GCT: Algebraic Geometry and Representation Theory in the P vs. NP Problem — E8 Intelligence Research

FINDING: Geometric complexity theory (GCT) reframes P vs. NP as an invariant-theoretic problem, using algebraic geometry and representation theory to separate complexity classes via orbit closures and their defining polynomials. | MATH: Core objects are orbit closures \(\overline{G \cdot v}\) under \(G = GL_N(\mathbb{C})\) acting on a representation \(V\). Separation requires finding a polynomial \(P\) that vanishes on one orbit closure but not the other: \(P|_{\overline{G \cdot v_1}} = 0\), \(P|_{\overline{G \cdot v_2}} \neq 0\). Key tools: Schur-Weyl duality, plethysm coefficients \(c_{\lambda,\mu}^\nu\) (from \(S^\nu(S^\mu \mathbb{C}^n)\)), and the moment polytope \(\Delta_G(v)\) (via Kirwan surjectivity). The "permanent vs. determinant" problem reduces to showing \(\overline{GL_{n^2} \cdot \text{perm}_n} \not\subset \overline{GL_{n^2} \cdot \text{det}_m}\) for \(m = \text{poly}(n)\). | CONNECTION: The moment polytope and its inequalities are governed by the root system of \(GL_N\) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com