{
  "abstract": "Let $$X_n=\\sum_{i=1}^s t_i A_i^{\\otimes n},$$ where $A_1,\\ldots,A_s\\in M_d(\\mathbb C)$ and $t_1,\\ldots,t_s\\in\\mathbb C$ are fixed, while $n$ grows. Direct computation of determinants or Schatten norms of $X_n$ is exponential in $n$. For a single tensor power these quantities are elementary, and even the determinant of a generic two-term combination admits a reduction to polynomially many scalar factors; however, no analogous elementary reduction is available for three or more terms. We give an exact representation-theoretic method which, for fixed $d$ and $s$, computes $\\|X_n\\|_p$, $0<p<\\infty$, and determinants in polynomial time in $n$. Schur--Weyl duality yields a simultaneous block decomposition, while Jacobi--Trudi identities in the Grothendieck ring replace Schur modules by signed combinations of tensor products of symmetric powers. For $d=3$, each irreducible contribution reduces to the difference of two explicitly computable symmetric-power terms, leading to an open-source implementation. In a single-thread CPU benchmark, a genuine three-term $3\\times3$ trace-norm problem with $n=18$ is evaluated in about $47$ seconds, whereas just storing the unreduced matrix would require approximately $2.4\\times10^{18}$ bytes. Direct and reduced computations agree to relative error below $3.4\\times10^{-15}$ throughout their common range $n\\leq9$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04341",
  "author_total": 2,
  "authors": [
    {
      "name": "Martin Áron Juhász"
    },
    {
      "name": "Mihály Weiner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/agyb9v55nmpz5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04341",
  "fulltext_url": null,
  "id": "agyb9v55nmpz5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04341",
  "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",
      "math.RT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7211911466",
  "pdf_url": "https://arxiv.org/pdf/2609.04341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/agyb9v55nmpz5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2609.04341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations",
  "title_text": "Schatten norms and determinants of linear combinations of matrix tensor powers via virtual representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/agyb9v55nmpz5",
  "venue": "arXiv",
  "versions": []
}