Degrees and rationality of characters in the principal block of $$A_n$$ A n

AbstractGiven two primes p and q, we study degrees and rationality of irreducible characters in the principal p-block of $${\mathfrak {S}}_n$$ S n and $${\mathfrak {A}}_n$$ A n , the symmetric and alternating groups. In particular, we show that such a block always admits an irreducible character of degree divisible by q. This extends and generalizes a recent result of Giannelli–Malle–Vallejo