Semistandard Young tableau
list | Cluster variable?
int64 |
|---|---|
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Dataset Card for Grassmannian cluster algebras and semistandard Young tableaux
The Grassmann manifold is the set of full-rank matrices up to equivalence of elementary row operations (equivalently the space whose points are -dimensional subspaces in ). Grassmannians are of fundamental geometric importance and are a central tool in a model of quantum field theory known as supersymmetric Yang-Mills theory.
Among the many algebraic-combinatorial properties of Grassmannians is an algebraic structure on its coordinate ring making it into an algebraic object called a cluster algebra. A recent result of Chang, Duan, Fraser, and Li [1] parameterize cluster variables of the Grassmannian coordinate ring in terms of equivalence classes of semistandard Young tableaux (SSYT). Not every SSYT indexes a cluster variable and a natural question to ask is whether there is a naive combinatorial rule that specifies whether a particular SSYT indexes a cluster variable. One necessary condition that is known is that the tableaux needs to be of rectangular shape.
The idea to apply machine learning to this problem was first posed by [2]. We follow their set up and use their code to generate positive examples (i.e., SSYT that index a cluster variable). However, we choose a different method of sampling tableaux that do not index cluster variables that we found makes the problem harder (and therefore forces the model to learn more robust features).
Dataset Details
In the cluster algebra associated with Grassmann manifold , each cluster variable is indexed by a rectangular SSYT with rows with entries drawn from (because this is a SSYT, the entries need to weakly increase as one moves left to right in the rows and strictly increase as one moves down each column). The rank of these rectangular SSYT (and the rank of the associated cluster algebras) is given by their number of columns. In this dataset we focus on and hence look at rectangular SSYT with 3 rows filled with entries drawn from . We further restrict to rank 4 SSYT. This leaves us with a collection of arrays whose entries increase weakly across rows and strictly down columns.
The dataset consists of a collection of rectangular SSYT each with a label indicating whether
it indexes a cluster variable or not. Those that do not index a cluster variable are labeled
with a 0 and those that do are labeled with a 1. Tableaux have shape , and
are filled with values drawn from .
We use braces and to demarcate rows of a tableaux, so that
[[2, 3, 4, 7], [3, 5, 6, 8], [4, 9, 11, 12]]
is the tableaux whose first row of 4 elements contains 2, 3, 4, and 7.
Data loaders can be found here.
Statistics
| SSYT indexes a cluster variable | SSYT does not index a cluster variable | Total number of instances | |
|---|---|---|---|
| Train | 74,329 | 74,329 | 148,658 |
| Test | 18,582 | 18,582 | 37,164 |
Data generation
The positive examples sampled for this dataset were generated using code from the paper
[2]. This code can be found at
https://github.com/edhirst/GrassmanniansML.
We generated our own negative examples which we believe are harder for the model to learn
and hence result in more robust features. To sample random rectangular SSYT, we took
advantage of the
random_element() method in the Tableaux class in Sage.
Task
Math question: The ultimate goal is find a concise characterization of those SSYT that index cluster variables.
New necessary or sufficient conditions are also of value.
ML task: Train a model that can predict whether a SSYT indexes a cluster variable or not.
Extract mathematical insights from a performant model.
Small model performance
| Architecture | Accuracy |
|---|---|
| Logistic regression | 65.7% |
| MLP | 99.3% 0.1% |
| Transformer | 99.5% 0.1% |
| Guessing '0' or '1' | 50% |
The signs indicate 95% confidence intervals from random weight initialization and training.
- Curated by: Herman Chau
- Funded by: Pacific Northwest National Laboratory
- Language(s) (NLP): NA
- License: CC-by-2.0
Dataset Sources
Data generation scripts for positive examples can be found here. Data generation scripts for negative examples 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, acdbenchdataset@gmail.com
References
[1] Chang, Wen, et al. "Quantum affine algebras and Grassmannians." Mathematische Zeitschrift 296.3 (2020): 1539-1583.
[2] Cheung, Man-Wai, et al. "Clustering cluster algebras with clusters." arXiv preprint arXiv:2212.09771 (2022).
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