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

We prove that deciding the vanishing of the character of the symmetric group is C₌ P-complete. We use this hardness result to prove that the absolute value and also the square of the character are not contained in #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 PP-complete under many-one reductions, and hence PH-hard under Turing reductions.