Lifts of matroid representations over partial fields

TL;DR

This paper introduces the Lift Theorem, unifying various results on matroid representability over different fields, and explores implications for ternary, GF(4), GF(5), and complex representations.

Contribution

The paper proves the Lift Theorem, providing a unified framework for understanding matroid representations over multiple fields and deriving new results in the theory.

Findings

Parts of Whittle's characterization follow from the theorem

Matroids representable over GF(4) and GF(5) are representable over the reals with golden ratio determinants

3-connected matroids with multiple GF(5) representations are representable over complex numbers

Abstract

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.