Divisibility of character values of the symmetric group by prime powers

TL;DR

The paper resolves Miller's conjecture by showing that for large $n$ and any prime power $q$ within a logarithmic bound, almost every entry of the symmetric group character table $\chi^{\lambda}_{\mu}$ is divisible by $q$. The authors extend prime-divisibility methods to prime powers by developing a new MN-based symmetry mechanism: after repeatedly merging $p^r$ equal parts in the cycle type and applying MN, they obtain a strong congruence $\chi^{\lambda}_{\mu} \equiv 0 \pmod{p^r}$ for typical partitions, leveraging core-structure and abacus techniques. A key technical engine combines two ingredients: (i) precise congruences under part-merging for characters, and (ii) a robust analysis of partitions that remain $t$-cores for many large $t$ via abaci and border-strip removals, plus partition-generating-function bounds. The resulting bound $O(p(n)^2 \exp(-(\\log\log n)^2))$ for the nondivisible entries yields a density tending to 1 of divisibility by any fixed prime power, marking a substantial advance in the probabilistic understanding of $S_n$ character values with potential connections to core-partition phenomena and analytic partition theory.

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.

Figures (3)

• Figure 1: The Young diagram of $(6,5,3,1,1,1)$
• Figure 2: Hook-lengths for $(6,5,3,1,1,1)$
• Figure 3: The Young diagram of $(6,2,1,1,1,1)$

Theorems & Definitions (25)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Theorem 2.2
• Proposition 2.3
• Lemma 2.4
• proof : Deducing Theorem \ref{['thm:main']}
• proof : Proof of Lemma \ref{['lem:combiningparts']}
• proof : Proof of Lemma \ref{['lem:tcores']}
• Theorem 4.1: The Murnaghan--Nakayama rule