The symmetrized determinant is #P-hard over polynomial-sized algebras and its polynomial family is VNP-complete in both non-commutative and commutative matrix algebra settings.
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On the Principal Minor Expansion and Complexity of the Symmetrized Determinant
The symmetrized determinant is #P-hard over polynomial-sized algebras and its polynomial family is VNP-complete in both non-commutative and commutative matrix algebra settings.