The CompGIT package: a computational tool for Geometric Invariant Theory quotients (2506.19431v2)

Abstract: We describe CompGIT, a SageMath package to describe Geometric Invariant Theory (GIT) quotients of projective space by simple groups. The implementation is based on algorithms described by Gallardo--Martinez-Garcia--Moon--Swinarski. In principle the package is sufficient to describe any GIT quotient of a projective variety by a simple group -- in practice it requires that the user can construct an equivariant embedding of the polarised variety into projective space. The package describes the non-stable and unstable loci up to conjugation by the group, as well as describing the strictly polystable loci. We discuss potential applications of the outputs of CompGIT to algebraic geometry problems, a well as suggesting directions for future developments.

• The paper introduces CompGIT, a SageMath package that computes explicit GIT quotients by determining unstable, non-stable, and strictly polystable loci in projective varieties.
• It employs advanced representation theory and combinatorial algorithms to efficiently analyze group actions on high-dimensional moduli spaces.
• Detailed numerical results and comparisons with prior tools highlight its practical utility and potential for future enhancements, including parallelization and broader group support.

The CompGIT Package: Computational Methods for GIT Quotients

The CompGIT package provides a computational framework for the explicit analysis of Geometric Invariant Theory (GIT) quotients of projective varieties by simple algebraic groups, implemented as a SageMath package. The package operationalizes algorithms from Gallardo, Martinez-Garcia, Moon, and Swinarski, focusing on the practical computation of unstable, non-stable, and strictly polystable loci in projective GIT problems. This work addresses a longstanding need for accessible, systematic computational tools in the paper of moduli spaces and invariant theory, particularly for problems involving high-dimensional representations and exceptional groups.

Core Functionality and Implementation

CompGIT is designed to handle GIT quotients of the form $\mathbb{P}^n / G$, where $G$ is a simple group acting linearly on projective space. The package leverages SageMath's infrastructure for root systems, Weyl groups, and representation theory, allowing users to specify $G$-representations via highest weights and to work directly with the combinatorial data underlying the group action.

The main computational tasks performed by CompGIT are:

• Computation of unstable loci: Identification of points in projective space that are unstable under the group action, using the Hilbert-Mumford numerical criterion.
• Computation of non-stable loci: Determination of points that are not stable, i.e., those with positive-dimensional stabilizers or non-closed orbits.
• Computation of strictly polystable loci: Identification of orbits that are polystable but not stable, with respect to a maximal torus.

The package is structured around a GITProblem class, which encapsulates the representation data and provides methods for solving and interpreting the GIT problem. The implementation is tightly integrated with SageMath's WeylCharacterRing and WeylGroup classes, enabling efficient manipulation of weights, orbits, and group symmetries.

Example Workflow

A typical workflow involves:

from CompGIT import * Phi = WeylCharacterRing("A2") representation = Phi(3,0,0) # e.g., plane cubics P = GITProblem(representation) P.solve_non_stable() P.print_solution_nonstable()

The output provides explicit descriptions of destabilizing one-parameter subgroups and the associated weight data, which can be interpreted in terms of geometric properties (e.g., singularities of curves).

Numerical Results and Complexity

The paper provides detailed tables quantifying the complexity of various GIT problems, including the number of weights, non-stable families, unstable families, and strictly polystable families for representations of classical and exceptional groups. For instance, in the case of $B_2$ acting on degree $d$ hypersurfaces, the number of non-stable families grows moderately with $d$, while for higher-rank or exceptional groups, the combinatorial complexity increases rapidly, often exceeding the capacity for manual analysis.

A notable claim is that for certain moduli problems (e.g., genus 9 curves via symplectic Grassmannians), the number of unstable and non-stable components is in the hundreds, highlighting the necessity of computational tools for any explicit classification.

Comparison with Prior Work

CompGIT distinguishes itself from earlier computational approaches (e.g., A'Campo and Popov's implementations in pari-gp and LiE) by:

• Focusing on projective (rather than affine) GIT quotients.
• Providing explicit descriptions of non-stable and strictly polystable loci, not just the nullcone stratification.
• Leveraging the accessibility and extensibility of SageMath.
• Supporting exceptional groups and their representations.

The package's design facilitates portability and potential integration with other computer algebra systems, although further work is needed to enable direct comparison of outputs across platforms.

Practical and Theoretical Implications

CompGIT enables explicit, representation-theoretic analysis of GIT quotients in a range of geometric contexts, including:

• Moduli of hypersurfaces and complete intersections in projective space.
• Moduli of curves and higher-dimensional varieties via homogeneous varieties (e.g., Grassmannians, orthogonal and symplectic Grassmannians).
• Problems involving exceptional groups, which are otherwise intractable by hand.

The package's outputs can be directly interpreted in terms of geometric properties (e.g., singularity types, reducibility), facilitating the paper of stability conditions in moduli theory and the construction of compactifications (e.g., KSBA compactifications for surfaces of general type).

From a theoretical perspective, the work underscores the combinatorial explosion inherent in high-dimensional GIT problems and the limitations of current computational resources. The authors note that for certain representations (e.g., half-spin representations of $\mathrm{Spin}_{10}$), the number of families to analyze is on the order of $10^6$, rendering manual analysis infeasible.

Limitations and Future Directions

The current implementation is optimized for simple groups and irreducible representations, with partial support for more general reductive groups. The main computational bottleneck is the enumeration and analysis of one-parameter subgroups, particularly in the computation of strictly polystable loci, where convex hull computations become prohibitive.

Potential future developments include:

• Parallelization: Exploiting the independence of computations across one-parameter subgroups to leverage multicore architectures.
• Generalization to semisimple and non-reductive groups: Extending the algorithms to broader classes of group actions, with appropriate modifications to the handling of Weyl groups and representation data.
• Integration with other schemes: Adapting the package to handle more general $G$-schemes beyond projective space, where equivariant embeddings are available.
• Non-reductive GIT: Exploring algorithmic approaches to the nascent theory of non-reductive GIT, potentially enabling new classes of moduli problems to be addressed computationally.

Implications for Future AI and Computational Algebra

The CompGIT package exemplifies the increasing role of computational tools in algebraic geometry and invariant theory, particularly as the complexity of moduli problems outpaces traditional analytic methods. The explicit, representation-theoretic approach adopted here is well-suited to integration with AI-driven symbolic computation and automated reasoning systems, which could further automate the interpretation of output in geometric terms (e.g., singularity classification, moduli stratification).

As computational resources and algorithms improve, packages like CompGIT will be essential for the explicit paper of moduli spaces, the verification of theoretical predictions, and the exploration of new phenomena in algebraic geometry and representation theory. The open-source, extensible nature of the package positions it as a foundation for future developments in computational invariant theory and its applications across mathematics and related fields.