Generalised Matroid Representations: Universality and Decidability

A matroid is a combinatorial object based on an abstraction of linear independence in vector spaces and forests in graphs. It is a classical question to determine whether a given matroid is representable as a vector configuration over a field. Such a matroid is called linear.

This talk addresses generalisations of such representations over division rings or matrix rings which are called skew linear and multilinear matroids respectively.We will describe a generalised Dowling geometry that encodes non commutative equations in matroids. This construction allows us to reduce word problem instances to skew linear or multilinear matroid representations.

The talk is based on joint work with Rudi Pendavingh and Geva Yashfe.