Reduced Kronecker coefficients and counter-examples to Mulmuleys strong saturation conjecture SH

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.

💡 Research Summary

The paper investigates two central conjectures in Geometric Complexity Theory (GCT) concerning Kronecker coefficients: the strong saturation hypothesis (Strong SH) and the strong positivity hypothesis (Strong PH2). Both conjectures were proposed by Mulmuley as part of a strategy to show that the decision problem ZeroKronecker (whether a Kronecker coefficient is zero) lies in P, thereby supporting the broader GCT program aimed at separating complexity classes.

The authors introduce reduced Kronecker coefficients, originally defined by Murnaghan (1938) and later named by Klyachko (2004). For three partitions α, β, γ, the sequence (g_{(n-|\gamma|,\gamma),(n-|\alpha|,\alpha),(n-|\beta|,\beta)}) stabilizes as n grows; the limiting value is denoted (\bar g_{\gamma,\alpha,\beta}). This stability allows the authors to relate ordinary Kronecker coefficients to their reduced counterparts and to study the stretching function (e g_{\lambda,\mu,\nu}(N)=g_{N\lambda,N\mu,N\nu}), which Mulmuley had shown to be a quasipolynomial in N.

The main result is an infinite family of explicit counter‑examples to Strong SH. Choose integers i>j>0 and k>2i+j and set \