Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction

TY - JOUR

T1 - Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction

AU - Dechant, Pierre-Philippe

N1 - This content is made available by the publisher under a Creative Commons CC-BY Licence.

PY - 2015/7

Y1 - 2015/7

N2 - In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. In a Clifford algebra framework, an even number of successive Coxeter reflections yields - via the Cartan-Dieudonne theorem - spinors that describe rotations. In three dimensions these spinors themselves have a natural four-dimensional Euclidean structure, and discrete spinor groups can therefore be interpreted as 4D polytopes. In fact, these polytopes have to be root systems, thereby inducing Coxeter groups of rank 4. For the corresponding case in two dimensions, the groups I_2(n) are shown to be self-dual.

AB - In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. In a Clifford algebra framework, an even number of successive Coxeter reflections yields - via the Cartan-Dieudonne theorem - spinors that describe rotations. In three dimensions these spinors themselves have a natural four-dimensional Euclidean structure, and discrete spinor groups can therefore be interpreted as 4D polytopes. In fact, these polytopes have to be root systems, thereby inducing Coxeter groups of rank 4. For the corresponding case in two dimensions, the groups I_2(n) are shown to be self-dual.

U2 - 10.1088/1742-6596/597/1/012027

DO - 10.1088/1742-6596/597/1/012027

M3 - Article

SN - 1742-6588

VL - 597

JO - Journal of Physics: Conference Series

JF - Journal of Physics: Conference Series

IS - conference 1

M1 - 012027

ER -