{
  "abstract": "For a polynomial $F(t) = 1 + a_1 t + a_2 t^2 +\\dots + a_d t^d$ with coefficients in a $\\mathbb{Q}$-algebra $\\mathcal{R}$, we consider Schur-positivity of the symmetric function $Ω_F(\\mathbf{x}) = \\prod_{i \\geq 1} F(x_i)$. Using work of Lindström and Gessel--Viennot, we give sufficient conditions for Schur-positivity in terms of directed networks. In the case when $F(t)$ is the multivariate independence polynomial of the incomparability graph $inc(P)$ of a natural unit interval order, we apply these ideas to give a new proof of Schur-positivity of the chromatic symmetric function $X_{inc(P)}(\\mathbf{x})$. Additionally, we prove Schur-positivity of certain sums of chromatic symmetric functions, refining recent work of Amdeberhan--Shareshian--Stanley. In the case when $\\mathcal{R} = \\mathbb{Q}[z_1,z_2,\\dots,z_n]$, we prove necessary and sufficient conditions for Schur-positivity in terms of symmetric function evaluations. In particular, we show for $\\mathbf{u}, \\mathbf{v} \\in \\mathbb{R}_{\\geq 0}^n$, $s_λ(\\mathbf{u}) \\geq s_λ(\\mathbf{v})$ for all partitions $λ$ if and only if $m_λ(\\mathbf{u}) \\geq m_λ(\\mathbf{v})$ for all partitions $λ$, and these inequalities are equivalent to $\\mathbf{u}$ log-majorizing $\\mathbf{v}$. Using representation theory of $GL(\\mathbb{C}^n)$, we prove Schur-positivity of $Ω_F(\\mathbf{x})$ when $F(t) = \\det(I + t\\sum_{i = 1}^m z_i \\mathbf{v}_i \\mathbf{v}_i^*)$ for a collection of vectors $\\mathbf{v}_1,\\mathbf{v}_2,\\dots, \\mathbf{v}_m \\in \\mathbb{C}^n$. As an application for this result, we introduce the Kirchhoff symmetric function $K_G(\\mathbf{x})$ of a loopless graph $G$ and show that it it always Schur-positive.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25036",
  "author_total": 1,
  "authors": [
    {
      "name": "Isaiah Siegl"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/22cdczd00r7yy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25036",
  "fulltext_url": null,
  "id": "22cdczd00r7yy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25036",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223035533",
  "pdf_url": "https://arxiv.org/pdf/2610.25036",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/22cdczd00r7yy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25036",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25036"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices",
  "title_text": "Schur-Positve Symmetric Functions from Products: Log-Majorization, Total-Positivity, and PSD Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/22cdczd00r7yy",
  "venue": "arXiv",
  "versions": []
}