No occurrence obstructions in geometric complexity theory

TL;DR

This paper proves that the method of using occurrence obstructions to separate orbit closures in geometric complexity theory is impossible, but it leaves open the potential of using multiplicity obstructions for the permanent versus determinant problem.

Contribution

It demonstrates the impossibility of using occurrence obstructions in geometric complexity theory to resolve the permanent versus determinant conjecture.

Findings

Occurrence obstructions cannot separate orbit closures.

The approach using occurrence obstructions is fundamentally limited.

Multiplicity obstructions remain a viable alternative.

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 GL_{n^2}(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.