{
  "abstract": "Orbit-span and dimension problems for designs have been studied in several classical association schemes using a variety of methods. Recently, through a detailed analysis of total trades, Ghorbani et al. showed that the orbit of a fixed combinatorial design asymptotically attains the full dimension permitted by the design equations. Analogous dimension-saturation results for the global spans of index-one designs in the bilinear forms and Grassmann schemes were obtained via laborious eigenvalue computations. In this paper, we work with the top fiber $X$ of a graded poset carrying a compatible transitive action of a finite group $G$, and assume that the induced Schurian association scheme on $X$ is commutative. For the multiplicity-free decomposition $\\mathbb{C}^{X}=U_{0}\\oplus U_{1}\\oplus\\cdots\\oplus U_{m}$, we prove that, under explicit spectral and quantitative conditions, the $G$-orbit of the characteristic vector of any $t$-design spans the maximal submodule $U_{0}\\oplus U_{t+1}\\oplus U_{t+2}\\oplus\\cdots\\oplus U_{m}$ allowed by the $t$-design definition. We verify these conditions asymptotically for the Hamming, bilinear forms, Johnson, and Grassmann schemes. This recovers the fixed-orbit saturation theorem of Ghorbani et al. for combinatorial designs, gives a new result for orthogonal arrays, and asymptotically extends the previous bilinear forms and Grassmann results to orbit spans of individual designs of arbitrary fixed index.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13965",
  "author_total": 2,
  "authors": [
    {
      "name": "Qilong Li"
    },
    {
      "name": "Yue Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7besafcbwskj8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13965",
  "fulltext_url": null,
  "id": "7besafcbwskj8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13965",
  "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"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7213353317",
  "pdf_url": "https://arxiv.org/pdf/2609.13965",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7besafcbwskj8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13965",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13965"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Orbit span of a design and some saturation theorems in commutative Schurian association schemes",
  "title_text": "Orbit span of a design and some saturation theorems in commutative Schurian association schemes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7besafcbwskj8",
  "venue": "arXiv",
  "versions": []
}