# On the Divisibility of Character Values of the Symmetric Group

## TL;DR

This paper proves that for large symmetric groups, the majority of irreducible characters evaluated at fixed cycle types are divisible by a given integer, revealing a deep asymptotic divisibility property.

## Contribution

It establishes that as the size of the symmetric group grows, the proportion of partitions with character values divisible by a fixed integer approaches one.

## Key findings

- Proportion of partitions with divisible character values tends to 1 as n increases.
- Provides asymptotic behavior of character divisibility in symmetric groups.
- Connects partition structure with divisibility properties of character values.

## Abstract

Fix a partition $\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let $\chi^\lambda_\mu$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type $(\mu_1,\dotsc,\mu_m,1^{n-k})$. We show that the proportion of partitions $\lambda$ of $n$ such that $\chi^\lambda_\mu$ is divisible by $d$ approaches $1$ as $n$ approaches infinity.