README.md

metadata

license: cc-by-2.0
pretty_name: Characters of Irreducible Representations of Symmetric Groups

Dataset Card for Characters of Irreducible Representations of Symmetric Groups, $S_{n}S\_n$, for $n=18,20,22n\; =\; 18,\; 20,\; 22$

One way to understand the algebraic structure of permutations (symmetric groups, $S_{n}S\_n$ is through their representation theory [1], which converts algebraic questions into linear algebra questions that are often easier to solve. A representation of group $GG$ on vector space $VV$, is a map $\varphi :G\to GL(V)\backslash phi:G\; \backslash rightarrow\; GL(V)$ that converts elements of $gg$ to invertible matrices on vector space $VV$ which respect the compositional structure of the group. A basic result in representation theory says that all representations of a finite group can be decomposed into atomic building blocks called irreducible representations. Amazingly, irreducible representations are themselves uniquely determined by the value of the trace, $\text{Tr}(\varphi (g))\backslash text\{Tr\}(\backslash phi(g))$, where $gg$ ranges over subsets of $GG$ called conjugacy classes. These values are called characters.

The representation theory of symmetric groups has rich combinatorial interpretations. Both irreducible representations of $S_{n}S\_n$ and the conjugacy classes of $S_{n}S\_n$ are indexed by partitions of $nn$ and thus the characters of irreducible representations of $S_{n}S\_n$ are indexed by pairs of partitions of $nn$. For $\lambda ,\mu ⊢n\backslash lambda,\backslash mu\; \backslash vdash\; n$ we write ${\chi }_{\mu }^{\lambda }\backslash chi^\backslash lambda\_\backslash mu$ for the associated character. This combinatorial connection is not superficial, some of the most famous algorithms for computation of irreducible characters (e.g., the Murnaghan-Nakayama rule) are completely combinatorial in nature).

Dataset Details

Dataset Description

Since the conjugacy classes of the symmetric group $S_{n}S\_n$ are indexed by integer partitions of $nn$, characters are constant on conjugacy classes, and the irreducible representations of $S_{n}S\_n$ are also indexed by integer partitions of $nn$, the task is to use a pair of integer partitions of $nn$ to predict the character of the corresponding irreducible representation of the symmetric group.

Within each file, two integer partitions are provided followed by an integer corresponding to the character. For instance, the line

[3,1,1],[2,2,1],-2

says that the character ${\chi }_{2,2,1}^{3,1,1}=-2\backslash chi^\{3,1,1\}\_\{2,2,1\}\; =\; -2$.

The datasets can also be found here. Data loaders can be found here.

In all cases the characters are heavily concentrated around 0 with very long tails. This likely contributes to the difficulty of the task and could be overcome with some simple pre- and post-processing. We have not chosen to do this in our baselines.

Statistics for the three datasets, $n=18,20,22n\; =\; 18,20,22$

Characters of $S_{18}S\_\{18\}$

	Number of instances
Train	118,580
Test	29,645

Maximum character value 16,336,320, minimum character value -1,223,040.

Characters of $S_{20}S\_\{20\}$

	Size
Train	298,661
Test	74,819

Maximum character value 249,420,600, minimum character value -17,592,960.

Characters of $S_{22}S\_\{22\}$

	Size
Train	763,109
Test	190,726

Maximum character value 5,462,865,408, minimum character value -279,734,796.

Math question (solved): The Murnaghan–Nakayama rule is an example of an algorithm for calculating the character of an irreducible representation of the symmetric group using only elementary operations on the corresponding pair of partitions.

ML task: Train a model that can take two partitions of $nn$, $\lambda \backslash lambda$ and $\mu \backslash mu$, and predict the corresponding character ${\chi }_{\mu }^{\lambda }\backslash chi^\{\backslash lambda\}\_\{\backslash mu\}$.

If a successful model is trained, it would be interesting to understand whether the model has learned an existing algorithm or whether it has discovered something new.

Small model performance

We provide some basic baselines for this task framed as regression. Benchmarking details can be found in the associated paper.

Size	Linear regression	MLP	Transformer	Guessing training label mean
$n=18n=\; 18$	$1.5920\times {10}^{10}1.5920\; \backslash times\; 10^\{10\}$	$2.7447\times {10}^8\pm 8.8602\times {10}^{8}2.7447\; \backslash times\; 10^\{8\}\; \backslash pm\; 8.8602\; \backslash times\; 10^8$	$2.4913\times {10}^{10}\pm 1.4350\times {10}^{7}2.4913\; \backslash times\; 10^\{10\}\; \backslash pm\; 1.4350\; \backslash times\; 10^7$	$1.5920\times {10}^{10}1.5920\; \backslash times\; 10^\{10\}$
$n=20n=\; 20$	$4.2007\times {10}^{12}4.2007\; \backslash times\; 10^\{12\}$	$4.2254\times {10}^{11}\pm 5.1236\times {10}^{11}4.2254\; \backslash times\; 10^\{11\}\; \backslash pm\; 5.1236\; \backslash times\; 10^\{11\}$	$5.3897\times {10}^{12}\pm 3.6464\times {10}^{11}5.3897\; \backslash times\; 10^\{12\}\; \backslash pm\; 3.6464\; \backslash times\; 10^\{11\}$	$4.2007\times {10}^{12}4.2007\; \backslash times\; 10^\{12\}$
$n=22n=\; 22$	$8.0395\times {10}^{14}8.0395\; \backslash times\; 10^\{14\}$	$1.1192\times {10}^{14}\pm 4.9321\times {10}^{12}1.1192\; \backslash times\; 10^\{14\}\; \backslash pm\; 4.9321\; \backslash times\; 10^\{12\}$	$1.3797\times {10}^{14}\pm 6.2799\times {10}^{12}1.3797\; \backslash times\; 10^\{14\}\; \backslash pm\; 6.2799\; \backslash times\; 10^\{12\}$	$8.0395\times {10}^{14}8.0395\; \backslash times\; 10^\{14\}$

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

• Curated by: Henry Kvinge
• 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

Uses

This dataset was generated to study ML model's ability to calculate the characters of irreducible representations of the symmetric group.

Direct Use

There are a range of tasks that could be performed using this dataset. The one we consider here is regression of characters from the two integer partitions that index it. That is, given $\lambda \backslash lambda$ and $\mu \backslash mu$, predict the corresponding character ${\chi }_{\mu }^{\lambda }\backslash chi^\{\backslash lambda\}\_\{\backslash mu\}$.

Out-of-Scope Use

None.

Dataset Structure

Within each file, two integer partitions are provided followed by an integer corresponding to the character. For instance, the line

[3,1,1],[2,2,1],-2

says that the character ${\chi }_{2,2,1}^{3,1,1}=-2\backslash chi^\{3,1,1\}\_\{2,2,1\}\; =\; -2$.

Dataset Creation

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

Curation Rationale

The dataset was generated because the discovery of algorithms that calculate the characters of irreducible representations of the symmetric group was a breakthrough in algebraic combinatorics and thus it is interesting if ML systems can re-discover this result.

Who are the source data producers?

Henry Kvinge used SageMath to generate this dataset.

Bias, Risks, and Limitations

We only provide characters for $n=18,20,22n\; =\; 18,20,22$. These characters exist for any $n>0n\; >\; 0$. We are happy to generate additional datasets upon request.

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]

References

[1] Sagan, Bruce E. The symmetric group: representations, combinatorial algorithms, and symmetric functions. Vol. 203. Springer Science & Business Media, 2013.