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

Dechant, Pierre-Philippe

Abstract

In this paper, we show that via a novel construction every rank-3 root system induces a root system of rank 4. Via the Cartan-Dieudonné theorem, an even number of successive Coxeter reflections yields rotations that in a Clifford algebra framework are described by spinors. 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, we show that these polytopes have to be root systems, thereby inducing Coxeter groups of rank 4, and that their automorphism groups include two factors of the respective discrete spinor groups trivially acting on the left and on the right by spinor multiplication. Special cases of this general theorem include the exceptional 4D groups D4, F4 and H4, which therefore opens up a new understanding of applications of these structures in terms of spinorial geometry. In particular, 4D groups are ubiquitous in high energy physics. For the corresponding case in two dimensions, the groups I2(n) are shown to be self-dual, whilst via a similar construction in terms of octonions each rank-3 root system induces a root system in dimension 8; this root system is in fact the direct sum of two copies of the corresponding induced 4D root system.

Citation

Dechant, P. (2015). Rank-3 root systems induce root systems of rank 4 via a new Clifford spinor construction. Journal of Physics: Conference Series, 597(1), Article 012027. https://doi.org/10.1088/1742-6596/597/1/012027

• Journal Article Type: Article
• Acceptance Date: Apr 14, 2015
• Publication Date: Apr 13, 2015
• Deposit Date: Apr 14, 2015
• Publicly Available Date: Apr 15, 2015
• Journal: Journal of Physics: Conference Series
• Print ISSN: 1742-6588
• Electronic ISSN: 1742-6596
• Publisher: IOP Publishing
• Peer Reviewed: Peer Reviewed
• Volume: 597
• Issue: 1
• Article Number: 012027
• DOI: https://doi.org/10.1088/1742-6596/597/1/012027
• Public URL: https://durham-repository.worktribe.com/output/1410093