## Projective Representation of Non-Representable Matroids (of Biased Graphs)

- Feb. 12, 2016
- 2:30 p.m.
- LeConte 312

## Abstract

A matroid is an axiomatization of the concept of linear dependence and linear independence. The independence concept is present in several areas of mathematics. Including graph theory, linear spaces and transcendental extension fields. In such cases we say that the matroid is graphic, linear or algebraic representable; respectively. However, a matroid that is representable in one area of mathematics may be no representable in another areas.

This is an unfinished project. Four years ago I gave a talk about this project here in this seminar. This time I going to discuss some advances that we have so far. After a short introduction of what a matroids is we discuss *biased expansion* of *K3* and the bias matroid representability. Given a quasi group *Q*, there is a complete graph *K3* with multiple edges corresponding to the elements of *Q*. This graph give rise to two rank-3 matroids --the *full frame matroid G(Q K3)* and *extended lift matroid L0(Q K3)*.

When *Q* is a subgroup of the multiplicative or additive group of a skew field *F*, the two mentioned matroids are representable in the projective plane over *F*. Thomas Zaslavsky and I are generalizing this standard theorem to arbitrary quasigroups (this is more complicated), and the role of *F* being taken by a planar ternary ring associated with a projective plane.