Betti numbers associated to the facet ideal of a matroid

TL;DR

This paper explores the relationship between Betti numbers and the structure of matroids, showing how these algebraic invariants are determined by the matroid's blocks and applying this to cycle matroids of cactus graphs.

Contribution

It establishes that Betti numbers and higher weight hierarchies of a matroid are determined by those of its blocks, and connects these invariants for cycle matroids of cactus graphs.

Findings

Betti numbers are determined by the blocks of a matroid.

Higher weight hierarchies are also determined by blocks.

Betti numbers and weight hierarchies coincide for cycle matroids of cactus graphs.

Abstract

To a matroid M with n edges, we associate the so-called facet ideal F(M) generated by monomials corresponding to bases of M. We show that the Betti numbers related to an N-graded minimal free resolution of 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.