A Clifford algebraic framework for Coxeter group theoretic computations. (English) Zbl 1301.20030

Dechant, Pierre-Philippe

Adv. Appl. Clifford Algebr. 24, No. 1, 89-108 (2014).

Summary: 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 \(H_3\).

MSC:

20F55	Reflection and Coxeter groups (group-theoretic aspects)
15A66	Clifford algebras, spinors
17B22	Root systems

Keywords:

Coxeter groups; Clifford algebras; root systems; versor computations; conformal geometry; Coxeter elements; Coxeter plane; spinors; complex structures; viruses; fullerenes; quasicrystals

Software:

CLUCalc

References:

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