Clifford spinors and root system induction: $H_4$ and the Grand Antiprism

Abstract

Recent work has shown that every 3D root system allows the construction of a correponding 4D root system via an `induction theorem'. In this paper, we look at the icosahedral case of $H_3\rightarrow H_4$ in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonn\'e theorem, giving a simple construction of the Pin and Spin covers. Using this connection with $H_3$ via the induction theorem sheds light on geometric aspects of the $H_4$ root system (the $600$-cell) as well as other related polytopes and their symmetries, such as the famous Grand Antiprism and the snub 24-cell. The uniform construction of root systems from 3D and the uniform procedure of splitting root systems with respect to subrootsystems into separate invariant sets allows further systematic insight into the underlying geometry. All calculations are performed in the even subalgebra of Cl(3), including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes, and are shared as supplementary computational work sheets. This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework.