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

TY - JOUR

T1 - Clifford Algebra Unveils a Surprising Geometric Significance of Quaternionic Root Systems of Coxeter Groups

AU - Dechant, Pierre-Philippe

N1 - © 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.

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - Clifford algebras

KW - Coexter groups

KW - root systems

KW - quaternions

KW - representations

KW - spinors

KW - binary polyhedral groups

U2 - 10.1007/s00006-012-0371-3

DO - 10.1007/s00006-012-0371-3

M3 - Article

SN - 0188-7009

VL - 23

SP - 301

EP - 321

JO - Advances in Applied Clifford Algebras

JF - Advances in Applied Clifford Algebras

IS - 2

ER -