Degrees and prime power order zeros of characters of symmetric and alternating groups
Eugenio Giannelli, Stacey Law, Eoghan McDowell
TL;DR
This work resolves Navarro's conjecture for symmetric groups by showing that the $p$-part of an irreducible character degree $\chi(1)$ is fully determined by the set of prime-power zeros $\operatorname{Van}_{\mathrm{pow}}(\chi)$ (and hence by zeros on a Sylow $p$-subgroup). The authors develop a framework based on partitions, hook lengths, cores/quotients, and the Murnaghan–Nakayama rule to extract the $p$-part, block defect, and $p$-height from the vanishing data, with explicit formulas linking $\nu_p(\chi(1))$ to the $p$-power weights $\mathbf{w}_{p^i}(\lambda)$. For alternating groups, odd $p$ yields essentially the same conclusions, while $p=2$ yields two possible outcomes and several non-uniqueness phenomena, demonstrated through explicit constructions. A key conceptual advance is the reduction to defect groups, showing that the meaningful vanishing information concentrates on defect-containing subgroups, enabling blockwise comparisons. The results illuminate the extent to which zeros control representation-theoretic statistics and expose clear boundaries where the vanishing data cannot distinguish certain characters, particularly in the $A_n$ case at $p=2$.
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.