Discretization of SU(2) and the Orthogonal Group Using Icosahedral Symmetries and the Golden Numbers

Abstract

The vertices of the four dimensional $120$-cell form a non-crystallographic root system whose corresponding symmetry group is the Coxeter group $H_4$. There are two special coordinate representations of this root system in which they and their corresponding Coxeter groups involve only rational numbers and the golden ratio $\tau$. The two are related by the conjugation $\tau \mapsto {\tau }^{\prime }=-1/\tau$. This paper investigates what happens when the two root systems are combined and the group generated by both versions of $H_4$ is allowed to operate on them. The result is a new, but infinite, `root system' $\Sigma$ which itself turns out to have a natural structure of the unitary group $SU(2,\mathcal{R})$ over the ring $\mathcal{R}=\mathbb{Z}[\frac{1}{2},\tau ]$ (called here golden numbers). Acting upon it is the naturally associated infinite reflection group $H^{\infty }$, which we prove is of index $2$ in the orthogonal group $O(4,\mathcal{R})$. The paper makes extensive use of the quaternions over $\mathcal{R}$ and leads to highly structured discretized filtration of $SU(2)$. We use this to offer a simple and effective way to approximate any element of $SU(2)$ to any degree of accuracy required using the repeated actions of just five fixed reflections, a process that may find application in computational methods in quantum mechanics.

arXiv comment24 pages and 4 figures