A 3D spinorial view of 4D exceptional phenomena

TL;DR

This paper introduces a Clifford algebra-based framework to analyze 4D exceptional phenomena, revealing new insights into symmetry groups, root systems, and automorphism groups through a spinorial perspective.

Contribution

It presents a novel Clifford algebra approach to construct 4D root systems from 3D ones, offering new understanding of exceptional symmetries and automorphisms.

Findings

Constructed 4D root systems from 3D ones using Clifford spinors

Explained automorphism groups of 4D root systems via spinorial perspective

Demonstrated concrete calculations for groups like H3 and E8 using Clifford algebra

Abstract

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via `sandwiching'. This extends to a description of orthogonal transformations in general by means of `sandwiching' with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflections by the Cartan-Dieudonn\'e theorem. We begin by viewing the largest non-crystallographic reflection/Coxeter group $H_4$ as a group of rotations in two different ways -- firstly via a folding from the largest exceptional group $E_8$, and secondly by induction from the icosahedral group $H_3$ via Clifford spinors. We then generalise the second way by presenting a construction of a 4D root system from any given 3D one. This affords a new -- spinorial -- perspective on 4D phenomena, in particular as the induced root systems are precisely the exceptional ones in 4D, and their unusual automorphism groups are easily explained in the spinorial picture; we discuss the wider context of Platonic solids, Arnold's trinities and the McKay correspondence. The multivector groups can be used to perform concrete group-theoretic calculations, e.g. those for $H_3$ and $E_8$, and we discuss how various representations can also be constructed in this Clifford framework; in particular, representations of quaternionic type arise very naturally.