_total": 3,
  "authors": [
    {
      "name": "Claus Koestler"
    },
    {
      "name": "Liam McQuay"
    },
    {
      "name": "Alexandru Nica"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b2pjkmeq7zb15.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24443",
  "fulltext_url": null,
  "id": "b2pjkmeq7zb15",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24443",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223010779",
  "pdf_url": "https://arxiv.org/pdf/2610.24443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b2pjkmeq7zb15.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On the $ω$-circular $*$-distribution for complex unimodular $ω$",
  "title_text": "On the ω-circular *-distribution for complex unimodular ω",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/b2pjkmeq7zb15",
  "venue": "arXiv",
  "versions": []
}