Confinement of matroid representations to subsets of partial fields

Let M be a matroid representable over a (partial) field and B a matrix representable over a sub-partial field . We say that B confines M to if, whenever a -representation matrix A of M has a submatrix B, A is a scaled -matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem [34]. A combination of the Confinement Theorem and the Lift Theorem from Pendavingh and Van Zwam [19] leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields [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 M and ...

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Rudi Pendavingh, Stefan H. M. van Zwam
• Keywords: Confinement Theorem, JCT 2010, Matroid, Partial Field