Matrix representations of matroids of biased graphs correspond to gain functions
نویسندگان
چکیده
Let M be a frame matroid or a lifted-graphic matroid and let (G,B) be a biased graph representing M . Given a field F, a canonical F-representation of M particular to (G,B) is a matrix A arising from a gain function over the multiplicative or additive group of F that realizes (G,B). First, for a biased graph (G,B) that is properly unbalanced, loopless, and vertically 2-connected, we show that two canonical F-representations particular to (G,B) are projectively equivalent iff their associated gain functions are switching equivalent. Second, when M has sufficient connectivity, we show that every F-representation of M is projectively equivalent to a canonical F-representation; furthermore, when (G,B) is properly unbalanced, the canonical representation is particular to and unique with respect to (G,B).
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