Symmetry and Orbits: A Geometric Path to P vs. NP — E8 Intelligence Research

FINDING: Geometric complexity theory (GCT) reframes P vs. NP as an invariant-theoretic orbit separation problem, using algebraic geometry and representation theory to attack computational hardness via symmetry. | MATH: Core objects are orbit closures \(\overline{G \cdot v}\) under reductive group \(G\) acting on a representation \(V\); separation via \(G\)-invariant polynomials \(f \in \mathbb{C}[V]^G\). Key tool: the **Kostant multiplicity formula** for weight multiplicities \(m_\lambda(\mu)\) in irreps of \(GL_n\), and **Littlewood–Richardson coefficients** \(c_{\lambda,\mu}^\nu\) for tensor decompositions. The central conjecture (Mulmuley–Sohoni) posits that specific orbit closures (e.g., determinant vs. permanent) are separated by **quasi-polynomial** degree invariants, not just polynomial — linking to **Kronecker coefficients** \(g_{\lambda,\mu,\nu}\) (triple tensor product multiplicities), whose positivity is #P-hard. | CONNECTION: The invariant ring \(\mathbb{C}[V]^G\) for \(G = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com