Linear Strands of Initial Ideals of Determinantal Facet 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.

16 pages; v2: added revisions in line with referee comments + DOI for published version. v1: significantly expanded version of contents originally appearing in 2006.14434. arXiv admin note: text overlap with arXiv:2006.14434

References in corpus (2)

• Prime splittings of Determinantal Ideals
• A proof for a conjecture on the regularity of binomial edge ideals