Biased Graphs: IV. Geometrical Realizations

A gain graph \Phi consists of a graph \Gamma and a gain function from E (\Gamma) to a group G , the gain group. A circle (edge set of a simple closed path) is balanced if its edge gain product is the identity; this determines a biased graph as studied in Parts I-III. On E (\Gamma) we have two matroids determined by \Phi in which every balanced circle is a circuit. The bias matroid G has connected circuits; the lift matroid L has circuits not necessarily connected. We investigate representations of these matroids. Each has a canonical vector representation * Research substantially assisted by grants from the National Science Foundation: in 1976--1977 at the Massachusetts Institute of Technology, in 1984--1985 under DMS-8407102 while I was visiting the University of Evansville, and in 1986 under DMS-8606102 at Binghamton. over any skew field F such that G ` F (in the case of G) or G ` F + (in the case of L). The bias matroid also has two projective representations that strongly..