Biased graphs IV: Geometrical realizations

A gain graph is a graph whose oriented edges are labelled invertibly from a group G, the gain group. A gain graph determines a biased graph and therefore has three natural matroids (as shown in Parts I and II): the bias matroid G has connected circuits; the complete lift matroid L0 and its restriction to the edge set, the lift matroid L, have circuits not necessarily connected. We investigate representations of these matroids. Each has a canonical vector representation over any skew field F such that G⊆F∗ (in the case of G) or G⊆F+ (in the case of L and L0). The representation of G is unique up to change of gains when the gain graph is full, but not in general. The representation of G or L is unique or semi-unique (up to changing the gains) for ‘thick’ biased graphs. The lift representations are unique (up to change of gains) for L0 but not for L. The bias matroid is representable also in other ways by points and hyperplanes; one of these representations dualizes the vector representation, while two, in projective space, strongly generalize the theorems of Menelaus and Ceva. (The latter specialize to properties of the geometry of midpoints and farpoints, and of median and edge-parallel hyperplanes, in an affine simplex.) The dual hyperplane representation can be abstracted away from fields to a kind of equational logic and permutation geometry that exist for every gain group. The lift matroids are representable by orthographic points and by linear, projective, and affinographic hyperplanes. L0 also has a metric hyperplanar representation that depends on the Pythagorean theorem. Incidental results are that Whitneyʹs 2-isomorphism operations preserve gains and, to an extent, matroids; and new definitions of the bias and lift matroids based on extremal properties of the rank functions.