Labelling the character tables of symmetric and alternating groups

Abstract

Let $X$ be a character table of the symmetric group $S_n$. It is shown that unless $n = 4$ or $n=6$, there is a unique way to assign partitions of $n$ to the rows and columns of $X$ so that for all $λ$ and $ν$, $X_{λν}$ is equal to $χ^λ(ν)$, the value of the irreducible character of $S_n$ labelled by $λ$ on elements of cycle type $ν$. Analogous results are proved for alternating groups, and for the Brauer character tables of symmetric and alternating groups.