README.md

---
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\\), for \\(n = 18, 20, 22\\)

One way to understand the algebraic structure of permutations (symmetric groups, \\(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 \\(G\\) on vector 
space \\(V\\), is a map \\(\phi:G \rightarrow GL(V)\\) that converts elements of \\(g\\) to invertible 
matrices on vector space \\(V\\) 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}(\phi(g))\\), 
where \\(g\\) ranges over subsets of \\(G\\) called conjugacy classes. These values are called *characters*. 

The representation theory of symmetric groups has rich combinatorial interpretations. Both 
irreducible representations of \\(S_n\\) and the conjugacy classes of \\(S_n\\) are indexed by 
partitions of \\(n\\) and thus the characters of irreducible representations of \\(S_n\\) are indexed
by pairs of partitions of \\(n\\). For \\(\lambda,\mu \vdash n\\) we write \\(\chi^\lambda_\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](https://en.wikipedia.org/wiki/Murnaghan–Nakayama_rule)) are completely 
combinatorial in nature).

## Dataset Details

### Dataset Description

Since the conjugacy classes of the symmetric group \\(S_n\\) are indexed by integer 
partitions of \\(n\\), characters are constant on conjugacy classes, and the irreducible 
representations of \\(S_n\\) are also indexed by integer partitions of \\(n\\), the task is 
to use a pair of integer partitions of \\(n\\) 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^{3,1,1}_{2,2,1} = −2\\). 

The datasets can also be found [here](https://drive.google.com/file/d/15AHAn9NnC7crzG_8BnaH3pp1aOGUUniV/view?usp=sharing).
Data loaders can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/symmetric_group_character).

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 = 18,20,22\\)

**Characters of \\(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}\\)**
|  | Size | 
|----------|----------|
| Train | 298,661 |
| Test  | 74,819 |

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

**Characters of \\(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](https://en.wikipedia.org/wiki/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 \\(n\\), \\(\lambda\\) and \\(\mu\\), 
and predict the corresponding character \\(\chi^{\lambda}_{\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= 18\\) | \\(1.5920 \times 10^{10}\\) | \\(2.7447 \times 10^{8} \pm 8.8602 \times 10^8\\) | \\(2.4913 \times 10^{10} \pm 1.4350 \times 10^7\\)| \\(1.5920 \times 10^{10}\\) |
| \\(n= 20\\) | \\(4.2007 \times 10^{12}\\) | \\(4.2254 \times 10^{11} \pm 5.1236 \times 10^{11}\\) | \\(5.3897 \times 10^{12} \pm 3.6464 \times 10^{11}\\) | \\(4.2007 \times 10^{12}\\) |
| \\(n= 22\\) | \\(8.0395 \times 10^{14}\\) | \\(1.1192 \times 10^{14} \pm 4.9321 \times 10^{12}\\) | \\(1.3797 \times 10^{14} \pm 6.2799 \times 10^{12}\\)| \\(8.0395 \times 10^{14}\\) |

The \\(\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](https://www.sagemath.org/). Data generation scripts can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/symmetric_group_character).

- **Repository:** [ACD Repo](https://github.com/pnnl/ML4AlgComb/tree/master/symmetric_group_character)
  
## 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\\) 
and \\(\mu\\), predict the corresponding character \\(\chi^{\lambda}_{\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^{3,1,1}_{2,2,1} = −2\\). 

## Dataset Creation

The dataset was generated using [SageMath](https://www.sagemath.org/). 
Data generation scripts can be found 
[here](https://github.com/pnnl/ML4AlgComb/tree/master/symmetric_group_character).

### 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](https://www.sagemath.org/) to generate this dataset.

## Bias, Risks, and Limitations

We only provide characters for \\(n = 18,20,22\\). These characters exist for any \\(n > 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.