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 VP ws and VNP. Mulmuley and Sohoni [SIAM J Comput 2001] suggested to study a strengthened version of this conjecture over the 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 GLn2(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 approach to the permanent versus determinant problem via multiplicity obstructions as proposed by in [32].