Linear Strands of Initial Ideals of Determinantal Facet Ideals

TL;DR

This paper constructs explicit linear strands for initial ideals of determinantal facet ideals, showing Betti number coincidences under certain conditions and applying results to binomial edge ideals and polyhedral complexes.

Contribution

It provides an explicit construction of linear strands for initial ideals of DFIs and proves Betti number coincidences when -nonfaces are absent, extending to binomial edge ideals.

Findings

Betti numbers of linear strands coincide under no 1-nonfaces.

Linear strands supported on polyhedral cell complexes.

Application to conjecture on binomial edge ideals.

Abstract

A determinantal facet ideal (DFI) is an ideal $J_\Delta$ generated by maximal minors of a generic matrix parametrized by an associated simplicial complex $\Delta$. In this paper, we construct an explicit linear strand for the initial ideal with respect to any diagonal term order $<$ of an arbitrary DFI. In particular, we show that if $\Delta$ has no \emph{1-nonfaces}, then the Betti numbers of the linear strand of $J_\Delta$ and its initial ideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibi on Betti numbers of closed binomial edge ideals in the case that the associated graph has at most $2$ maximal cliques. More generally, we show that the linear strand of the initial ideal (with respect to $<$) of \emph{any} DFI is supported on a polyhedral cell complex obtained as an induced subcomplex of the \emph{complex of boxes}, introduced by Nagel and Reiner.