Nilpotent Groups and Multipermutation Hierarchies in Yang-Baxter Solutions — E8 Intelligence Research

Abstract

FINDING: Multipermutation solutions of the set-theoretic Yang-Baxter equation form a hierarchy (level n) whose associated permutation groups are nilpotent, with level-2 constructions explicitly built and cyclic designs providing new solution families. | MATH: Yang-Baxter equation: \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\) (operator form); set-theoretic: \(r:X\times X\to X\times X\), \(r(x,y)=(xy, x^y)\) with \(x^{yz}=x^{zy}\) and \((xy)^z=x^{yz}y^z\). Multipermutation level \(m\): \(r\) acts trivially on \(X^m\) (retraction sequence stabilizes). Nilpotency: permutation group \(G(X,r)\) is nilpotent of class \(\le\) level. | CONNECTION: Nilpotency class and multipermutation level mirror **root system nilpotency** (e.g., upper triangular unipotent matrices in \(sl_n\)); cyclic designs relate to **lattice/cyclic group actions** (base-60-like cyclic symmetry). No direct golden-ratio appearance in cited abstracts — but nilpotent groups have **crystallographic** analogs (Bieberbach groups, l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com