Character degrees in principal blocks for distinct primes

TL;DR

This paper proves that certain finite groups, including symmetric, alternating, and classical simple groups, have irreducible characters of degrees not divisible by two distinct primes, lying in both their principal blocks, extending previous results.

Contribution

It extends earlier results by Navarro-Rizo-Schaeffer Fry, completing the proof of a conjecture for symmetric, alternating, and classical groups regarding character degrees in principal blocks.

Findings

Existence of non-trivial irreducible characters with degrees prime to both p and q in specified groups.

Extension of previous results to new classes of groups, including classical simple groups.

Completion of a conjecture case for symmetric and alternating groups.

Abstract

Let $G$ be a finite group of order divisible by two distinct primes $p$ and $q$. We show that $G$ possesses a non-trivial irreducible character of degree not divisible by $p$ nor $q$ lying in both the principal $p$- and $q$-block whenever $G$ is one of the following: an alternating group $\mathfrak{A}_n$, $n\geq 4$, a symmetric group $\mathfrak{S}_n$, $n\geq 3$, or a finite simple classical group of type A, B, or C, defined in characteristic distinct from $p$ and $q$. This extends earlier results of Navarro-Rizo-Schaeffer Fry for $2\in\{p,q\}$, and in particular completes the proof of an instance of a conjecture of the same authors, e.g., in the case of symmetric and alternating groups.