The free $m$-cone of a matroid and its $\mathcal{G}$-invariant

TL;DR

This paper introduces the free m-cone of a matroid, demonstrating how it preserves the G-invariant while enabling the construction of nonisomorphic matroids with identical G-invariants but different configurations.

Contribution

It defines the free m-cone of a matroid and proves its properties, including how it relates to the G-invariant and configuration, providing new examples of matroids with identical G-invariants.

Findings

The G-invariant of M determines that of Q_m(M).

The configuration of Q_m(M) determines M.

Q_m(M) and Q_m(N) can have the same G-invariant but different configurations.

Abstract

For a matroid $M$, 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 $Q_m(M)$.