Positivity of the symmetric group characters is as hard as the polynomial time hierarchy
arxiv(2022)
摘要
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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