$p$-vanishing conjugacy classes of symmetric groups

TL;DR

This paper classifies 2-vanishing and 3-vanishing conjugacy classes in symmetric groups, providing complete results for these primes and partial insights for larger primes, addressing a question posed by Navarro.

Contribution

It offers a complete classification of 2- and 3-vanishing conjugacy classes in symmetric groups and advances understanding for larger primes, partially answering Navarro's question.

Findings

Complete classification for p=2 and p=3.

Partial classification for p≥5.

Addresses Navarro's question for p=2 and p=3.

Abstract

For a prime $p$, we say that a conjugacy class of a finite group $G$ is $p$-vanishing if every irreducible character of $G$ of degree divisible by $p$ takes value 0 on that conjugacy class. In this paper we completely classify 2-vanishing and 3-vanishing conjugacy classes for the symmetric group and do some work in the classification of $p$-vanishing conjugacy classes of the symmetric group for $p\geq 5$. This answers a question by Navarro for $p=2$ and $p=3$ and partly answers it for $p\geq 5$.