{
  "abstract": "We study gauge-invariant two-matrix polynomials of degree $L$, corresponding to the states in the $su(2)$ sector of $\\mathcal{N}=4$ super Yang-Mills theory with gauge group $U(N)$ or $SU(N)$ at finite $N$. For the $U(N)$ theory, the state space has the structure of an algebra $\\mathcal{A}_N(L)$, which is a direct sum of permutation centraliser algebras, i.e. sub-algebras of the group algebra $\\mathbb{C}[S_L]$ of $S_L$. For $N \\ge L$, $\\mathcal{A}_N (L) = \\mathcal{A} (L)$ is independent of $N$, and the one-loop dilatation operator acts on $\\mathcal{A}(L)$ with $N$ as a parameter. For $N < L$, $\\mathcal{A}_N (L)$ is a subspace of $\\mathcal{A}(L)$ defined using a finite-$N$ cutoff on Young diagrams with $L$ boxes. We establish two results on the eigenspaces of the one-loop dilatation operator. Our first main result is that the characteristic polynomial of the one-loop dilatation operator on $\\mathcal{A}_N(L)$ divides the characteristic polynomial from $\\mathcal{A}(L)$, continued to $N < L$. Our second result is that each eigenstate of the one-loop dilatation operator in the $SU(N)$ theory generates an infinite tower of eigenstates in the $U(N)$ theory, through the multiplication of single-letter traces. These towers define an eigenspace factorisation of the $U(N)$ theory labelled by the number of such traces. Further, we conjecture that the space of $SU(N)$-invariant polynomials in two traceless matrices at finite $N$ is isomorphic to the intersection of $\\mathcal{A}_N(L)$ with the subspace of $\\mathbb{C}[S_L]$ spanned by derangements. This intersection conjecture is supported by dimension counting. Finally, we propose a one-loop dilatation operator on the intersection, thus providing a direct permutation framework for the $SU(N)$ spectral problem at finite $N$. The computed spectra based on the proposal agree with the prediction from the eigenspace factorisation in the $U(N)$ theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10745",
  "author_total": 3,
  "authors": [
    {
      "name": "Adrian Padellaro"
    },
    {
      "name": "Sanjaye Ramgoolam"
    },
    {
      "name": "Ryo Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dmr9t4p0fnpqy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10745",
  "fulltext_url": null,
  "id": "dmr9t4p0fnpqy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10745",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7221162562",
  "pdf_url": "https://arxiv.org/pdf/2610.10745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dmr9t4p0fnpqy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:54Z",
      "native_id": "2610.10745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Polynomial and Eigenspace Factorisations for the One-loop Dilatation Operator at Finite $N$",
  "title_text": "Polynomial and Eigenspace Factorisations for the One-loop Dilatation Operator at Finite N",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/dmr9t4p0fnpqy",
  "venue": "arXiv",
  "versions": []
}