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.
Additional Metadata
Keywords Matroids, Representations, Partial fields, Homomorphisms
MSC Matroids, geometric lattices (msc 05B35)
THEME Logistics (theme 3)
Publisher Academic Press
Journal Journal of Combinatorial Theory - Series B
Project Matroid Structure for Efficiency
Citation
Pendavingh, R, & van Zwam, S.H.M. (2010). Confinement of matroid representations to subsets of partial fields. Journal of Combinatorial Theory - Series B, 100(6), 510–545.