Permanent v. determinant: an exponential lower bound assumingsymmetry and a potential path towards Valiant's conjecture

TL;DR

This paper explores symmetric determinantal representations of the permanent, establishing optimality within certain symmetry constraints and linking this to Valiant's conjecture on permanent versus determinant complexity.

Contribution

It demonstrates the optimality of Grenet's representation under specific symmetries and connects symmetry-respecting representations to Valiant's conjecture.

Findings

Grenet's representation is optimal among symmetric determinantal representations

Symmetry constraints relate to the validity of Valiant's conjecture

Potential path towards proving Valiant's conjecture through symmetry analysis

Abstract

We initiate a study of determinantal representations with symmetry. We show that Grenet's determinantal representation for the permanent is optimal among determinantal representations respecting left multiplication by permutation and diagonal matrices (roughly half the symmetry group of the permanent). In particular, if any optimal determinantal representation of the permanent must be polynomially related to one with such symmetry, then Valiant's conjecture on permanent v. determinant is true.