About the Quiver Mutation Database

The Quiver Mutation Database (QMD) is an open research infrastructure for quivers, exchange matrices, and mutation-equivalence classes arising in cluster algebras, representation theory, and related areas of algebra and geometry.

Quiver mutation is a central operation in modern algebraic combinatorics, but its combinatorial complexity grows rapidly. Even for modest numbers of vertices, mutation-equivalence classes can be large, infinite, or difficult to explore systematically. As a result, much of the existing knowledge about mutation classes, invariants, and exceptional phenomena is scattered across papers, personal code, and ad hoc datasets.

QMD is designed to address this gap by providing a curated, citable, and reproducible database of quivers and mutation classes, together with rigorously defined invariants and transparent computational provenance. The project is inspired by successful community resources such as the L-functions and Modular Forms Database (LMFDB) and House of Graphs, and aims to bring similar infrastructure principles to the study of quiver mutation.

Design principles

FAIR: Findable, Accessible, Interoperable, Reusable
QMD is designed in accordance with the FAIR Data Principles. A database that is not findable, accessible to non-experts, integratable with existing mathematical infrastructure, or reusable is of little use to the mathematical community.

Stable identifiers
Each quiver is assigned a permanent identifier determined by a canonical representative up to relabeling. Mutation-equivalence classes are likewise identified by canonical representatives, with clear documentation of exploration limits when full enumeration is not possible.

Reproducibility and provenance
All computed data are accompanied by metadata describing how the values were obtained, including algorithmic assumptions, software versions, and references where applicable.

Forward-facing with room to grow
New invariants, frozen vertices, and skew-symmetrizable matrices are objects of interest that will be included. The database is built so that future expansion is a core design principle, not an afterthought.

Accessibility to a mathematical audience
The data should be easily filterable, searchable, viewable, and downloadable. The goal of the database is to promote data-driven research in the mathematical community, and that starts with a low-barrier-to-entry public-facing interface.

Data and provenance

QMD's current corpus is generated exhaustively, not imported: it is produced by a single reproducible pipeline rather than assembled from external files. As the database grows, curated objects from the literature and from systems such as SageMath will be incorporated, with their origin recorded alongside them.

What is included
The present release covers skew-symmetric integer quivers on 1–4 vertices whose exchange-matrix entries are bounded by |b_ij| ≤ 2 (the maximum edge weight).

How it is built
The pipeline runs three deterministic steps:

1. Seed enumeration. For each rank up to four, every skew-symmetric matrix with entries in {−2, …, 2} is enumerated, and one canonical representative is kept per quiver isomorphism class — the lexicographically minimal matrix over all vertex relabelings.
2. Mutation exploration. From each seed, the mutation class is explored by breadth-first search, applying quiver mutation until the class closes up or a mutation would push an entry past the bound.
3. Class assembly and identifiers. Orbits that share a common quiver are merged into a single mutation class, and every quiver and class is given a permanent identifier derived from a SHA-256 hash of its canonical form.

Assumptions and limits
These bounds matter when interpreting the data:

• Edge-weight bound. Enumeration and mutation are both capped at |b_ij| ≤ 2. A class is recorded as open when a mutation would exceed this bound, so “open” means the class escapes the current exploration window — this includes genuinely mutation-infinite classes as well as any that would only be captured at a higher bound. Classes marked finite type were fully enumerated within the bound and are complete.
• Matrix type. Only skew-symmetric integer matrices are included. Skew-symmetrizable matrices and frozen vertices are planned but not yet present.
• Rank. Only ranks 1–4 are currently generated; higher ranks are planned (see the Roadmap).
• Reproducibility. The pipeline is deterministic: canonical forms and identifiers depend only on the matrices themselves, so re-running it reproduces the same database exactly.

This follows QMD's soundness-first principle: every recorded value is either fully computed within stated bounds or explicitly flagged as partial.

Contact and contributions

QMD is intended as a community resource. Suggestions, corrections, and expressions of interest are welcome.

Blake Jackson
ICARM, Carnegie Mellon University
jackson@icarm.io