Confinement of matroid representations to subsets of partial fields

Abstract

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.

45 pages

References in corpus (3)

• Lifts of matroid representations over partial fields
• The Hydra-k partial fields
• The excluded minors for near-regular matroids

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• The excluded minors for near-regular matroids
• Foundations of matroids I: Matroids without large uniform minors
• The Templates for Some Classes of Quaternary Matroids
• The excluded minors for GF(5)-representable matroids on ten elements
• Stability, fragility, and Rota's Conjecture
• Representability of matroids with a large projective geometry minor
• An obstacle to a decomposition theorem for near-regular matroids