Matrix Representations of Frame and Lifted-Graphic Matroids Correspond To Gain Functions

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Let M be a 3-connected matroid and let F" role="presentation"> be a field. Let A be a matrix over F" role="presentation"> representing M and let (G,B)" role="presentation"> be a biased graph representing M. We characterize the relationship between A and (G,B)" role="presentation">, settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation (G,B)" role="presentation"> of M, A is projectively equivalent to a canonical matrix representation arising from G as a gain graph over F+" role="presentation"> or F×" role="presentation"> realizing B" role="presentation">. 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)" role="presentation">, except in one degenerate case.



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