Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

Abstract

We prove that deciding the vanishing of the character of the symmetric group is $\textsf{C}_= \textsf{P}$-complete. We use this hardness result to prove that the absolute value and also the square of the character are not contained in $\textsf{#P}$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof, we conclude that deciding positivity of the character is $\textsf{PP}$-complete under many-one reductions, and hence $\textsf{PH}$-hard under Turing reductions.