A Family of Monomial Gr\"obner Degenerations of Determinantal Ideals with Minimal Cellular Resolutions

Abstract

We explore a family of monomial ideals derived as Gr\"obner degenerations of determinantal ideals. These ideals, previously examined as block diagonal matching field ideals within the realm of toric degenerations of Grassmannians, are identified as monomial initial ideals of determinantal ideals. We establish their linear quotient property and calculate their Betti numbers, demonstrating that their minimal free resolution is supported on a regular CW complex, equivalently, they have a cellular resolution. On the combinatorial front, we examine a range of weight vectors that yield diverse monomial initial ideals for determinantal ideals. We show that despite variations in weight vectors, all these ideals share identical Betti numbers. However, altering a weight vector results in distinct cellular complexes supporting their minimal free resolutions.