Linear Strands of Initial Ideals of Determinantal Facet Ideals

A determinantal facet ideal (DFI) is an ideal $J_Δ$ generated by maximal minors of a generic matrix parametrized by an associated simplicial complex $Δ$. 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 $Δ$ has no \emph{1-nonfaces}, then the Betti numbers of the linear strand of $J_Δ$ 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.

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