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

The purpose of this note is to give an insertion scheme proof of the formula, $$p_\mu = \sum_{\lambda\vdash k} \chi^\lambda(\mu)s_\lambda,\formula$$ where $p_\mu$ is the power sum symmetric function, $s_\lambda$ is the Schur function and $\chi^\lambda(\mu)$ is the irreducible character of the symmetric group $S_k$ indexed by the partition $\lambda$ and evaluated at a permutation of cycle type $\mu=(\mu_1,\ldots,\mu_\ell)$. The proof of this formula is by direct application of the Robinson-Schensted-Knuth insertion scheme and a recent formula of Roichman.