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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 x A_1 x 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 rank3 groups A_1 x A_1 x A_1, A_3, B_3 and H_3 yield a new simple construction of binary polyhedral groups. These in turn generate the groups A_1 x A_1 x A_1 x A_1, D_4, F_4 and 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 rank4 groups from the rank3 groups. In particular, the groups D_4, F_4 and H_4 are exceptional structures, which our study sheds new light on.
Original language  English 

Pages (fromto)  301321 
Number of pages  20 
Journal  Advances in Applied Clifford Algebras 
Volume  23 
Issue number  2 
DOIs  
Publication status  Published  Jun 2013 
Bibliographical note
© 2012 Springer Basel. This is an author produced version of a paper published in Advances in Applied Clifford Algebras. Uploaded in accordance with the publisher's selfarchiving policy.Keywords
 Clifford algebras
 Coexter groups
 root systems
 quaternions
 representations
 spinors
 binary polyhedral groups
Activities
 1 Workshop

Yau Institute: Tsinghua Summer Workshop in Geometry and Physics 2017
PierrePhilippe Dechant (Keynote/plenary speaker)
6 Aug 2017 → 12 Aug 2017Activity: Talk or presentation › Workshop