Matroid liftability and Oxley--Wang derived matroids, with connections to scene analysis

Abstract

Liftability was first introduced for paving matroids, where it was used to systematically construct polynomials in the ideals of the associated matroid varieties. In this paper, we generalise the notion of liftability to matroids with realisable truncations. We additionally introduce the notion of generic realisability of a matroid; a strong notion of realisability that implies both realisability and liftability of the matroid. We study the behaviour of liftability and generic realisability under the weak order of matroids within the lattice of matroids with fixed truncation. We develop a unified framework for studying liftability of matroids through Oxley--Wang derived matroid of generic matroid realisations. We characterise liftability and generic realisability in terms of Oxley--Wang derived matroids. Moreover, we give an algebraic description of the bases of the generic Oxley--Wang derived matroids in terms of coordinate sections of matroid realisation spaces and relate our constructions to slack realisation spaces of matroids. As an application of our framework, we obtain a combinatorial characterisation of liftable and generically realisable paving matroids and connect it with Whiteley's theory of scene analysis. In particular, our approach gives a new proof, via Oxley--Wang derived matroids, of Whiteley's characterisation of hypergraphs admitting non-trivial (sharp) scenes. Finally, we compute the generic Oxley--Wang derived matroid for a special class of sparse paving matroids.