Vanishing pairs of conjugacy classes for the symmetric group

TL;DR

This paper classifies pairs of conjugacy classes in the symmetric group where every non-linear irreducible character vanishes on at least one class, advancing understanding of character behavior in symmetric groups.

Contribution

It provides a complete classification of such conjugacy class pairs, a novel result in the representation theory of symmetric groups.

Findings

Identifies all pairs of conjugacy classes with the vanishing property.

Clarifies the relationship between conjugacy classes and irreducible characters.

Enhances understanding of character vanishing patterns in symmetric groups.

Abstract

In this short note, we classify pairs of conjugacy classes of the symmetric group such that any non-linear irreducible character of the symmetric group vanishes on at least one of them.