Symmetric Powers of Matroids

Abstract

The study of matroid products has become an active area of research, owing to their connections with tropical ideals and linear representability. In this paper, we study matroidal abstractions of the multilinearity of symmetric powers of vector spaces, using a duality between symmetric powers of matroids and abstract rigidity. These observations allow us to solve Mason's conjecture concerning the equivalence of two definitions of a symmetric power of a matroid. We show that Mason's conjecture holds for second symmetric powers of matroids whereas it fails for third symmetric powers.