Tutte Colloquium - Tamas Schwarcz- Interaction between skew-representability, tensor products, extension properties, and rank inequalities

Abstract:  The study of matroid tensor products dates back to the 1970s, extending the tensor operation from linear algebra to the combinatorial setting. While any two matroids representable over the same field admit a tensor product via the Kronecker product of matrices, Las Vergnas showed that such products do not exist for matroids in general, leaving the area underexplored. In this work, we utilize this operation to study skew-representability — representation over division rings that need not be commutative — by proving that a matroid is skew-representable if and only if it admits iterated tensor products with specific test matroids. A key consequence is the existence of algorithmic certificates for non-representability. We further show that every rank-3 matroid admits a tensor product with any uniform matroid, constructing the unique freest such product. Finally, we demonstrate the power of this framework by deriving the first known linear rank inequality for (folded skew-)representable matroids that is independent of the common information property.