Hyperpfaffians and Geometric Complexity Theory

TL;DR

This paper investigates the hyperpfaffian polynomial, establishing its uniqueness as the smallest degree SL-invariant for higher order tensors and proving its VNP-completeness, thereby challenging existing conjectures in geometric complexity theory.

Contribution

It proves the hyperpfaffian's uniqueness as the minimal degree SL-invariant and establishes its VNP-completeness, disproving Mulmuley's conjecture.

Findings

Hyperpfaffian is the unique smallest degree SL-invariant.

Hyperpfaffian is VNP-complete.

Disproves Mulmuley's conjecture in geometric complexity theory.

Abstract

The hyperpfaffian polynomial was introduced by Barvinok in 1995 as a natural generalization of the well-known Pfaffian polynomial to higher order tensors. We prove that the hyperpfaffian is the unique smallest degree SL-invariant on the space of higher order tensors. We then study the hyperpfaffian's computational complexity and prove that it is VNP-complete. This disproves a conjecture of Mulmuley in geometric complexity theory about the computational complexity of invariant rings.