Weak Maps and Stabilizers of Classes of Matroids

LetFbe a field and letNbe a matroid in a class N ofF-representable matroids that is closed under minors and the taking of duals. ThenNis anF-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 itsN-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 thatNwill be anF'-stabilizer for N for every fieldF' over which members of N are representable. It is shown that, just as withF-stabilizers, one can establish whether or notNis a universal stabilizer for N by an elementary finite check. IfNis a universal stabilizer for N, we determine additional conditions onNand N that ensure that ifNis not a strict rank-preserving weak-map image of any matroid in N, then no connected matroid in N with anN-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 thatU"2","5is a universal stabilizer for the class of quaternary matroids with noU"3","6-minor. Moreover, ifM"1andM"2are distinct quaternary matroids withU"2","5-minors but noU"3","6-minors andM"1is connected whileM"2is 3-connected, thenM"1is not a rank-preserving weak-map image ofM"2.

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