Geometric Complexity Theory: Lie Groups and Orbit Closures in P vs. NP — E8 Intelligence Research

Abstract

FINDING: Geometric Complexity Theory (GCT) uses Lie group representations and orbit closures to separate complexity classes (e.g., P vs. NP, determinant vs. permanent). | MATH: Permanent vs. determinant: \(\text{per}(X) \in \mathbb{C}[x_{ij}]\) degree \(n\); \(\det(X)\) degree \(n\). GCT reduces lower bounds to showing \(\overline{\text{GL}_{n^2} \cdot \text{per}_m}\) does not lie in \(\overline{\text{GL}_{n^2} \cdot \det_n}\) for \(m = n^{\omega(1)}\). Key invariants: multiplicities of irreducible representations (plethysm coefficients) in coordinate rings of orbit closures. | CONNECTION: Root systems of Lie groups (e.g., \(A_{n-1}\) for \(\text{SL}_n\)) define symmetries of orbit closures. Crystallographic root lattices (e.g., \(A_n, D_n\)) appear in representation multiplicities. No direct ratio (0.618, etc.) but structural harmony in Weyl group symmetries and Dynkin diagrams. | DEPTH: 8 — Foundational link between algebraic geometry, representation theory, and computational complex Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com