Weak Maps and Stabilizers of Classes of Matroids

✍ Scribed by James Geelen; James Oxley; Dirk Vertigan; Geoff Whittle

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 338 KB
• Volume: 21
• Category: Article
• ISSN: 0196-8858
• DOI: 10.1006/aama.1998.0600

✦ Synopsis

Let F be a field and let N be a matroid in a class N N of F-representable matroids that is closed under minors and the taking of duals. Then N is an F-stabilizer for N N if every representation of a 3-connected member of N 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 FЈ-stabilizer for N N for every field FЈ over which members of N 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 N by an elementary finite check. If N is a universal stabilizer for N N, we determine additional conditions on N and N N that ensure that if N is not a strict rank-preserv-