Robinson-Schensted-Knuth insertion and characters of symmetric groups and Iwahori-Hecke algebras of type A

Abstract

The purpose of this note is to give an insertion scheme proof of the formula, $$p_μ= \sum_{λ\vdash k} χ^λ(μ)s_λ,\formula$$ where $p_μ$ is the power sum symmetric function, $s_λ$ is the Schur function and $χ^λ(μ)$ is the irreducible character of the symmetric group $S_k$ indexed by the partition $λ$ and evaluated at a permutation of cycle type $μ=(μ_1,\ldots,μ_\ell)$. The proof of this formula is by direct application of the Robinson-Schensted-Knuth insertion scheme and a recent formula of Roichman.