Nikolaus Conference 2022

Let \(p\) and \(q\) be different primes. In 2022, Navarro, Rizo, and Schaeffer Fry proposed a conjecture about finite groups possessing non-trivial characters of degree not divisible by \(p\) and \(q\) which lie in both the principal \(p\)- and \(q\)-block of the group. They relate this property to the Sylow \(p\)- and \(q\)-structure of the group. The authors established a reduction strategy which let them prove the conjecture in the special case \(2 \in \{p, q\}\). In this talk, I give further evidence that this conjecture holds, by verifying it for the symmetric and alternating groups in full generality.