Determinantal Facet Ideals for Smaller Minors

TL;DR

This paper introduces and studies generalized determinantal facet ideals (r-DFIs), exploring their algebraic properties, Gr"obner bases, Cohen-Macaulay conditions, and Betti number conjectures, extending previous work on determinantal ideals.

Contribution

It generalizes determinantal facet ideals to r-DFIs, characterizes Gr"obner bases for lcm-closed r-DFIs, and establishes Cohen-Macaulay conditions and Betti number conjectures.

Findings

Minors form a reduced Gr"obner basis for lcm-closed r-DFIs.

Lcm-closedness generalizes previous conditions and is conjectured necessary for Gr"obner bases when r=n.

Conditions on maximal cliques ensure Cohen-Macaulayness of r-DFIs.

Abstract

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\leq m$ indexed by the facets of a simplicial complex $\Delta$. 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 $\Delta$ for some $1\leq r\leq n$. We define and study so-called lcm-closed and unit interval $r$-DFIs, and show that the minors parametrized by the facets of $\Delta$ form a reduced Gr\"obner basis with respect to \emph{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 Gr\"obner basis. We also give conditions on the maximal cliques of $\Delta$ 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.