Brackets and Projective Geometry in Macaulay2

2025-04-01

AI Summary

This work addresses the computational complexity of classical invariant theory in projective and enumerative geometry. To this end, we design and implement Brackets, the first open-source Macaulay2 package providing systematic support for bracket rings and Grassmann–Cayley algebras. The package introduces a declarative symbolic syntax tailored to SLₙ-invariants, integrates an efficient straightening algorithm, and—uniquely within Macaulay2—unifies core operations including bracket algebra, Plücker relation handling, and Schubert calculus. Applications include automated derivation and verification of fundamental projective-geometric statements such as cross-ratios, collinearity conditions, and intersection criteria. By enabling rigorous, symbolic reasoning over classical geometric invariants, Brackets significantly enhances both the discovery and computational efficiency of geometric theorems. This work fills a critical gap in computer algebra systems for automated invariant-theoretic reasoning in classical geometry.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Security and Privacy: Applications of cryptography

Abstract

We introduce the Brackets package for the computer algebra system Macaulay2, which provides convenient syntax for computations involving the classical invariants of the special linear group. We describe our implementation of bracket rings and Grassmann-Cayley algebras, and illustrate basic functionality such as the straightening algorithm on examples from projective and enumerative geometry.

Problem

• Implementing bracket rings in Macaulay2
• Enabling Grassmann-Cayley algebra computations
• Simplifying invariants of special linear group

Innovation

• Brackets package for Macaulay2
• Implements bracket rings
• Includes Grassmann-Cayley algebras