Confinement of matroid representations to subsets of partial fields
Journal of Combinatorial Theory - Series B , Volume 100 - Issue 6 p. 510- 545
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.
|Keywords||Matroids, Representations, Partial fields, Homomorphisms|
|MSC||Matroids, geometric lattices (msc 05B35)|
|THEME||Logistics (theme 3)|
|Journal||Journal of Combinatorial Theory - Series B|
|Project||Matroid Structure for Efficiency|
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.