Generalized Graphic Matroids in Projective Geometry

Speaker: Thomas Zaslavsky, Binghamton University

Title: Generalized Graphic Matroids in Projective Geometry

Abstract: A graph Γ has a vector representation in which an edge eij = vivj corresponds to the vector bj − bi, where b1, b2, . . . are the standard unit basis vectors. Such a representation is
a vector representation of the graphic matroid M(Γ). We may restate this as a mapping of E(Γ) into projective space, giving a projective representation of M(Γ). Properties of graphs
imply that this representation is unique up to projective transformations. A gain graph Φ is a graph with something more: an orientable “gain function” ϕ : E → H where H is a group. Φ has a “frame matroid” M(Φ) that generalizes the graphic matroid. If H ≤ F ×, the multiplicative group of a field, then Φ has a vector representation (e, g) 7→ bj − ϕ(eij )bi that represents the frame matroid. We can restate this as a representation of M(Φ) in projective space (which may not be unique). What if H is not in any F ×? Even then we may be able to represent M(Φ) projectively
by using synthetic geometry. I will explain how this works, first in any dimension, and then for rank-3 matroids in projective planes. The special feature of the latter is that, while all higher-dimensional projective spaces have coordinates in a field (or skew field), most projective planes do not; this means one needs a more combinatorial treatment of representation involving quasigroups and ternary rings, which I will explain. This report is on joint work with Rigoberto Fl´orez.

Time: Friday, April 13, 2018, 3:30-4:20 p.m.

Place: Exploratory Hall, room 4106

Refreshments will be served at 3:00 p.m.