README.md

metadata

license: cc-by-2.0
pretty_name: Mutation equivalence of quivers

Mutation Equivalence of Quivers

Quivers and quiver mutations are central to the combinatorial study of cluster algebras, algebraic structures with connections to Poisson Geometry, string theory, and Teichmuller theory. Quivers are directed (multi)graphs, and a quiver mutation is a local transformation centered at a chosen node of the graph that involves adding, deleting, and reversing the orientation of specific edges based on a set of combinatorial rules. A fundamental open problem in this area is finding an algorithm that determines whether two quivers are mutation equivalent (one can traverse from one quiver to another by applying mutations). Currently, such algorithms only exist for special cases, including types $AA$ [1], $DD$ [2], and $\tilde{B}\backslash tilde\{B\}$, $\tilde{C}\backslash tilde\{C\}$, and $\tilde{D}\backslash tilde\{D\}$ [3]. The $\tilde{B}\backslash tilde\{B\}$ and $\tilde{C}\backslash tilde\{C\}$ types correspond to the classes $BDBD$ and $BBBB$ in our dataset, respectively. Consistent with Sage we use the naive notation, which specifies a quiver by indicating the two ends of the diagram, which are joined by a path [7]. To our knowledge, the remaining classes in this dataset ( $EE$, $DEDE$, $BEBE$) lack characterizations.

Recent work has explored whether deep learning models can learn to correctly predict if two quivers are mutation equivalent [4]. [5] utilized an alternative version of this dataset to re-discover known characterization theorems. The dataset consists of adjacency matrices for quivers drawn from 7 different mutation equivalence classes ( $AA$, $DD$, $EE$, $DEDE$, $BEBE$, $BDBD$, and $BBBB$).

Dataset

The task associated with this dataset involves identifying whether two quivers are mutation equivalent. Thus, the inputs are quivers (directed multigraphs). We chose to use examples with $1111$ nodes (though one could reasonably have chosen another number). They are encoded by their $11\times 1111\; \backslash times\; 11$ adjacency matrices and the labels are one of $77$ different equivalence classes: $A_{11},B{B}_{11},B{D}_{11},B{E}_{11},D_{11},D{E}_{11},E_{11}A\_\{11\},BB\_\{11\},BD\_\{11\},BE\_\{11\},D\_\{11\},DE\_\{11\},E\_\{11\}$. For the quiver mutation classes that are not mutation finite (that is, the mutation equivalence class has an infinite number of elements), the datasets contain quivers generated up to a certain depth, which is the distance from the original quiver, measured by number of mutations. The depths for those classes which are infinite are listed below and were chosen to balance the sizes of different classes.

Mutation equivalance class	Sampling depth
$B{B}_{11}BB\_\{11\}$	10
$B{D}_{11}BD\_\{11\}$	9
$B{E}_{11}BE\_\{11\}$	8
$D{E}_{11}DE\_\{11\}$	9
$E_{11}E\_\{11\}$	9

Dataset statistics are as follows:

	$A_{11}A\_\{11\}$	$B{B}_{11}BB\_\{11\}$	$B{D}_{11}BD\_\{11\}$	$B{E}_{11}BE\_\{11\}$	$D_{11}D\_\{11\}$	$D{E}_{11}DE\_\{11\}$	$E_{11}E\_\{11\}$	Total
Training	11,940	27,410	23,651	22,615	25,653	23,528	28,998	163,795
Test	2,984	6,852	5,912	5,653	6,413	5,881	7,249	40,944

Data generation

All mutation classes were generated using Sage [6], and the script can be found here.

Task

Math question: Find simple rules to determine whether or not a quiver belongs to a specific mutation equivalence class (out of classes $A_{11},B{B}_{11},B{D}_{11},B{E}_{11},D_{11},D{E}_{11},E_{11}A\_\{11\},BB\_\{11\},BD\_\{11\},BE\_\{11\},\; D\_\{11\},DE\_\{11\},E\_\{11\}$). Note that rules for $A_{11}A\_\{11\}$ and $D_{11}D\_\{11\}$ are known.

ML task: Train a model that can predict a quiver's mutation equivalence class out of the 7 options above.

See the work [5] for an example of how a model trained on a variant of this dataset was used to re-discover known theorems.

Small model performance

	Accuracy
Logistic regression	$40.3\mathrm{\%}40.3\backslash \%$
MLP	$86.5\mathrm{\%}\pm 1.9\mathrm{\%}86.5\backslash \%\; \backslash pm\; 1.9\backslash \%$
Transformer	$92.9\mathrm{\%}\pm 0.5\mathrm{\%}92.9\backslash \%\; \backslash pm\; 0.5\backslash \%$
Guessing largest class	$17.7\mathrm{\%}17.7\backslash \%$

The $\pm \backslash pm$ signs indicate 95% confidence intervals from random weight initialization and training.

Further information

• Curated by: Helen Jenne
• Funded by: Pacific Northwest National Laboratory
• Language(s) (NLP): NA
• License: CC-by-2.0

Dataset Sources

The dataset was generated using SageMath. Data generation scripts can be found here.

• Repository: ACD Repo

Citation

BibTeX:

@article{chau2025machine,
    title={Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics},
    author={Chau, Herman and Jenne, Helen and Brown, Davis and He, Jesse and Raugas, Mark and Billey, Sara and Kvinge, Henry},
    journal={arXiv preprint arXiv:2503.06366},
    year={2025}
}

APA:

Chau, H., Jenne, H., Brown, D., He, J., Raugas, M., Billey, S., & Kvinge, H. (2025). Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics. arXiv preprint arXiv:2503.06366.

Dataset Card Contact

Henry Kvinge, [email protected]

[1] Buan, Aslak Bakke, and Dagfinn F. Vatne. "Derived equivalence classification for cluster-tilted algebras of type $A_n$." Journal of Algebra 319.7 (2008): 2723-2738.
[2] Vatne, Dagfinn F. "The mutation class of $D_n$ quivers." Communications in Algebra 38.3 (2010): 1137-1146.
[3] Henrich, Thilo. "Mutation classes of diagrams via infinite graphs." Mathematische Nachrichten 284.17‐18 (2011): 2184-2205.
[4] Bao, Jiakang, et al. "Machine learning algebraic geometry for physics." arXiv preprint arXiv:2204.10334 (2022).
[5] He, Jesse, et al. "Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes." arXiv preprint arXiv:2411.07467 (2024).
[6] Stein, William. "Sage: Open source mathematical software." (2008). [7] Musiker, Gregg, and Christian Stump. "A compendium on the cluster algebra and quiver package in Sage." arXiv preprint arXiv:1102.4844 (2011).