No occurrence obstructions in geometric complexity theory

Abstract

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes V P w s \mathrm {VP}_{\mathrm {ws}} and V N P \mathrm {VNP} . In 2001 Mulmuley and Sohoni suggested studying a strengthened version of this conjecture over complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of G L n 2 ( C ) \mathrm {GL}_{n^2}(\mathbb {C}) , which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni in 2001.