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

FINDING: Computational complexity classes (P, NP, BPP, etc.) define fundamental limits on problem-solving, with P vs. NP as the central open question; geometric complexity theory (GCT) uses algebraic geometry and representation theory to attack lower bounds. MATH: - P ⊆ NP ⊆ PSPACE ⊆ EXP; BPP ⊆ P/poly (Adleman's theorem). - Key open: P ≠ NP? (no proven equation, but implies NP-complete problems require exponential time). - GCT uses plethysm coefficients and Littlewood–Richardson coefficients to model circuit lower bounds via symmetry groups (GL_n, S_n). - No specific numeric constants or ratios emerge directly. CONNECTION: - GCT explicitly links to root systems and representation theory of Lie groups (e.g., GL_n), which underlie crystallographic symmetries and lattice structures. - The complexity zoo's hierarchy mirrors nested symmetry groups (e.g., containment of classes analogous to subgroup chains). - No direct golden ratio or base-60 connections; however, the algebr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com