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- W2523428152 abstract "Let $M$ be a frame matroid or a lifted-graphic matroid and let $(G,mathcal{B})$ be a biased graph representing $M$. Given a field $mathbb{F}$, a canonical $mathbb{F}$-representation of $M$ particular to $(G,mathcal{B})$ is a matrix $A$ arising from a gain function over the multiplicative or additive group of $mathbb{F}$ that realizes $(G,mathcal{B})$. First, for a biased graph $(G,mathcal{B})$ that is properly unbalanced, loopless, and vertically 2-connected, we show that two canonical $mathbb{F}$-representations particular to $(G,mathcal{B})$ are projectively equivalent iff their associated gain functions are switching equivalent. Second, when $M$ has sufficient connectivity, we show that every $mathbb{F}$-representation of $M$ is projectively equivalent to a canonical $mathbb{F}$-representation; furthermore, when $(G,mathcal{B})$ is properly unbalanced, the canonical representation is particular to and unique with respect to $(G,mathcal{B})$." @default.
- W2523428152 created "2016-09-30" @default.
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- W2523428152 date "2016-09-19" @default.
- W2523428152 modified "2023-09-27" @default.
- W2523428152 title "Matrix representations of matroids of biased graphs correspond to gain functions" @default.
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