Discretization of SU(2) and the orthogonal group using icosahedral symmetries and the golden numbers
| Autor: | Jun Morita, Robert V. Moody |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Discretization Icosahedral symmetry Coxeter group Group Theory (math.GR) 010103 numerical & computational mathematics Symmetry group 01 natural sciences 20F55 81P68 0103 physical sciences Homogeneous space FOS: Mathematics Orthogonal group 010307 mathematical physics 0101 mathematics Quaternion Mathematics - Group Theory Special unitary group Mathematics |
| Zdroj: | Communications in Algebra. 46:2510-2533 |
| ISSN: | 1532-4125 0092-7872 |
| DOI: | 10.1080/00927872.2017.1388815 |
| Popis: | 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 $\tau \mapsto\tau' = -1/\tau$. 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. Comment: 24 pages and 4 figures |
| Databáze: | OpenAIRE |
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