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