Algorithmically Distinguishing Irreducible Characters of the Symmetric Group

Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial $n$, constructs $\pi\in S_n$ such $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. $f = \chi_\lambda$ for some character $S_n$, but do not know $\lambda$, given only oracle access to $f$. determines using number queries $f$ $n$. Each query can be computed $n$ by someone who knows $\lambda$.