Universal representations of matroids

Stefan van Zwam

 A matroid is a pair (E,B), where E is a finite set and B a collection of subsets of E, the "bases", satisfying some axioms (notably a "basis exchange axiom"). An important example is the following. Let A be a rank-r matrix over some field, and let E be the set of columns of A. Then B consists of all subsets of E that have size r and that are linearly independent. Not all matroids arise in this way, and important questions in matroid theory are whether a matroid can be represented by a matrix, and if so, over which fields. For a few finite fields F structural results are known: a matroid is representable over F unless it has a certain "minor". We will approach this problem from an algebraic angle: for a fixed matroid M we define a ring R and a matrix A with entries in this ring such that every representation of M over a field F is the image of A under some ring homomorphism f: R → F. We will give several examples of this construction.