A 3D Spinorial View of 4D Exceptional Phenomena

Abstract

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via ‘sandwiching’. This extends to a description of orthogonal transformations in general by means of ‘sandwiching’ with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflections by the Cartan-Dieudonné theorem. We begin by viewing the largest non-crystallographic reflection/Coxeter group $H_4$ as a group of rotations in two different ways—firstly via a folding from the largest exceptional group $E_8$, and secondly by induction from the icosahedral group $H_3$ via Clifford spinors. We then generalise the second way by presenting a construction of a 4D root system from any given 3D one. This affords a new, spinorial, perspective on 4D phenomena, in particular as the induced root systems are precisely the exceptional ones in 4D, and their unusual automorphism groups are easily explained in the spinorial picture; we discuss the wider context of Platonic solids, Arnold’s trinities and the McKay correspondence. The multivector groups can be used to perform concrete group-theoretic calculations, e.g. those for $H_3$ and $E_8$, and we discuss how various representations can also be constructed in this Clifford framework; in particular, representations of quaternionic type arise very naturally.