{
  "abstract": "Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\\ast$-subalgebras of the complex matrix algebra ${\\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every pair of elements, one from each subalgebra. Writing such a pair as ${\\sf D}_n$ and $U{\\sf D}_nU^*$ with $U$ unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition $|\\det U[I,J]|^2=\\binom{n}{r}^{-1}$ for every $1\\le r\\le n$ and all $I,J$ with $|I|=|J|=r$. We show that this condition holds if and only if $n\\le3$ and $\\sqrt{n} U$ is a complex Hadamard matrix. To quantify the failure of exact realization for $n\\ge 4$, we introduce the uniform-minor discrepancy $δ_r(U)$. We identify it with the mean-square error in the $r$-th coefficient of finite free multiplicative convolution for two diagonal matrices whose diagonal entries are independent and uniformly distributed on the unit circle. We establish the symmetry $δ_r(U)=δ_{n-r}(U)$ and the monotonicity $δ_1(U)\\leδ_2(U)\\le\\cdots\\le δ_{\\lfloor n/2\\rfloor}(U)$. For flat unitaries, we derive an explicit formula for $δ_2$, yielding $δ_r(U)\\ge\\frac{n-3}{2n}$ for $2\\le r\\le n-2$. Equality for $r=2$ holds precisely when the entrywise square of $\\sqrt{n}U$ is also complex Hadamard.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39677",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuki Ueda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a0dwjwqy4hssv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39677",
  "fulltext_url": null,
  "id": "a0dwjwqy4hssv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39677",
  "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.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7215081623",
  "pdf_url": "https://arxiv.org/pdf/2609.39677",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a0dwjwqy4hssv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39677",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39677"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Finite free position of maximal abelian $\\ast$-subalgebras of the matrix algebra",
  "title_text": "Finite free position of maximal abelian ∗-subalgebras of the matrix algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/a0dwjwqy4hssv",
  "venue": "arXiv",
  "versions": []
}