Tensor Products of Matroids

Definitions and mathematical statement

For a matroid $M$, write $E(M)$ for its ground set. A minor of $M$ is a matroid obtained by deletion and contraction, considered up to isomorphism. The supplied record does not specify whether the matroids under consideration must be finite.

The term tensor product of matroids is not defined in the supplied record. To display the logical structure without selecting an unsupported definition, let $\mathcal M$ denote the intended class of matroids and let $\mathsf T(M,N;P)$ be a placeholder for the assertion that $P$ is a tensor product of $M$ and $N$ in the intended sense. Define, schematically, \[ \mathcal C_{\mathsf T} =\left\{(M,N)\in\mathcal M\times\mathcal M: \exists P\in\mathcal M\;\mathsf T(M,N;P)\right\}. \] This notation supplies no tensor-product axioms and makes no assertion of existence or uniqueness.

Problem, subject to the unresolved definitions. Construct further examples of pairs in $\mathcal C_{\mathsf T}$; give combinatorial descriptions of tensor products for particular classes of matroids; and characterize obstructions to membership in $\mathcal C_{\mathsf T}$ by forbidden minors.

More explicitly, once the intended notion of tensor product and the relevant minor relation $\preceq$ on pairs have been specified, a forbidden-minor characterization would seek a family $\mathcal F$ of obstructing pairs such that, for every $(M,N)\in\mathcal M\times\mathcal M$, \[ (M,N)\in\mathcal C_{\mathsf T} \quad\Longleftrightarrow\quad \nexists (A,B)\in\mathcal F\text{ with }(A,B)\preceq(M,N). \] This is a conditional formulation of the requested obstruction theory: the source does not specify that obstructions are pairs, define the relevant minor relation, establish the required closure under minors, or request a finite obstruction family. In particular, componentwise deletion and contraction must not be assumed to be the intended relation without checking the source. The missing tensor-product definition and obstruction convention prevent a fully specified mathematical statement.

Short English statement

Determine which pairs of matroids admit the intended tensor product, construct additional examples, describe products combinatorially for selected matroid classes, and develop a forbidden-minor characterization. The supplied record leaves the tensor-product definition, matroid domain, and obstruction convention unspecified, so this remains a schematic statement requiring source clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30005720$, title Tensor Products of Matroids, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2023.
• Supplied report citation: Geometric, Algebraic and Topological Combinatorics (2024), Oberwolfach Reports, pp. 3249--3318, DOI 10.4171/owr/2023/58: https://doi.org/10.4171/owr/2023/58. The original statement includes a page marker 3311 and an unresolved reference to [1].
• Anderson, Matroid Products in Tropical Geometry: https://arxiv.org/abs/2306.14771. This is a supplied literature reference; its contents have not been independently checked here.
• Berczi et al., Matroid Products via Submodular Coupling: https://arxiv.org/abs/2411.02197. This is a supplied literature reference; its contents have not been independently checked here.
• Supplied literature assessment, checked 2026-08-22: status partially solved. The supplied assessment states that realizable matroids and some minor-closed tensor-product classes are understood, while the requested combinatorial descriptions and forbidden-minor theory remain incomplete. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not verification performed for this draft.