Matrix representations of frame and lifted-graphic matroids correspond to gain functions

Abstract

Let M be a 3-connected matroid and let F be a field. Let A be a matrix over F representing M and let (G,B) be a biased graph representing M. We characterize the relationship between A and (G,B), settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation (G,B) of M, A is projectively equivalent to a canonical matrix representation arising from G as a gain graph over F+ or F× realizing B. Further, we show that the projective equivalence classes of matrix representations of M are in one-to-one correspondence with the switching equivalence classes of gain graphs arising from (G,B), except in one degenerate case.