Covering Numbers of Some Irreducible Characters of the Symmetric Group

TL;DR

This paper calculates the covering numbers for specific irreducible characters of the symmetric group, focusing on characters associated with certain two-row and hook partitions, providing explicit values for these cases.

Contribution

It determines the covering numbers for irreducible characters of symmetric groups indexed by particular two-row and hook partitions, extending understanding of character coverings.

Findings

Covering numbers for characters indexed by (n-2,2) and ((n+1)/2, (n-1)/2) when n is odd.

Covering numbers for characters indexed by (n-2,1^2) and specific hook partitions.

Explicit values of covering numbers for these classes of irreducible characters.

Abstract

The covering number of a non-linear character $\chi$ of a finite group $G$ is the least positive integer $k$ such that every irreducible character of $G$ occurs in $\chi^k$. We determine the covering numbers of irreducible characters of the symmetric group $S_n$ indexed by certain two-row partitions (and their conjugates), namely $(n-2,2)$ and $((n+1)/2, (n-1)/2)$ when $n$ is odd. We also determine the covering numbers of irreducible characters indexed by certain hook-partitions (and their conjugates), namely $(n-2,1^2)$, the almost self-conjugate hooks $(n/2+1, 1^{n/2-1})$ when $n$ is even, and the self-conjugate hooks $((n+1)/2, 1^{(n-1)/2})$ when $n$ is odd.