Confinement of matroid representations to subsets of partial fields

R.A. Pendavingh, S.H.M. Zwam, van

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Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' in 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 (Whittle, 1999 [34]). A combination of the Confinement Theorem and the Lift Theorem from Pendavingh and Van Zwam (2010) [19] leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields (Whittle, 1997 [33]). 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 matrix A over PM such that every representation of M over a partial field P is equal to f(A) for some homomorphism f: PM --> P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen, Oxley, Vertigan, and Whittle (2002) [12]. Keywords: Matroids; Representations; Partial fields; Homomorphisms.
Original languageEnglish
Pages (from-to)510-545
JournalJournal of Combinatorial Theory, Series B
Issue number6
Publication statusPublished - 2010


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