A connected matroid is skew-representable if and only if it admits tensor products with the uniform matroid U_{2,3} at every order, giving verifiable certificates for non-representability and a new rank inequality.
Tensor Product of Polymatroids and Common Information
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A new connection between two different necessary conditions for a polymatroid to be linearly representable is presented. Specifically, we prove that the existence of a tensor product with the uniform matroid of rank two on three elements implies the existence of a common information extension for every pair of subsets of the ground set.
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Interaction between skew-representability, tensor products, extension properties, and rank inequalities
A connected matroid is skew-representable if and only if it admits tensor products with the uniform matroid U_{2,3} at every order, giving verifiable certificates for non-representability and a new rank inequality.