Weak Maps and Stabilizers of Classes of Matroids

Authors: Geelen J.¹; Oxley J.²; Vertigan D.²; Whittle G.³

Source: Advances in Applied Mathematics, Volume 21, Number 2, August 1998 , pp. 305-341(37)

Publisher: Academic Press

Abstract:

Let F be a field and let N be a matroid in a class N of F-representable matroids that is closed under minors and the taking of duals. Then N is an F-stabilizer for N if every representation of a 3-connected member of N is determined up to elementary row operations and column scaling by a representation of any one of its N-minors. The study of stabilizers was initiated by Whittle. This paper extends that study by examining certain types of stabilizers and considering the connection with weak maps.

The notion of a universal stabilizer is introduced to identify the underlying matroid structure that guarantees that N will be an F-stabilizer for N for every field F over which members of N are representable. It is shown that, just as with F-stabilizers, one can establish whether or not N is a universal stabilizer for N by an elementary finite check. If N is a universal stabilizer for N, we determine additional conditions on N and N that ensure that if N is not a strict rank-preserving weak-map image of any matroid in N, then no connected matroid in N with an N-minor is a strict rank-preserving weak-map image of any 3-connected matroid in N.

Applications of the theory are given for quaternary matroids. For example, it is shown that U2, 5 is a universal stabilizer for the class of quaternary matroids with no U3, 6-minor. Moreover, if M1 and M2 are distinct quaternary matroids with U2, 5-minors but no U3, 6-minors and M1 is connected while M2 is 3-connected, then M1 is not a rank-preserving weak-map image of M2. Copyright 1998 Academic Press.

Language: English

Document Type: Research article

Affiliations: 1: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada 2: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, 70803-4918 3: School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand

Publication date: 1998-08-01

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• In this Subject: Mathematics and Statistics
• By this author: Geelen J. ;  Oxley J. ;  Vertigan D. ;  Whittle G.

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