Positivity of the symmetric group characters is as hard as the polynomial time hierarchy

TL;DR

This paper establishes the computational complexity of symmetric group character problems, showing that deciding their positivity and vanishing is as hard as some of the most difficult problems in theoretical computer science, implying no simple combinatorial descriptions are likely.

Contribution

It proves the complexity of symmetric group character positivity and vanishing problems, linking them to the polynomial hierarchy and complexity classes like #P and PP.

Findings

Deciding vanishing of symmetric group characters is C_=P-complete.

The square of the character is not in #P unless the polynomial hierarchy collapses.

Deciding positivity of the character is PP-complete and PH-hard.

Abstract

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 the square of the character is 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.