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Abstract

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic n× m matrix where n≤ m indexed by the facets of a simplicial complex Δ. We consider the more general notion of an r-DFI, which is generated by a subset of r-minors of a generic matrix indexed by the facets of Δ for some 1 ≤ r≤ n. We define and study so-called lcm-closed and unit interval r-DFIs, and show that the minors parametrized by the facets of Δ form a reduced Gröbner basis with respect to any term order for an lcm-closed r-DFI. We also see that being lcm-closed generalizes conditions previously introduced in the literature, and conjecture that in the case r= n, lcm-closedness is necessary for being a Gröbner basis. We also give conditions on the maximal cliques of Δ ensuring that lcm-closed and unit interval r-DFIs are Cohen-Macaulay. Finally, we conclude with a variant of a conjecture of Ene, Herzog, and Hibi on the Betti numbers of certain types of r-DFIs, and provide a proof of this conjecture for Cohen-Macaulay unit interval DFIs.

Document Type

Article

Publication Date

3-1-2022

Notes/Citation Information

Publisher Copyright: © 2022, Springer Nature Switzerland AG.

Digital Object Identifier (DOI)

10.1007/s00013-021-01692-z

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