Grobner bases for determinantal facet ideals of simplicial complexes

Abstract

We provide necessary and sufficient conditions for simplicial complexes whose determinantal facet ideals admit reduced Grobner bases under diagonal term orders. Building on and extending foundational results for binomial edge ideals and determinantal ideals, we introduce two new classes of simplicial complexes-strong closed and poor closed-that generalize the notion of closedness in higher dimensions. Our main theorem offers a unified framework that recovers and refines several known results, including those for unit interval graphs and determinantal ideals of complete graphs. In particular, we correct and generalize prior characterizations of Grobner bases for determinantal facet ideals, establishing radicality for strong closed complexes and providing a new proof for the Grobner basis of maximal minors.