The Free m-Cone of a Matroid and Its $${\mathcal {G}}$$-Invariant

Abstract

For a matroid, its configuration determines its \({\mathcal {G}}\)-invariant. Few examples are known of pairs of matroids with the same \({\mathcal {G}}\)-invariant but different configurations. In order to produce new examples, we introduce the free m-cone \(Q_m(M)\) of a loopless matroid M, where m is a positive integer. We show that the \({\mathcal {G}}\)-invariant of M determines the \({\mathcal {G}}\)-invariant of \(Q_m(M)\), and that the configuration of \(Q_m(M)\) determines M; so if M and N are nonisomorphic and have the same \({\mathcal {G}}\)-invariant, then \(Q_m(M)\) and \(Q_m(N)\) have the same \({\mathcal {G}}\)-invariant but different configurations. We prove analogous results for several variants of the free m-cone. We also define a new matroid invariant of M, and show that it determines the Tutte polynomial of \(Q_m(M)\).