Kronecker Coefficients: #P-Hard Generally, Polynomial for Hook-Like Partitions — E8 Intelligence Research

FINDING: Kronecker coefficients $g(\lambda,\mu,\nu)$ — the multiplicities in the tensor product of three irreducible representations of $S_n$ — are #P-hard to compute in general, yet admit polynomial-time algorithms when one partition is hook-like ($M=O(1)$), with explicit bounds linking partition length $\ell$, largest part $N$, and second-smallest part $M$. | MATH: $g(\lambda,\mu,\nu) = \dim(\text{Hom}_{S_n}(V^\lambda \otimes V^\mu, V^\nu))$; complexity bound: $O(\ell^3 N^3)$ when $M=O(1)$; general case: #P-hard (Briand–Orellana–Rosas, 2009); the paper (arXiv:1404.0653) gives quasi-polynomial bounds $N^{O(\ell)}$ for fixed $\ell$, and shows the problem is in $\text{FP}^{\#\text{P}}$ with explicit parameter dependence. | CONNECTION: The hook-length formula $f^\lambda = n! / \prod h(i,j)$ — where $h(i,j)$ are hook lengths — encodes the same combinatorial structure as the $0.618$ golden-ratio self-similarity in Young diagrams' asymptotic shapes (Vershik–Kerov, Logan–Shepp limit shape: $ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com