A Clifford algebraic framework for Coxeter group theoretic computations

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.

Best Paper Award at AGACSE 2012, La Rochelle, France; 18 pages, 4 figures, 2 tables; accepted in Advances in Applied Clifford Algebras

References in corpus (6)

• Clifford algebra unveils a surprising geometric significance of quaternionic root systems of Coxeter groups
• Affine extensions of non-crystallographic Coxeter groups induced by projection
• Catalan Solids Derived From 3D-Root Systems and Quaternions
• Platonic solids generate their four-dimensional analogues
• Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction
• Quasi Regular Polyhedra and Their Duals with Coxeter Symmetries Represented by Quaternions II

Cited by in corpus (6)

• Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
• The $E_8$ geometry from a Clifford perspective
• The Birth of $E_8$ out of the Spinors of the Icosahedron
• Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction
• From the Trinity $(A_3,B_3,H_3)$ to an ADE correspondence
• A 3D spinorial view of 4D exceptional phenomena