Linear matroid products: a synthetic approach

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 {K₄,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 ℛ₂ and ℋ₂ are the only linearly representable $2$-rigidity families.

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Cite

BibTeX RIS

BibTeX

@misc{tyomkyn2026linear,
  title = {{Linear matroid products: a synthetic approach}},
  author = {Mykhaylo Tyomkyn},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.16747},
  eprint = {2609.16747},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16747},
}

RIS

TY  - UNPB
TI  - Linear matroid products: a synthetic approach
AU  - Mykhaylo Tyomkyn
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.16747
UR  - https://arxiv.org/abs/2609.16747
AB  - 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 ℛ_2 and ℋ_2 are the only linearly representable $2$-rigidity families.
ER  -

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