Clifford Algebra is the Natural Framework for Root Systems and Coxeter Groups. Group Theory: Coxeter, Conformal and Modular Groups

Abstract

In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan-Dieudonn\'e theorem all the transformations of interest can be written as products of reflections and thus via `sandwiching' with Clifford algebra multivectors. These multivector groups can be used to perform concrete calculations in different groups, e.g. the various types of polyhedral groups, and we treat the example of the tetrahedral group $A_3$ in detail. As an aside, this gives a constructive result that induces from every 3D root system a root system in dimension four, which hinges on the facts that the group of spinors provides a double cover of the rotations, the space of 3D spinors has a 4D euclidean inner product, and with respect to this inner product the group of spinors can be shown to be closed under reflections. In particular the 4D root systems/Coxeter groups induced in this way are precisely the exceptional ones, with the 3D spinorial point of view also explaining their unusual automorphism groups. This construction simplifies Arnold's trinities and puts the McKay correspondence into a wider framework. We finally discuss extending the conformal geometric algebra approach to the 2D conformal and modular groups, which could have interesting novel applications in conformal field theory, string theory and modular form theory.