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arXiv · 2504.02686

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

Eugenio Giannelli·Stacey Law·Eoghan McDowell
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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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Eugenio Giannelli, Stacey Law, Eoghan McDowell. 2025-04-03. Degrees and prime power order zeros of characters of symmetric and alternating groups. https://doi.org/10.1112/blms.70160

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