Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH

TL;DR

This paper presents counter-examples to Mulmuley's strong saturation conjecture for Kronecker coefficients, demonstrating limitations in the approach to polynomial-time decision problems in Geometric Complexity Theory, and proves the #P-hardness of computing these coefficients.

Contribution

It provides the first counter-examples to the strong saturation conjecture and offers a short proof of the #P-hardness of computing Kronecker coefficients.

Findings

Counter-examples to strong saturation conjecture for Kronecker coefficients.

Proof of #P-hardness of computing Kronecker coefficients.

Connections established between Kronecker and reduced Kronecker coefficients.

Abstract

We provide counter-examples to Mulmuley's strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynomial time. We also provide a short proof of the #P-hardness of computing the Kronecker coefficients. Both results rely on the connections between the Kronecker coefficients and another family of structural constants in the representation theory of the symmetric groups: Murnaghan's reduced Kronecker coefficients. An appendix by Mulmuley introduces a relaxed form of the saturation hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.