The GeometricDecomposability package for Macaulay2

INTRODUCTION.The geometric vertex decomposition of an ideal was first introduced by Knutson, Miller, and Yong [9] as part of their study of vexillary matrix Schubert varieties.Geometric vertex decomposition can be viewed as a generalization of a vertex decomposition of a simplicial complex.Using the notion of a geometric vertex decomposition, Klein and Rajchgot [7] introduced geometrically vertex decomposable ideals.These ideals, which are defined recursively, were partially inspired by the definition of a vertex decomposable simplicial complex, a recursively defined family of simplicial complexes.As shown by both [9] and [7], ideals that have a geometric vertex decomposition, or are geometrically vertex decomposable, have other desirable properties.As one such example, Klein and Rajchgot [7, Corollary 4.8] have shown that homogeneous geometrically vertex decomposable ideals are glicci, i.e., these ideals belong to the Gorenstein liaison class of a complete intersection.Further properties of geometrically vertex decomposable ideals have been developed in [4; 6; 8; 5].Due to their recent introduction, there are many features of geometrically vertex decomposable ideals that are still not known.To facilitate further experimentation and exploration, we have created GeometricDecomposability, a package for Macaulay2 that enables researchers to test and search for ideals that are geometrically vertex decomposable.In particular, our package allows the user to check if a given ideal satisfies the geometric vertex decomposition property of [9] or the geometrically vertex decomposable property (or its variants) as found in [7].This note reviews the needed mathematical background, summarizes the main features of our packages, and provides some illustrative examples.2. MATHEMATICAL BACKGROUND.We summarize the mathematical background to define the geometric vertex decomposition property and geometrically vertex decomposable ideals.Throughout, k denotes a field.