Degrees and prime power order zeros of characters of symmetric and alternating groups

2025/04/03 by Eugenio Giannelli, Stacey Law, Giannelli, Eugenio +3
Mathematics · Computer Science · #Graph theory and applications #Matrix Theory and Algorithms #Finite Group Theory Research

paper · doi:10.48550/arxiv.2504.02686

Abstract

We show that the p-part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of p-power order. As a corollary we deduce that the set of zeros of prime power order controls the degree of such a character. The same problem is analysed for alternating groups, where we show that when p=2 this data can only be determined up to two possibilities. We prove analogous statements for the defect of the p-block containing the character and for the p-height of the character.

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• Character degrees in principal blocks for distinct primes
• Orders of elements and zeros and heights of characters in a finite group
• Character degrees of some p-groups
• On the irreducible character degrees of symmetric groups and their multiplicities
• On the p-defect of character degrees of finite groups of Lie type