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 Whittleʹs Stabilizer Theorem (Whittle, 1999 [34]). ination of the Confinement Theorem and the Lift Theorem from Pendavingh and Van Zwam (2010) [19] leads to a short proof of Whittleʹs characterization of the matroids representable over GF ( 3 ) and other fields (Whittle, 1997 [33]). o 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 . onally we give, for a fixed matroid M, an algebraic construction of a partial field P M and a representation matrix A over P M such that every representation of M over a partial field P is equal to ϕ ( A ) for some homomorphism ϕ : P M → P . Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen, Oxley, Vertigan, and Whittle (2002) [12].