{
  "abstract": "We introduce a Schubert bialgebra $\\mathcal{A}=\\bigoplus_{n\\geq 0}\\mathbb{Z}[x_1,\\ldots,x_n]$, in which polynomials in different numbers of variables multiply to zero and the coproduct splits the variables into two consecutive blocks, and study its graded dual $\\mathcal{D}$, whose product is concatenation of weak compositions. The Schubert polynomials $\\mathfrak{S}_u^n$ form a basis of $\\mathcal{A}$; we give an explicit formula for the dual Schubert basis $Ξ_u^n$ of $\\mathcal{D}$ via a transition formula to an elementary symmetric basis, and show that the Schubert structure constants $c_{u,v}^w$ are the coefficients of the coproduct of $\\mathcal{D}$ in the dual Schubert basis. The product of $\\mathcal{D}$ is then lifted to an associative product $\\sqcup$ on the free abelian group $\\mathcal{BRC}$ of bounded RC graphs, built from a transition map $\\mathcal{Z}$ on RC graphs that realizes the Lascoux-Schützenberger transition formula and generalizes Little's bumping algorithm. Quotienting $\\mathcal{BRC}$ by ideals coming from Demazure crystal equivalence, Nadeau-Tewari forest equivalence, and quasi-Yamanouchi equivalence yields manifestly positive Littlewood-Richardson rules for the dual Schubert, dual key, dual forest, and dual slide bases of $\\mathcal{D}$. Read on $\\mathcal{A}$, these are branching rules: enumerative formulas for splitting a Schubert, key, forest, or slide polynomial across two blocks of variables, and in the Schubert case they compute the Levi branching coefficients of Ressayre-Richmond for $\\mathrm{GL}_p\\times\\mathrm{GL}_q\\hookrightarrow\\mathrm{GL}_{p+q}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22582",
  "author_total": 1,
  "authors": [
    {
      "name": "Matthew J. Samuel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9wmzba1agxp6w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22582",
  "fulltext_url": null,
  "id": "9wmzba1agxp6w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22582",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222975876",
  "pdf_url": "https://arxiv.org/pdf/2610.22582",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9wmzba1agxp6w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22582",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22582"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Schubert bialgebra",
  "title_text": "The Schubert bialgebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/9wmzba1agxp6w",
  "venue": "arXiv",
  "versions": []
}