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 [Whi99]. A combination of the Confinement Theorem and the Lift Theorem from [PZ] leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields [Whi97]. 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 A over PM such that every representation of M over a partial field P is equal to ¿(A) for some homomorphism ¿:PM¿P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.
|Number of pages||39|
|Publication status||Published - 2008|