Symmetry Groups and Orbit Closures: The Geometric Core of P vs NP — E8 Intelligence Research

FINDING: The core mathematical structure of P vs NP reduces to symmetry groups and representation theory — specifically, the permanent vs determinant problem embeds in GL(n) orbit closures, making the complexity question a geometric invariant theory problem. | MATH: Permanent (Perm) vs determinant (Det): Perm_n(X) = Σ_{σ∈S_n} Π_i x_{i,σ(i)}; Det_n(X) = Σ_{σ∈S_n} sgn(σ) Π_i x_{i,σ(i)}. GCT approach: show Perm_n ∉ closure of GL(n²)·[Det_m] orbit for m = poly(n). Key objects: GL(n²) acting on Sym^d(C^n ⊗ C^n), weight polytopes, Littlewood-Richardson coefficients, Kronecker coefficients c_{λ,μ,ν} = dim(Hom_{S_n}(V_λ ⊗ V_μ, V_ν)). | CONNECTION: Root system A_{n-1} (type A) governs GL(n) — its weight lattice is Z^n/⟨(1,...,1)⟩, with simple roots α_i = e_i − e_{i+1}. The permanent's symmetry group is S_n (Weyl group of A_{n-1}), while determinant's is S_n × S_n × C_2 — this symmetry gap is the geometric crux. The relevant ratios: the dimension of the ambient space for n×n matrices is n², and Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com