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A Clifford algebraic framework for Coxeter group theoretic computations

Dechant, Pierre-Philippe

A Clifford algebraic framework for Coxeter group theoretic computations Thumbnail


Authors

Pierre-Philippe Dechant



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 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 twodimensional non-crystallographic Coxeter groups I2(n) and the threedimensional groups A3, B3, as well as the icosahedral group H3. IPPP/12/49, DCPT/12/98

Citation

Dechant, P. (2014). A Clifford algebraic framework for Coxeter group theoretic computations. Advances in Applied Clifford Algebras, 24(1), 89-108. https://doi.org/10.1007/s00006-013-0422-4

Journal Article Type Article
Publication Date Mar 1, 2014
Deposit Date Jan 20, 2014
Publicly Available Date Jan 24, 2014
Journal Advances in Applied Clifford Algebras
Print ISSN 0188-7009
Electronic ISSN 1661-4909
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 24
Issue 1
Pages 89-108
DOI https://doi.org/10.1007/s00006-013-0422-4
Keywords Coxeter groups, Clifford algebras, Root systems, Versor computations, Conformal geometry, Coxeter element, Coxeter plane, Spinors, Complex structures, Viruses, Fullerenes, Quasicrystals.

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Copyright Statement
The final publication is available at Springer via http://dx.doi.org/10.1007/s00006-013-0422-4.





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