{
  "abstract": "We study the Schur-positive cone in $\\Rspace_{n,\\mathbb R}\\coloneqq \\operatorname{span}_{\\mathbb R}\\{p_d^{n/d}:d\\mid n\\}$ through its basis $Q_{n,d}\\coloneqq\\ell_{n/d}^{(1)}[p_d]$, where $\\ell_m^{(1)}$ is the Frobenius characteristic of the representation induced to $S_m$ from a faithful linear character of the subgroup generated by an $m$-cycle; brackets denote plethysm. A Boolean $Q$-sum is a sum of distinct elements of this basis. We give a unified proof of four conjectures of Sundaram on Schur positivity by classifying all Schur-positive Boolean $Q$-sums; the case of sums over divisors up to a prescribed bound recovers Hou's theorem. Specifically, for a nonempty set $J$ of divisors of $n$, the sum $\\sum_{d\\in J}Q_{n,d}$ is Schur-positive exactly when $1\\in J$ and, for even $n$, $n\\in J$ implies $n/2\\in J$. The same character estimates prove the Ramanujan-square conjecture of Shareshian and Sundaram: the function $\\sum_{d\\mid n}c_d(n/d)^2p_d^{n/d}$, where $c_d(r)$ is the Ramanujan sum, has a positive coefficient of $s_λ$ for every $n\\ge1$ and $λ\\vdash n$, except when $n\\equiv2\\pmod4$ and $λ=(1^n)$, in which case the coefficient is zero. We prove that an element of this space has integral Schur coefficients if and only if its $Q$-coordinates are integral. The Boolean classification also determines the convex hull of the Schur-positive Boolean points with $Q_{n,1}$-coordinate $1$. We compute its Ehrhart polynomial and volume, prove its integer decomposition property, and determine the Hilbert basis of its cone. For $n\\ge18$, we prove that setting the coefficient of $s_{(n)}$ or $s_{(1^n)}$ equal to $0$ or $1$ defines a facet of the section of the Schur-positive cone with $Q_{n,1}$-coordinate $1$. The positivity results and coordinate formulas also yield inequalities for major-index residue multiplicities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29757",
  "author_total": 1,
  "authors": [
    {
      "name": "Young-Tak Oh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5gkhcz8f317gk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29757",
  "fulltext_url": null,
  "id": "5gkhcz8f317gk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29757",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204955606",
  "pdf_url": "https://arxiv.org/pdf/2608.29757",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5gkhcz8f317gk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2608.29757",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29757"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure",
  "title_text": "The cyclic-induction Schur cone: Boolean sums, Ramanujan-square positivity, and integral structure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/5gkhcz8f317gk",
  "venue": "arXiv",
  "versions": []
}