GeometricDecomposability

A Macaulay2 package to check whether an ideal is geometrically vertex decomposable, developed by Mike Cummings and Adam Van Tuyl.

For further information, see: The GeometricDecomposability package for Macaulay2.

We also maintain a changelog that tracks changes by version.

Installation

Using version 1.21 or higher of Macaulay2, run the command loadPackage "GeometricDecomposability". We recommend using the most recent version of Macaulay2. The warning message that appeared upon loading the package in Macaulay2 version 1.21 (which, in most cases, can be safely ignored) has been fixed in version 1.22 of Maculay2.

Alternatively, this package can be installed to Macaulay2 by copying the file GeoetricDecomposability.m2 to your working directory from which you launch Macaulay2. Then in M2, run the command installPackage "GeometricDecomposability" and use the package as you would any other.

After the package is loaded, you can read the documentation by running viewHelp GeometricDecomposability, which will open the documentation in your web browser.

Background

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.

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.

Acknowledgements

We thank Sergio Da Silva, Megumi Harada, Patricia Klein, and Jenna Rajchgot for feedback and improvements. Cummings was partially supported by an NSERC USRA and CGS-M and the Milos Novotny Fellowship. Van Tuyl's research is partially supported by NSERC Discovery Grant 2019-05412.

References

[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.