Skip to main content

Research Repository

Advanced Search

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

Dechant, Pierre-Philippe

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


Authors

Pierre-Philippe Dechant



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

Files

Published Journal Article (829 Kb)
PDF

Publisher Licence URL
http://creativecommons.org/licenses/by/4.0/

Copyright Statement
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.





You might also like



Downloadable Citations