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

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 $Q_m(M)$.

Figures (4)

• Figure 1: Two rank-$3$ sparse paving matroids.
• Figure 2: Parts (a) and (b) show the lattice of cyclic flats of $M_1$ and $M_2$. Replacing each set by just its size and rank gives the configuration of $M_1$ and $M_2$, shown in part (c).
• Figure 3: The free $1$-cones of the matroids $M_1$ and $M_2$, on $E=[6]$, in Figure \ref{['fig:continuing examples']}. Each of $M_1$ and $M_2$ is shown in a face of a tetrahderon. The tip $a$ is at the vertex opposite that face. For each $e\in [6]$, the set $T_e$ is $\{\overline{e}\}$.
• Figure 4: The matroids above have the same Tutte polynomial, but $\mathcal{G}(N_1)\ne \mathcal{G}(N_2)$. Their free $1$-cones have different Tutte polynomials.

Theorems & Definitions (35)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Corollary 3.5