The E 8 Geometry from a Clifford Perspective

Abstract

This paper considers the geometry of E8 from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system H3 gives rise to the largest (and therefore exceptional) non-crystallographic root system H4. Arnold’s trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and E8. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the 120 elements of the icosahedral group H3 are doubly covered by 240 8-component objects, which endowed with a ‘reduced inner product’ are exactly the E8 root system. It was previously known that E8 splits into H4-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, D6 as well as E8, whose Coxeter versor factorises as W=exp(π30BC)exp(11π30B2)exp(7π30B3)exp(13π30B4). This explicitly describes 30-fold rotations in 4 orthogonal planes with the correct exponents { 1 , 7 , 11 , 13 , 17 , 19 , 23 , 29 } arising completely algebraically from the factorisation.