Von Staudt Constructions for Skew-Linear and Multilinear Matroids

TL;DR

This paper explores the algebraic properties of skew-linear and multilinear matroids using von Staudt constructions, revealing undecidability results and examples distinguishing these classes.

Contribution

It introduces a simple von Staudt construction variant and demonstrates undecidability and new examples in the representation theory of these matroids.

Findings

Undecidability of matroid representation problems over division rings

Existence of a matroid with an infinite multilinear characteristic set but not multilinear in characteristic 0

Existence of a skew-linear matroid that is not multilinear

Abstract

This paper compares skew-linear and multilinear matroid representations. These are matroids that are representable over division rings and (roughly speaking) invertible matrices, respectively. The main tool is the von Staudt construction, by which we translate our problems to algebra. After giving an exposition of a simple variant of the von Staudt construction we present the following results: $\bullet$ Undecidability of several matroid representation problems over division rings. $\bullet$ An example of a matroid with an infinite multilinear characteristic set, but which is not multilinear in characteristic $0$. $\bullet$ An example of a skew-linear matroid that is not multilinear.