Covering numbers for characters of symmetric groups

TL;DR

This paper investigates the covering numbers of characters in symmetric groups, establishing conditions under which the set of irreducible constituents covers all irreducible characters for certain powers.

Contribution

It provides a characterization of when the irreducible constituents of character powers cover the entire set of irreducible characters in symmetric groups.

Findings

For $n>4$, the set of irreducible constituents of $ heta^k$ equals all irreducible characters iff $k \\geq n-1$.

The result applies to nonlinear irreducible characters of symmetric groups.

It advances understanding of the structure of character powers in symmetric groups.

Abstract

If $n>4$ and $c(\theta)$ denotes the set of irreducible constituents of a character $\theta$, then $c(\chi^k)={\rm Irr}(S_n)$ for all nonlinear $\chi\in {\rm Irr}(S_n)$ if and only if $k\geq n-1$.