Combinatorics Seminar- A New Perspective on the G-Invariant of a Matroid

Speaker: Joe Bonin, GWU
Date and time: Wednesday, September 27, 4-5pm
Place: Rome 771
Title:  A New Perspective on the G-Invariant of a Matroid

Abstract:The G-invariant of a matroid was introduced by Derksen (2009), who showed that it properly generalizes the Tutte polynomial.  Derksen and Fink (2010) showed that the G-invariant is universal among invariants that satisfy an inclusion/exclusion-like property (defining valuations) on matroid base polytopes. In this joint work with Joseph Kung, we give a new view of this invariant and explore its implications.  We show that the G-invariant of a matroid M is equivalent to recording the size-increases along all maximal flags of flats of M.  With this, we can determine the effect of some matroid constructions, such as taking q-cones, on the G-invariant, and we can identify some of the information that the G-invariant picks up that the Tutte polynomial does not, such as the number of circuits and cocircuits of any given size, and the number of cyclic flats of any given rank and size.  From its G-invariant, we can tell whether a matroid is a free product of two other matroids (other than free extensions and coextensions).  Also, the G-invariant of a matroid can be reconstructed from the multi-set of G-invariants of the restrictions to hyperplanes.  Still, extending what J.~Eberhardt (2014) proved for the Tutte polynomial, the G-invariant is determined just by the isomorphism type of the lattice of cyclic flats along with the rank and size of each cyclic flat.