A bijection proving orthogonality of the characters of \(S_ n\) (Q790811)

The author gives a combinatorial proof of the orthogonality relation \(\sum \chi^{\lambda}\!_{\rho}\chi^{\lambda}\!_{\beta}=\delta_{\rho \beta}1^{j_ 1}j_ 1!2^{j_ 2}j_ 2!...,\) where \(\chi^{\lambda}\!_{\rho}\) is the irreducible character \(\lambda\) of the symmetric group \(S_ n\) evaluated at an element of type \(\rho =1^{j_ 1}2^{j_ 2}....\) The summation is over all partitions \(\lambda\) of n. The argument generalizes the Schensted correspondence between pairs of standard tableaux and permutations.