Betti numbers associated to the facet ideal of a matroid
To a matroid M with n edges, we associate the so-called facet ideal F(M)⊂k[x1,…,xn] , 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 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.
Accepted manuscript version. The final publication is available at Springer via http://doi.org/10.1007/s00574-014-0071-9.