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

TL;DR

This paper introduces a novel Clifford algebra-based construction that transforms rank-3 root systems into rank-4 root systems, revealing new geometric and algebraic insights with applications in physics.

Contribution

The paper presents a new Clifford spinor construction that induces rank-4 root systems from rank-3 systems, expanding understanding of Coxeter groups and their geometric representations.

Findings

Rank-3 root systems induce rank-4 root systems via Clifford spinors.

Exceptional 4D groups $D_4$, $F_4$, $H_4$ are explained through this construction.

Self-duality of $I_2(n)$ groups and 8D root systems from octonions are demonstrated.

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-Dieudonn\'e 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 $D_4$, $F_4$ and $H_4$, 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 $I_2(n)$ 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.