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The GeometricDecomposability package for Macaulay2

Mike Cummings, Adam Van Tuyl

TL;DR

The Macaulay2 package GeometricDecomposability is introduced which provides a suite of tools to experiment and test the geometric vertex decomposability property of an ideal.

Abstract

Using the geometric vertex decomposition property first defined by Knutson, Miller, and Yong, a recursive definition for geometrically vertex decomposable ideals was given by Klein and Rajchgot. We introduce the Macaulay2 package GeometricDecomposability which provides a suite of tools to experiment and test the geometric vertex decomposability property of an ideal.

The GeometricDecomposability package for Macaulay2

TL;DR

The Macaulay2 package GeometricDecomposability is introduced which provides a suite of tools to experiment and test the geometric vertex decomposability property of an ideal.

Abstract

Using the geometric vertex decomposition property first defined by Knutson, Miller, and Yong, a recursive definition for geometrically vertex decomposable ideals was given by Klein and Rajchgot. We introduce the Macaulay2 package GeometricDecomposability which provides a suite of tools to experiment and test the geometric vertex decomposability property of an ideal.
Paper Structure (6 sections, 1 theorem, 3 equations, 1 figure, 1 table)

This paper contains 6 sections, 1 theorem, 3 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical background
  3. The package
  4. Examples
  5. Square-free monomial ideals
  6. Toric ideals of graphs

Key Result

Theorem 4.2

Let $\Delta$ be a simplicial complex on $V = \{x_1,\ldots,x_n\}$. Then $\Delta$ is vertex decomposable if and only if $I_\Delta$ is geometrically vertex decomposable.

Figures (1)

  • Figure 1: The unique graph $G$ with 8 edges whose toric ideal is not GVD (on the left) and the unique graph $H$ with 9 edges whose toric ideal is weakly GVD but not GVD (on the right)

Theorems & Definitions (6)

  • Definition 2.1: KMY
  • Definition 2.2: KR
  • Definition 2.3: KR
  • Definition 2.4: KR
  • Definition 4.1
  • Theorem 4.2: KR