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

Understand

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.