Character degrees in $2$-blocks of $\mathfrak{S}_n$ and $\mathfrak{A}_n$

TL;DR

This paper proves that for large symmetric and alternating groups, each 2-block contains an irreducible character with degree divisible by an odd prime p, and classifies rational characters in most cases.

Contribution

It establishes the existence of degree-divisible characters in 2-blocks of symmetric and alternating groups and classifies rational characters in most blocks.

Findings

Every 2-block of $rak{S}_n$ and $rak{A}_n$ contains a character with degree divisible by p for large n.

Most 2-blocks of $rak{A}_n$ contain a rational valued character with degree divisible by p.

The results hold for sufficiently large n and odd prime p.

Abstract

Let $p$ be an odd prime. We show that for sufficiently large $n$, every $2$-block of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ contains an ordinary irreducible character of degree divisible by $p$. For almost all $2$-blocks of $\mathfrak{A}_n$, we classify whether it contains a rational valued ordinary irreducible character of degree divisible by $p$.