We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutteʼs definition, using chain groups. We show how such representations behave under duality and minors, we extend Tutteʼs 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.
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Advances in Applied Mathematics
Matroid Structure for Efficiency
Networks and Optimization

van Zwam, S., & Pendavingh, R. (2013). Representing some non-representable matroids. Advances in Applied Mathematics, 50(1), 201–227. doi:10.1016/j.aam.2011.08.003