Confinement of matroid representations to subsets of partial fields

TL;DR

This paper introduces a confinement concept for matroid representations over partial fields, providing finite verification methods, new characterizations of matroids' representability, and algebraic constructions that unify various representation theories.

Contribution

It establishes the Confinement Theorem, offering finite checks for confinement, and applies it to characterize matroids over GF(3) and GF(5), also constructing a universal partial field for a given matroid.

Findings

Finite check for confinement under certain conditions.

Characterization of 3-connected matroids with multiple GF(5) representations.

Construction of a universal partial field for any matroid.

Abstract

Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem. A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields. We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6. Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.