Lifts of matroid representations over partial fields

Details Lifts of matroid representations over partial fields Lifts of matroid representations over partial fields

2008-04-21

arxiv.org/abs/0804.3263v2

Journal of Combinatorial Theory, Series B, vol. 100, Issue 1, pp. 36-67, 2010

Mathematics

Combinatorics

41 pages, 1 figure. Contains minor revisions, the most substantial being a new Section 4.1 that shows how Tutte's characteriza

Scientific paper

10.1016/j.jctb.2009.03.006

There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First, parts of Whittle's characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a power of the golden ratio. Third, we give a characterization of the 3-connected matroids having at least two inequivalent representations over GF(5). We show that these are representable over the complex numbers. Additionally we provide an algebraic construction that, for any set of fields F_1,...,F_k, gives the best possible result that can be proven using the Lift Theorem.

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