Betti numbers associated to the facet ideal of a matroid

Abstract

To a matroid M with n edges, we associate the so-called facet ideal $\mathcal{F}(M) \subset \Bbbk [x_1 , \ldots ,x_n ]$, generated by monomials corresponding to bases of M. We show that when M is a graph, the Betti numbers related to an ℕ₀-graded minimal free resolution of $\mathcal{F}(M)$ are determined by the Betti numbers related to the blocks of M. Similarly, we show that the higher weight hierarchy of M is determined by the weight hierarchies of the blocks, as well. Drawing on these results, we show that when M is the cycle matroid of a cactus graph, the Betti numbers determine the higher weight hierarchy — and vice versa. Finally, we demonstrate by way of counterexamples that this fails to hold for outerplanar graphs in general.