Confinement of matroid representations to subsets of partial fields

Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P′ ⊆ 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 [34]. A combination of the Confinement Theorem and the Lift Theorem from Pendavingh and Van Zwam [19] leads to a short proof of Whittle’s characterization of the matroids representable over GF(3) and other fields [33]. 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 PM and a representation matrix A over PM such that every representation of M over a partial field P is equal to φ(A) for some homomorphism φ : PM → P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen, Oxley, Vertigan, and Whittle [12].