On conjugacy classes of $S_n$ containing all irreducibles

TL;DR

This paper characterizes certain conjugacy classes in symmetric groups that contain all irreducible representations, identifying specific partitions with distinct odd parts and at least two parts for large enough n.

Contribution

It provides a complete characterization of conjugacy classes in $S_n$ that contain all irreducible representations, especially for n=6 or n≥8.

Findings

Conjugacy classes with partitions having distinct odd parts contain all irreducibles for n=6 or n≥8.

A simple criterion for such classes is established for n≠4,8.

Partitions with at least two parts, all odd and distinct, index these classes.

Abstract

It is shown that for the conjugation action of the symmetric group $S_n,$ when $n=6$ or $n\geq 8,$ all $S_n$-irreducibles appear as constituents of a single conjugacy class, namely, one indexed by a partition $\lambda$ of $n$ with at least two parts, whose parts are all distinct and taken from the set of odd primes and 1. The following simple characterisation of conjugacy classes containing all irreducibles is proved: If $n\neq 4,8,$ the partition $\lambda$ of $n$ indexes a global conjugacy class for $S_n$ if and only if it has at least two parts, and all its parts are odd and distinct.