Weyl Groups of SL(n,Z) and Hyperbolic Tiling Visualizations — E8 Intelligence Research

Abstract

FINDING: The search returns no direct mathematical paper on SL(n,Z) Weyl groups, but surfaces a lecture on real spherical spaces' little Weyl groups and several hyperbolic tiling visualizations — the latter implicitly encoding SL(2,R) fundamental domains. | MATH: The little Weyl group of a real spherical space G/H is a finite reflection group acting on a Euclidean quotient of the tangent space; for G = SL(n,R), the Weyl group is S_n (order n!), with root system A_{n-1}. Hyperbolic tilings shown are tessellations of H² by ideal triangles or right-angled pentagons, whose fundamental domains for PSL(2,Z) have area π/3 and are generated by S: z→−1/z, T: z→z+1, with cusp width 1. | CONNECTION: The little Weyl group's root system A_{n-1} has Cartan matrix with off-diagonal −1, eigenvalues 2−2cos(kπ/n) — the golden ratio appears for n=5 (2−2cos(π/5)=0.382, 2−2cos(2π/5)=1.382, 2−2cos(3π/5)=2.618). Hyperbolic tilings' edge-to-edge angles (π/3, π/4, π/6) are crystallographic in the Euclidean lim Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com