Interaction between skew-representability, tensor products, extension properties, and rank inequalities
Skew-representable matroids form a fundamental class in matroid theory, bridging combinatorics and linear algebra. They play an important role in areas such as coding theory, optimization, and combinatorial geometry, where linear structure is crucial for both theoretical insights and algorithmic applications. Since skew-representability is undecidable even for rank-3 matroids, structural characterizations and explicit certificates of non-skew-representability are particularly interesting. In this paper, we introduce an approach to studying skew-representability and structural properties of matroids and polymatroid functions via tensor products. We characterize skew-representable matroids, as well as matroids representable over skew fields of a prescribed characteristic, in terms of iterated tensor products. In particular, a connected matroid is non-skew-representable if and only if, for some positive integer $k$, no $k$-fold iterated tensor product with $U_{2,3}$ exists. Thus, non-skew-representability admits a finite, computably verifiable matroid-theoretic obstruction; an analogous statement holds when the characteristic is prescribed. We also prove that every rank-3 matroid admits a tensor product with every uniform matroid and give a construction yielding the unique freest tensor product in this setting. Finally, as an application of the tensor product framework, we give a new proof of Ingleton's inequality and, more importantly, derive the first known linear rank inequality for folded skew-representable matroids that does not follow from the common information property.