The QuiverTools package for SageMath and Julia (2506.19432v1)

Abstract: We introduce QuiverTools, a new software package, available in both a SageMath and Julia version, to study quivers and their moduli spaces of representations. Its key features are the computation of general subdimension vectors, leading to canonical decompositions, and checking the existence of (semi)stable representations, as well as the enumeration of Harder-Narasimhan types and related calculations for Teleman quantization. Computations related to intersection theory on quiver moduli are also implemented.

• The paper introduces QuiverTools, a software package in SageMath and Julia enabling explicit algorithmic computations in quiver representation theory and moduli geometry.
• The package provides algorithms for fundamental tasks such as computing canonical decompositions, checking stability, and calculating geometric invariants for quiver moduli.
• Availability in SageMath and Julia enables computational verification of theoretical results and facilitates experimental mathematics in quiver representation theory.

The QuiverTools Package for SageMath and Julia: Algorithmic Approaches to Quiver Moduli

The paper presents QuiverTools, a software package implemented for both SageMath and Julia, designed to facilitate explicit computations in the representation theory of quivers and the geometry of their moduli spaces. The package provides algorithmic tools for tasks such as computing general subdimension vectors, canonical decompositions, (semi)stability checks, enumeration of Harder–Narasimhan types, calculations relevant to Teleman quantization, and intersection theory on quiver moduli. The implementation leverages the combinatorial and linear-algebraic nature of quiver representation theory, making previously theoretical results accessible for computational experimentation and verification.

Core Features and Algorithms

The package centers on the explicit computation of invariants and structures associated with quivers:

• General Subdimension Vectors and Canonical Decomposition:

QuiverTools implements Schofield’s recursive characterization of general subdimension vectors, allowing the determination of which subrepresentations appear generically in a given dimension vector. The canonical decomposition, following Kac’s theorem, is computed by recursively decomposing dimension vectors into Schur roots, using the vanishing of general extension groups as a criterion. This enables the identification of indecomposable summands in generic representations.

• (Semi)Stability and Moduli Space Non-emptiness:

The package provides algorithms to check the existence of (semi)stable representations for a given stability parameter $\theta$, by verifying the relevant inequalities for all general subdimension vectors. This is essential for determining the non-emptiness of moduli spaces of (semi)stable representations, a central question in geometric invariant theory (GIT) for quivers.

• Enumeration of Harder–Narasimhan Types:

By leveraging the effective description of Harder–Narasimhan filtrations in the quiver context, QuiverTools enumerates all possible types, which correspond to the Hesselink strata in the unstable locus of the moduli problem. This enumeration is crucial for stratification-based computations, such as those required for cohomological and intersection-theoretic invariants.

• Teleman Quantization and Cohomology Vanishing:

The package implements the computation of the weights and bounds required for the application of Teleman’s quantization theorem in the context of quiver moduli. This enables the verification of vanishing results for higher cohomology of equivariant sheaves, and the explicit calculation of the relevant weights for universal bundles and canonical line bundles.

• Intersection Theory and Chow Rings:

For smooth projective quiver moduli admitting universal families, QuiverTools computes presentations of the Chow ring, Todd class, and point class, enabling the calculation of Euler characteristics, Hilbert series, and degrees of line bundles. These computations are grounded in the explicit generators and relations for the Chow ring in terms of the Chern classes of the universal bundle summands.

Numerical Results and Implementation Examples

The paper provides concrete computational examples using the 4-Kronecker quiver with dimension vector $(2,3)$, demonstrating the package’s capabilities in both SageMath and Julia. Notable numerical results include:

• Canonical Decomposition:

For the 4-Kronecker quiver and $(2,3)$, the canonical decomposition is trivial, indicating the generic representation is indecomposable.

• Stability Checks:

The existence of stable representations is shown to depend sensitively on the choice of stability parameter, with the canonical parameter yielding a non-empty moduli space.

• Geometric Invariants:

For the same example, the moduli space has dimension 12, Picard rank 1, and index 4.

• Harder–Narasimhan Types and Betti Numbers:

The enumeration of Harder–Narasimhan types leads to the explicit computation of Betti numbers for the moduli space, e.g., [1, 0, 1, 0, 3, 0, 4, 0, 7, 0, 8, 0, 10, 0, 8, 0, 7, 0, 4, 0, 3, 0, 1, 0, 1].

• Teleman Bounds and Weights:

The package computes the Teleman bounds $\eta_{hn}$ for each Harder–Narasimhan type, as well as the weights of universal bundles and the canonical bundle, facilitating the application of vanishing theorems.

• Intersection Numbers:

The degree of the anticanonical bundle raised to the dimension of the moduli space is computed as $1996824248320$ for the running example.

Practical and Theoretical Implications

The availability of QuiverTools in both SageMath and Julia significantly lowers the barrier for researchers to perform explicit computations in quiver representation theory and moduli geometry. This has several implications:

• Algorithmic Verification of Theoretical Results:

The package enables the direct verification of results concerning canonical decompositions, stability, and cohomological vanishing, which were previously accessible only through intricate theoretical arguments.

• Facilitation of Experimental Mathematics:

Researchers can now explore conjectures and phenomena in quiver moduli by computational experimentation, potentially leading to new insights or counterexamples.

• Integration with Broader Mathematical Software Ecosystems:

The dual implementation in SageMath and Julia allows integration with existing algebraic and geometric computation frameworks, broadening the user base and applicability.

• Support for Intersection-Theoretic and Cohomological Studies:

The explicit computation of Chow rings, Todd classes, and Betti numbers supports research in enumerative geometry and the paper of derived categories of quiver moduli.

Future Directions

The modular and extensible design of QuiverTools suggests several avenues for further development:

• Extension to Quivers with Potential and Enumerative Invariants:

While the current package does not address quivers with potential or enumerative invariants, integration with packages such as CoulombHiggs and msinvar could provide a more comprehensive computational environment.

• Optimization and Parallelization:

As the complexity of quiver moduli problems grows rapidly with the size of the quiver and dimension vectors, further optimization and parallelization of algorithms will be essential for scaling to larger examples.

• Bridging to Derived and Noncommutative Geometry:

The explicit computations enabled by QuiverTools may facilitate new research at the interface of quiver moduli, derived categories, and noncommutative algebraic geometry.

• Educational and Expository Applications:

The package’s explicit and accessible interface makes it a valuable tool for teaching and expository work in modern algebraic geometry and representation theory.

Conclusion

QuiverTools represents a significant step in the algorithmic paper of quiver representations and their moduli spaces, providing a robust computational foundation for both theoretical research and practical applications. Its implementation of key algorithms from the literature, combined with explicit numerical capabilities, positions it as a valuable resource for the mathematical community engaged in the paper of quivers, moduli, and related geometric structures.