### Abstract

Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P'= 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 [Whi99]. A combination of the Confinement Theorem and the Lift Theorem from [PZ] leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields [Whi97]. 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 A over PM such that every representation of M over a partial field P is equal to ¿(A) for some homomorphism ¿:PM¿P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al.

Original language | English |
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Publisher | s.n. |

Number of pages | 39 |

Publication status | Published - 2008 |

### Publication series

Name | arXiv.org [math.CO] |
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Volume | 0806.4487 |

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## Cite this

Pendavingh, R. A., & Zwam, van, S. H. M. (2008).

*Confinement of matroid representations to subsets of partial fields*. (arXiv.org [math.CO]; Vol. 0806.4487). s.n.