Determinantal Facet Ideals for Smaller Minors

A determinantal facet ideal (DFI) is generated by a subset of the maximal minors of a generic $n\times m$ matrix where $n\le m$ indexed by the facets of a simplicial complex $\mathrm{\Delta }$. 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 $\mathrm{\Delta }$ for some $1\le r\le n$. We define and study so-called lcm-closed and unit interval $r$-DFIs, and show that the minors parametrized by the facets of $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ 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.