By the same authors

From the same journal

From the same journal

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

Research output: Contribution to journalArticle

Author(s)

Department/unit(s)

Publication details

JournalAdvances in Applied Clifford Algebras
DatePublished - Jun 2013
Issue number2
Volume23
Number of pages20
Pages (from-to)301-321
Original languageEnglish

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 rank-3 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 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.

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 self-archiving policy.

    Research areas

  • Clifford algebras, Coexter groups, root systems, quaternions, representations, spinors, binary polyhedral groups

Activities

Discover related content

Find related publications, people, projects, datasets and more using interactive charts.

View graph of relations