{
  "abstract": "We study Fulton's universal Schubert polynomials $\\mathfrak S_w(c)$ as regular functions on the upper unitriangular group $U_N$. The standard triangular cluster structure associates a Schubert $\\sf g$-vector to every permutation. Their convex hull is unimodularly equivalent to $Δ_1\\times\\cdots\\timesΔ_{N-1}$, and their root-degree fibers are parabolic Bruhat intervals realized by the strata of staircase quiver Grassmannians. The geometric (i.e., generic, canonical, and Mirković--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that $\\mathfrak S_w(c)$ is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of $\\mathfrak S_w(c)$ implies $Λ_Q$-rigidity of $Z_{\\sf g_w}$. We also classify simultaneously the unit columns of the geometric-to-Schubert transitions, determine support components of the PBW-to-geometric and code-to-Schubert transitions, and exhibit a permutation $w\\in S_{10}$ for which the three geometric basis elements are distinct.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29125",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiarui Fei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52ygevwsj9bcv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29125",
  "fulltext_url": null,
  "id": "52ygevwsj9bcv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29125",
  "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": "W7204954163",
  "pdf_url": "https://arxiv.org/pdf/2608.29125",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52ygevwsj9bcv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29125",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29125"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions",
  "title_text": "Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/52ygevwsj9bcv",
  "venue": "arXiv",
  "versions": []
}