Kronecker Coefficient Complexity Bounds Depend on Partition Shape Parameters — E8 Intelligence Research

FINDING: Kronecker coefficients \(g(\lambda,\mu,\nu)\) — central to Geometric Complexity Theory (GCT) — resist efficient computation; explicit complexity bounds depend on partition shape parameters, with hook-like cases (small second part \(M\)) yielding tractable regimes. | MATH: Kronecker coefficient \(g(\lambda,\mu,\nu) = \dim (S^\lambda \otimes S^\mu \otimes S^\nu)^{GL_n}\) (triple tensor product invariant multiplicity). Complexity bounds: \(\ell\) (number of parts), \(N\) (largest part), \(M\) (smallest second part). When \(M = O(1)\) (hook-like), computation is polynomial-time; general case suspected #P-hard. Plethysm coefficients \(p_\lambda(\mu)\) similarly hard. GCT uses orbit closure separation: \(\overline{GL_n \cdot [\det_n]} \not\subseteq \overline{GL_n \cdot [\text{perm}_n]}\) to prove \(\text{VP} \neq \text{VNP}\). | CONNECTION: The partitions \(\lambda,\mu,\nu\) index irreducible representations of \(GL_n\) — root systems of type \(A_{n-1}\). Hook-like partitions corres Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com