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 × . 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).