{
  "abstract": "We study weighted power means on the Lie group $\\operatorname{UT}_n$ of real upper triangular matrices with positive diagonal entries. For $t\\in\\mathbb R\\setminus\\{0\\}$, the weighted power mean $P_t(\\mathbf{w}; A_1,\\ldots,A_m)$ is defined by \\[ X=\\sum_{i=1}^m w_i(X\\#_tA_i), \\qquad X\\#_tA:=X(X^{-1}A)^t. \\] Using divided differences, analytic matrix functions, and the Parlett recurrence, we prove existence, uniqueness, and real-analytic dependence of $P_t$ on the data and the weights. We derive finite entrywise and block-recursive algorithms, establish structural properties of the mean, and obtain explicit formulas for the cases $\\operatorname{UT}_2$ and $\\operatorname{UT}_3$. We prove that $P_t$ converges to the weighted Karcher mean as $t\\to 0$. We also establish a Lie--Trotter formula and derive second-order asymptotic expansions for the power, Karcher, and log-Euclidean means, showing that the parameter $t$ first enters the local behavior at second order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24821",
  "author_total": 2,
  "authors": [
    {
      "name": "Huajun Huang"
    },
    {
      "name": "Jimmie Lawson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45037vpbzeh2j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24821",
  "fulltext_url": null,
  "id": "45037vpbzeh2j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24821",
  "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.FA"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7223061299",
  "pdf_url": "https://arxiv.org/pdf/2610.24821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45037vpbzeh2j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.24821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weighted Power Means on Upper Triangular Matrix Groups",
  "title_text": "Weighted Power Means on Upper Triangular Matrix Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/45037vpbzeh2j",
  "venue": "arXiv",
  "versions": []
}