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

2025-07-14

AI Summary

This paper addresses the problem of characterizing skew-representability of matroids and determining representability over skew fields of prime characteristic. Methodologically, it introduces a novel framework based on tensor products—first systematically characterizing skew-representability and yielding verifiable negative certificates for non-representability. It constructs matroid tensor products, including explicit constructions for rank-3 and uniform matroids, as well as their most general (freest) forms. The framework establishes deep connections between skew-representability, extension properties, and linear rank inequalities, leading to the first folded linear rank inequality for skew-representable matroids not derivable from common information-theoretic principles. Furthermore, it provides a new algebraic proof of the Ingleton inequality. The approach integrates tensor algebra, matroid function analysis, and recursively enumerable verification techniques, combining structural insights with explicit constructions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Characterize skew-representable matroids via tensor products
• Decide skew-representability with co-recursive enumerability
• Derive linear rank inequalities for folded skew-representable matroids

Innovation

• Characterizes skew-representable matroids via tensor products
• Decides skew-representability via co-recursive enumeration
• Proves new linear rank inequality for folded matroids