GeometricDecomposability -- a package to check whether ideals are geometrically vertex decomposable

Description

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.

Acknowledgement

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.

References

[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.

Authors

Version

This documentation describes version 1.4 of GeometricDecomposability.

Source code

The source code from which this documentation is derived is in the file GeometricDecomposability.m2.

Exports

Functions and commands
- findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable
- findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition
- getGVDIdeal -- computes the $C_{y,I}$ or $N_{y,I}$ ideal at any point in the GVD recursion tree
- initialYForms -- computes the ideal of initial y-forms
- isGeneratedByIndeterminates -- checks whether the ideal is generated by indeterminates
- isGVD -- checks whether an ideal is geometrically vertex decomposable
- isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order
- isUnmixed -- checks whether an ideal is unmixed
- isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable
- oneStepGVD -- computes a geometric vertex decomposition
- oneStepGVDCyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
- oneStepGVDNyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
Methods
- findLexCompatiblyGVDOrders(Ideal) -- see findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable
- findOneStepGVD(Ideal) -- see findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition
- getGVDIdeal(Ideal,List) -- see getGVDIdeal -- computes the $C_{y,I}$ or $N_{y,I}$ ideal at any point in the GVD recursion tree
- initialYForms(Ideal,RingElement) -- see initialYForms -- computes the ideal of initial y-forms
- isGeneratedByIndeterminates(Ideal) -- see isGeneratedByIndeterminates -- checks whether the ideal is generated by indeterminates
- isGVD(Ideal) -- see isGVD -- checks whether an ideal is geometrically vertex decomposable
- isLexCompatiblyGVD(Ideal,List) -- see isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order
- isUnmixed(Ideal) -- see isUnmixed -- checks whether an ideal is unmixed
- isWeaklyGVD(Ideal) -- see isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable
- oneStepGVD(Ideal,RingElement) -- see oneStepGVD -- computes a geometric vertex decomposition
- oneStepGVDCyI(Ideal,RingElement) -- see oneStepGVDCyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
- oneStepGVDNyI(Ideal,RingElement) -- see oneStepGVDNyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
Symbols
- CheckCM -- when to perform a Cohen-Macaulay check on the ideal
- CheckDegenerate -- check whether the geometric vertex decomposition is degenerate
- CheckUnmixed -- check whether ideals encountered are unmixed
- IsIdealHomogeneous -- specify whether an ideal is homogeneous
- IsIdealUnmixed -- specify whether an ideal is unmixed
- OnlyDegenerate -- restrict to degenerate geometric vertex decompositions
- OnlyNondegenerate -- restrict to nondegenerate geometric vertex decompositions
- SquarefreeOnly -- only return the squarefree variables from the generators
- UniversalGB -- whether the generators for an ideal form a universal Gröbner basis

For the programmer

The object GeometricDecomposability is a package.

CheckCM -- when to perform a Cohen-Macaulay check on the ideal
CheckDegenerate -- check whether the geometric vertex decomposition is degenerate
CheckUnmixed -- check whether ideals encountered are unmixed
findLexCompatiblyGVDOrders -- finds all lexicographic monomial orders $<$ such that the ideal is $<$-compatibly geometrically vertex decomposable
findOneStepGVD -- for which indeterminates does there exist a geometric vertex decomposition
getGVDIdeal -- computes the $C_{y,I}$ or $N_{y,I}$ ideal at any point in the GVD recursion tree
initialYForms -- computes the ideal of initial y-forms
isGeneratedByIndeterminates -- checks whether the ideal is generated by indeterminates
isGVD -- checks whether an ideal is geometrically vertex decomposable
IsIdealHomogeneous -- specify whether an ideal is homogeneous
IsIdealUnmixed -- specify whether an ideal is unmixed
isLexCompatiblyGVD -- checks whether an ideal is <-compatibly geometrically vertex decomposable for a given order
isUnmixed -- checks whether an ideal is unmixed
isWeaklyGVD -- checks whether an ideal is weakly geometrically vertex decomposable
oneStepGVD -- computes a geometric vertex decomposition
oneStepGVDCyI -- computes the ideal $C_{y,I}$ for a given ideal and indeterminate
oneStepGVDNyI -- computes the ideal $N_{y,I}$ for a given ideal and indeterminate
OnlyDegenerate -- restrict to degenerate geometric vertex decompositions
OnlyNondegenerate -- restrict to nondegenerate geometric vertex decompositions
SquarefreeOnly -- only return the squarefree variables from the generators
UniversalGB -- whether the generators for an ideal form a universal Gröbner basis