Given two primes p and q, we study degrees and rationality of irreducible characters in the principal p-block of $${\mathfrak {S}}_n$$ and $${\mathfrak {A}}_n$$ , 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.

中文翻译：

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$$A_n$$主块字符的度数和合理性

给定两个素数 p 和 q，我们研究 $${\mathfrak {S}}_n$$ 和 $${\mathfrak {A}}_n$$ 的主 p 块中不可约字符的度和合理性，对称和交替组。特别地，我们证明了这样的块总是允许被 q 整除的不可约性质。这扩展并概括了 Giannelli-Malle-Vallejo 的最新结果。