Skew partial fields, multilinear representations of matroids, and a matrix tree theorem

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends [email protected]?s definition, using chain groups. We show how such representations behave under duality and minors, we extend [email protected]?s representability criterion to this new class, and we study the generator matrices of the chain groups. An example shows that the class of matroids representable over a skew partial field properly contains the class of matroids representable over a skew field. Next, we show that every multilinear representation of a matroid can be seen as a representation over a skew partial field. Finally we study a class of matroids called quaternionic unimodular. We prove a generalization of the matrix tree theorem for this class.