A Clifford algebraic framework for Coxeter group theoretic computations

TL;DR

This paper introduces a Clifford algebraic framework for Coxeter group computations, enabling unified treatment of symmetries, group generation, and geometric analysis in physical systems with reflective and rotational symmetries.

Contribution

It develops a Clifford algebra-based method to generate and analyze Coxeter groups, including binary polyhedral and Lie groups, revealing connections across dimensions and extending to conformal geometric algebra.

Findings

Unified Coxeter group generation using versors

Construction of binary polyhedral groups as spinor groups

Insights into Coxeter element geometry and applications to quasicrystals

Abstract

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D_4, F_4 and H_4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the two-dimensional non-crystallographic Coxeter groups I_2(n) and the three-dimensional groups A_3, B_3, as well as the icosahedral group H_3.