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

TL;DR

This paper uses Clifford algebra to explicitly construct and analyze 4D root systems like $H_4$ from 3D systems such as $H_3$, revealing geometric insights into polytopes like the Grand Antiprism.

Contribution

It provides a systematic Clifford algebra-based method for inducing 4D root systems from 3D ones, with explicit calculations and visualizations of complex polytopes.

Findings

Explicit construction of $H_4$ root system from $H_3$ using Clifford algebra

Visualization of polytopes like the Grand Antiprism and snub 24-cell

A systematic algebraic framework for analyzing reflection groups and root systems

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.