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 .

The E 8 Geometry from a Clifford Perspective

“…We therefore systematically factorise Coxeter versors of root systems in the Clifford algebra, which gives the eigenplanes and exponents algebraically. We briefly discuss the 2D case of the two-dimensional family of non-crystallographic Coxeter groups I 2 (n), followed by the three-dimensional groups A 3 , B 3 and H 3 [8], before discussing the higher-dimensional examples. The simple roots for I 2 (n) can be taken as α 1 = e 1 , α 2 = − cos π n e 1 + sin π n e 2 , which yields the Coxeter versor W describing the n-fold rotation encoded by the I 2 (n) Coxeter element via v → wv =W vW as…”

“…At the last AGACSE in 2012, I presented a new link between the geometries of three and four dimensions; in particular, I have shown that any reflection group in three dimensions induces a corresponding symmetry group in four dimensions, via a new Clifford spinor construction [8,9]. This connection had been overlooked for centuries (usually one assumes the larger groups are more fundamental) but the new construction derives all the exceptional phenomena in 4D-D 4 , F 4 and H 4 -via induction from the 3D symmetry groups of the Platonic solids A 3 , B 3 and H 3 .…”

“…This previous construction worked along the following lines: each 3D root system-which generates the corresponding reflection symmetry group via the reflections in the hyperplanes orthogonal to the root vectors-allows one to form a group of spinors by multiplying together even numbers of the reflection generating root vectors in the Clifford algebra; these spinors have four components (the usual 1 scalar and 3 bivector components) and one can endow these with a 4D Euclidean metric. One can then show that The E 8 Geometry from a Clifford Perspective 399 the resulting set of spinors reinterpreted as 4D vectors satisfies the axioms for a 4D root system, thereby generating a symmetry group in 4D. For instance, starting with the root system H 3 which generates the symmetries of the icosahedron, one generates a group 2I of 120 spinors (the binary icosahedral group) via multiplication in the Clifford algebra.…”

A 3D Spinorial View of 4D Exceptional Phenomena

“…Likewise, a reflection (continuous O(3) or the discrete subgroup, the full icosahedral group the Coxeter group H 3 ) corresponds to sandwiching with the versor A, whilst the versors single-sidedly form a multiplicative group (the Pin(3) group or the discrete analogue, the double cover of H 3 , which we denote Pin(H 3 )). In the conformal geometric algebra setup one uses the fact that the conformal group C(p, q) is homomorphic to SO(p + 1, q + 1) to treat translations as well as rotations in a unified versor framework [24,16,5,8,9]. [8,9] also discuss reflections, inversions, translations and modular transformations in this way.…”

Viruses and fullerenes – symmetry as a common thread?

“…Via an affine extension, the symmetry point group is augmented so that it can describe structures at different radial levels collectively (Dechant et al, 2012(Dechant et al, , 2013Dechant, 2014). It therefore lends itself to the modelling of carbon onions (Patera & Twarock, 2002;Twarock, 2002).…”