Divisibility of character values of the symmetric group by prime powers

TL;DR

This paper proves that for large symmetric groups, almost all character table entries are divisible by any specified prime power, extending previous prime divisibility results.

Contribution

It generalizes earlier prime divisibility results to prime powers, confirming a conjecture of Miller for large symmetric groups.

Findings

Almost all entries in the character table of S_n are divisible by any given prime power as n grows large.

Extends previous work on prime divisibility to prime power divisibility.

Supports Miller's conjecture for large symmetric groups.

Abstract

Proving a conjecture of Miller, we show that as $n$ tends to infinity almost all entries in the character table of $S_n$ are divisible by any given prime power. This extends our earlier work which treated divisibility by primes.