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.
Additional Metadata
Keywords Matroids, Representations, Skew partial field, Multilinear representations, matrix tree theorem
MSC Matroids, geometric lattices (msc 05B35)
THEME Other (theme 6)
Publisher Elsevier
Persistent URL dx.doi.org/10.1016/j.aam.2011.08.003
Journal Advances in Applied Mathematics
Project Matroid Structure for Efficiency
Note Special issue in honor of Geoff Whittle on his 60th birthday
Citation
van Zwam, S.H.M, & Pendavingh, R. (2013). Representing some non-representable matroids. Advances in Applied Mathematics, 50(1), 201–227. doi:10.1016/j.aam.2011.08.003