This package includes routines to check whether an ideal is geometrically vertex decomposable.
Geometrically vertex decomposable ideals can be viewed as a generalization of the properties of the Stanley-Reisner ideal of a vertex decomposable simplicial complex. This family of ideals is based upon the geometric vertex decomposition property defined by Knutson, Miller, and Yong [KMY]. Klein and Rajchgot then gave a recursive definition for geometrically vertex decomposable ideals in [KR] using this notion.
An unmixed ideal $I$ in a polynomial ring $R$ is geometrically vertex decomposable if it is the zero ideal, the unit ideal, an ideal generated by indeterminates, or if there is a indeterminate $y$ of $R$ such that two ideals $C_{y,I}$ and $N_{y,I}$ constructed from $I$ are both geometrically vertex decomposable. For the complete definition, see isGVD.
Observe that a geometrically vertex decomposable ideal is recursively defined. The complexity of verifying that an ideal is geometrically vertex decomposable will increase as the number of indeterminates appearing in the ideal increases.
We thank Sergio Da Silva, Megumi Harada, Patricia Klein, and Jenna Rajchgot for feedback and suggestions. Additionally, we thank the anonymous referees of the paper [CVT] for their concrete suggestions that significantly improved that manuscript and this package. Cummings was partially supported by an NSERC USRA and CGS-M and a Milos Novotny Fellowship. Van Tuyl's research is partially supported by NSERC Discovery Grant 2019-05412.
[CDSRVT] Mike Cummings, Sergio Da Silva, Jenna Rajchgot, and Adam Van Tuyl. Geometric vertex decomposition and liaison for toric ideals of graphs. Algebr. Comb., 6(4):965–997, 2023.
[CVT] Mike Cummings and Adam Van Tuyl. The GeometricDecomposability package for Macaulay2. Preprint, available at arXiv:2211.02471, 2022.
[DSH] Sergio Da Silva and Megumi Harada. Geometric vertex decomposition, Gröbner bases, and Frobenius splittings for regular nilpotent Hessenberg Varieties. Transform. Groups, 2023.
[KMY] Allen Knutson, Ezra Miller, and Alexander Yong. Gröbner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math. 630 (2009) 1–31.
[KR] Patricia Klein and Jenna Rajchgot. Geometric vertex decomposition and liaison. Forum Math. Sigma, 9 (2021) e70:1-23.
[SM] Hero Saremi and Amir Mafi. Unmixedness and arithmetic properties of matroidal ideals. Arch. Math. 114 (2020) 299–304.
Version 1.2 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2024-01-23, in the article The GeometricDecomposability package for Macaulay2 (DOI: 10.2140/jsag.2024.14.41). That version can be obtained from the journal or from the Macaulay2 source code repository.
This documentation describes version 1.4.1 of GeometricDecomposability.
The source code from which this documentation is derived is in the file GeometricDecomposability.m2.
The object GeometricDecomposability is a package.