Linear strands of initial ideals of determinantal facet ideals

Abstract

We construct an explicit linear strand for the initial ideal of any determinantal facet ideal (DFI) with respect to any diagonal term order. We show that if the clique complex of Δ has no 1-nonfaces larger than a certain cardinality, then the Betti numbers of the linear strand of and its initial ideal coincide. This confirms a conjecture of Ene–Herzog–Hibi for closed graphs with at most 2 maximal cliques. Additionally, we show that the linear strand of the initial ideal of any DFI is supported on an induced subcomplex of the complex of boxes introduced by Nagel–Reiner.