Kronecker Coefficients Cannot Yield Superpolynomial Permanent-Determinant Lower Bounds — E8 Intelligence Research

FINDING: The vanishing of rectangular Kronecker coefficients cannot yield superpolynomial lower bounds for permanent vs. determinant, but their positivity holds for a large regime — a structural negative result in GCT. | MATH: Let \(s_n = _n\) (n×n determinant) and \(perm_n\) (n×n permanent). GCT seeks to separate \(perm_n\) from \(s_n\) via representation theory: if \(perm_n\) lies in the orbit closure \({GLn^2 · s_n}\), then for every partition \(λ\), the plethysm coefficient \(cλ(n^2, _n)\) must be ≥ the Kronecker coefficient \(kλ((n^2), (n), (n))\) (or rectangular variants). The paper (arXiv:1512.03798) proves: for rectangular Kronecker coefficients \(kλ(μ, ν, ρ)\) with \(μ = (a^m)\), \(ν = (b^m)\), \(ρ = (c^m)\) (rectangular partitions), if \(a+b+c\) is large relative to \(m\), then \(kλ > 0\) for all \(λ\) in a certain range — and crucially, the *vanishing* of these coefficients cannot occur in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com