Lori Koban (Binghamton)

Abstracts for the Combinatorics and Number Theory Seminar

How to Characterize Matroid Circuit Signatures by Modular Triples

2003 September 24

A circuit signature of a matroid is obtained by signing (either + or -) each element of every circuit. Matroid orientations, weak orientations, and ternary signatures are all examples of circuit signatures. They have been characterized in many ways, such as by forbidden minors, circuit elimination, and orthogonality. I will use modular triples of circuits (an abstraction of theta-graphs) to give a new characterization of these signatures. I will briefly mention the motivation for looking at modular triples, but a detailed explanation will be postponed until my talk next week.

Using Gains to Lift Ternary Matroids

2003 October 1

For a matroid M, Dowling and Kelly showed how to use a special class of its circuits, which we call balanced circuits, to construct a lift of M (a related matroid of rank 1 greater).  In the case where M comes from a graph, Zaslavsky defined the balanced circuits in terms of gains, which are group elements that label the edges of the graph.  The resulting lift matroid is helpful in studying particular hyperplane arrangements.  I will use gains to define the balanced circuits of an arbitrary matroid and show that the lift construction can be done precisely when the matroid is representable over GF(3).