{
  "abstract": "The recently established duality between tensor, symmetric and exterior products of uniform matroids on the one hand, and abstract (bi-)rigidity matroids on the other hand, connects theorems and open questions from both areas, previously thought unrelated. In particular, a 1981 question of Mason asks if any two uniform matroids admit a freest product. Via duality, its reformulation due to Cruickshank, Jackson, Jordán and Tanigawa asks if the generic birigidity matroid is the freest abstract birigidity matroid for its parameters. A related conjecture of Jackson and Tanigawa asks if the generic $2$-hyperconnectivity matroid is the freest $\\{K_4,K_{3,3}\\}$-matroid. We show that these problems can be addressed effectively, and often solved completely for the class of linearly representable matroids. To this end, we introduce star-basis normal forms that allow to compare representations of abstract (bi-)rigidity matroids over the same field, resulting in a refinement of the weak order relation on the underlying matroids. Utilizing it, we prove that the generic birigidity matroid is the freest linearly representable abstract $(a,b)$-birigidity matroid. This confirms Mason's conjecture for linear matroids. We prove that every representable abstract $2$-rigidity matroid admits a rigidity matrix representation. This is a strengthening of the maximality property of the generic $2$-rigidity matroid. We also prove that every representable abstract $2$-rigidity matroid in which every copy of $K_{3,3}$ is a circuit admits a hyperconnectivity matrix representation. It follows that the generic rigidity and hyperconnectivity families $\\mathcal{R}_2$ and $\\mathcal{H}_2$ are the only linearly representable $2$-rigidity families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16747",
  "author_total": 1,
  "authors": [
    {
      "name": "Mykhaylo Tyomkyn"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82f2twe7ygzcs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16747",
  "fulltext_url": null,
  "id": "82f2twe7ygzcs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16747",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213436375",
  "pdf_url": "https://arxiv.org/pdf/2609.16747",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82f2twe7ygzcs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.16747",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16747"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Linear matroid products: a synthetic approach",
  "title_text": "Linear matroid products: a synthetic approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/82f2twe7ygzcs",
  "venue": "arXiv",
  "versions": []
}