Interaction between skew-representability, tensor products, extension properties, and rank inequalities

TL;DR

This paper introduces a tensor product framework to characterize skew-representable matroids, providing new insights into their structure, decidability, and associated rank inequalities, with implications for combinatorics and linear algebra.

Contribution

It offers a novel tensor product approach to characterize skew-representability and derive new rank inequalities, advancing understanding of matroid representability over skew fields.

Findings

Characterization of skew-representable matroids via tensor products

Decidability results: co-recursive enumerability of non-representability

New linear rank inequality for folded skew-representable matroids

Abstract

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 deciding skew-representability is computationally intractable, much effort has been focused on identifying necessary or sufficient conditions for a matroid to be skew-representable. In this paper, we introduce a novel approach to studying skew-representability and structural properties of matroids and polymatroid functions via tensor products. We provide a characterization of skew-representable matroids, as well as of those representable over skew fields of a given prime characteristic, in terms of tensor products. As an algorithmic consequence, we show that deciding skew-representability, or representability over a skew field of fixed prime characteristic, is co-recursively enumerable: that is, certificates of non-skew-representability -- in general or over a fixed prime characteristic -- can be verified. We also prove that every rank-3 matroid admits a tensor product with any 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.