Demystifiying quaternionic representations

What is it about?

It has been noticed that certain 4-dimensional root systems have representations in terms of quaternions (generalisations of complex numbers, in some sense), and that certain 3-dimensional ones have representations in terms of pure quaternions (quaternions with no real part). However, this is merely an algebraic approach, without any insight into why and when that should be the case. It turns out this is intimately linked to the geometry of three dimensions and reflections and rotations therein.

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Why is it important?

This paper shows how a Clifford spinorial construction gives these quaternionic representations in a uniform way, with a clear geometric understanding. It also shows that the 3D representations work only as long as the corresponding Coxeter group contains the inversion. In particular, the uniformity of the construction shows how to derive a quaternionic 4D representation from any 3D root system, relating the two sets and explaining the existence of certain exceptional phenomena in 4D.