Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups

Abstract

Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E8, have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A1 × A1 × A1, A3, B3 and H3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A1 × A1 × A1, A3, B3 and H3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A1 × A1 × A1 × A1, D4, F4 and H4 of rank 4 and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D4, F4 and H4 are exceptional structures, which our study sheds new light on. IPPP/12/26, DCPT/12/52