Betti numbers associated to the facet ideal of a matroid

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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.

Keywords

matroids facet ideals Betti numbers higher weights blocks cactus graphs 

Mathematical subject classification

05B35 05E40 13D02 05C25 

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Copyright information

© Sociedade Brasileira de Matemática 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TromsøTromsøNorway

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