A Clifford Algebraic Framework for Coxeter Group Theoretic Computations

Dechant, Pierre-Philippe (2012) A Clifford Algebraic Framework for Coxeter Group Theoretic Computations. In: Applied Geometric Algebras in Computer Science and Engineering, 2-4 July 2012, La Rochelle, France.

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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 D4, F4 and H4. 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-crystallographicCoxeter groups I2(n) and the three-dimensional groups A3, B3, as well as the icosahedral group H3.
IPPP/12/49, DCPT/12/98

Item Type: Conference or Workshop Item (Lecture)
Status: Published
Related URLs:
Subjects: Q Science > Q Science (General)
Q Science > QA Mathematics
School/Department: STEM
URI: http://ray.yorksj.ac.uk/id/eprint/3358

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