Representing some non-representable matroids
Advances in Applied Mathematics , Volume 50 - Issue 1 p. 201- 227
We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends TutteÊ¼s definition, using chain groups. We show how such representations behave under duality and minors, we extend TutteÊ¼s 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|
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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