Symmetry, Young Diagrams, and Orbit Closures in Geometric Complexity Theory — E8 Intelligence Research

FINDING: Geometric Complexity Theory (GCT) uses Young diagrams, representation theory of GL_n/S_n, and orbit closures to attack P vs NP, revealing that complexity lower bounds are encoded in the algebraic geometry of symmetry groups. | MATH: Key objects: Young diagrams (partitions λ ⊢ n), Schur modules S_λ(V) for GL_n, Specht modules for S_n, orbit closures \(\overline{G \cdot v}\) in representation spaces, plethysm coefficients \(c_{\lambda,\mu}^\nu\) (from \(S_\nu(S_\mu(V))\)), and the permanent vs determinant problem via \(det_n \in \overline{GL_{n^2} \cdot perm_m}\). The central conjecture (Mulmuley–Sohoni) posits that if \(perm_m\) lies in the orbit closure of \(det_n\) for \(n = poly(m)\), then certain representation multiplicities (Kronecker coefficients) must satisfy specific inequalities — a purely combinatorial statement about Young diagrams. | CONNECTION: Young diagrams are partitions, whose hook-length formula gives dimensions of irreps: \( \dim S_\lambda = \prod_{(i,j)\in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com