Representing matroids via pasture morphisms

TL;DR

This paper introduces algorithms based on pasture morphisms to compute matroid foundations and all morphisms between pastures, enabling efficient solutions to various matroid representation problems.

Contribution

It develops algorithms for computing matroid foundations and pasture morphisms, advancing the understanding of matroid representability and related properties.

Findings

Efficient algorithms for matroid foundation computation

Method to determine all pasture morphisms between two pastures

Applications to orientability and non-representability of matroids

Abstract

Using the framework of pastures and foundations of matroids developed by Baker-Lorscheid, we give algorithms to: (i) compute the foundation of a matroid, and (ii) compute all morphisms between two pastures. Together, these provide an efficient method of solving many questions of interest in matroid representations, including orientability, non-representability, and computing all representations of a matroid over a finite field.