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arXiv · 2207.05423

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

Christian Ikenmeyer·Igor Pak·Greta Panova
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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.

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Christian Ikenmeyer, Igor Pak, Greta Panova. 2022-07-12. Positivity of the symmetric group characters is as hard as the polynomial time hierarchy. https://doi.org/10.1093/imrn%2Frnad273

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