Affine extensions of non-crystallographic Coxeter groups induced by projection

2013

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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 .

Section: Coxeter Formulationmentioning

confidence: 99%

A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

2013

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“…Here, we demonstrate how in fact the rank-4 groups can be derived from the rank-3 groups via the geometric product of Clifford Geometric Algebra. This is complementary to the top-down approaches of projection, for instance from E 8 to H 4 [35,32,27,25,6], or of generating subgroups by deleting nodes in Coxeter-Dynkin diagrams.…”

Section: Introductionmentioning

confidence: 99%

Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups

2012

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Abstract. Quaternionic representations of Coxeter (reflection) groups of ranks 3 and 4, as well as those of E 8 , have been used extensively in the literature. The present paper analyses such Coxeter groups in the Clifford Geometric Algebra framework, which affords a simple way of performing reflections and rotations whilst exposing more clearly the underlying geometry. The Clifford approach shows that the quaternionic representations in fact have very simple geometric interpretations. The representations of the groups A 1 × A 1 × A 1 , A 3 , B 3 and H 3 of rank 3 in terms of pure quaternions are shown to be simply the Hodge dualised root vectors, which determine the reflection planes of the Coxeter groups. Two successive reflections result in a rotation, described by the geometric product of the two reflection vectors, giving a Clifford spinor. The spinors for the rank-3 groups A 1 × A 1 × A 1 , A 3 , B 3 and H 3 yield a new simple construction of binary polyhedral groups. These in turn generate the groupsand H 4 of rank 4, and their widely used quaternionic representations are shown to be spinors in disguise. Therefore, the Clifford geometric product in fact induces the rank-4 groups from the rank-3 groups. In particular, the groups D 4 , F 4 and H 4 are exceptional structures, which our study sheds new light on.IPPP/12/26, DCPT/12/52Mathematics Subject Classification (2010). Primary 51F15, 20F55; Secondary 15A66, 52B15.

“…9). This is also related to the fact that on the level of the root system there is a projection which maps the 240 roots of E 8 onto the 120 roots of H 4 and their τ -multiples in one of the H 4 -invariant 4-subspaces [5,21] (cf. previous section).…”

Section: H 4 As a Group Of Rotations Rather Than Reflections IImentioning

confidence: 99%

The E 8 Geometry from a Clifford Perspective

2016

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Abstract. This paper considers the geometry of E8 from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system H3 gives rise to the largest (and therefore exceptional) non-crystallographic root system H4. Arnold's trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and E8. Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the 120 elements of the icosahedral group H3 are doubly covered by 240 8-component objects, which endowed with a 'reduced inner product' are exactly the E8 root system. It was previously known that E8 splits into H4-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, D6 as well as E8, whose Coxeter versor factorises as W = exp( π 30