README.md

---
license: cc-by-2.0
pretty_name: Characters of Irreducible Representations of Symmetric Groups, n = 22
---

# Characters of Irreducible Representations of the Symmetric Group, \\(S_{22}\\)

One way to understand the algebraic structure of the set of permutations of \\(n\\) elements
(the *symmetric group*, \\(S_n\\)) is 
through its 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

Each instance of the dataset consists of two integer partitions of \\(22\\) (one 
corresponding to the irreducible
representation and one corresponding to the conjugacy class) and the
corresponding character (which is always an integer). For a small \\(n = 5\\) example, 
if the first partition is `[3,1,1]`, the second partition is `[2,2,1]`, and the character
is `-2`, then this says that the character \\(\chi^{3,1,1}_{2,2,1} = −2\\). 

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.

**Characters of \\(S_{22}\\)**
|  | Number of instances | 
|----------|----------|
| 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 one 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 22, \\(\lambda\\) and \\(\mu\\), 
and predict the corresponding character \\(\chi^{\lambda}_{\mu}\\). Identify the decision
process that a performant model is running. Is this a known algorithm or a new algorithm?

## 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= 22\\) | \\(8.0395\times 10^{14}\\) | \\(1.1192\times10^{14} \pm 4.9321\times10^{12}\\) | \\(1.3797\times10^{14} \pm 6.2799\times10^{12}\\)| \\(8.0395\times10^{14}\\) |

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

## Further information

- **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)
  
## 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

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