Discretization of SU(2) and the orthogonal group using icosahedral symmetries and the golden numbers. Issue 6 (3rd June 2018)

• Record Type: Journal Article
• Authors: Moody, Robert V.; Morita, Jun
• Is Part Of: Communications in algebra. Volume 46:Issue 6(2018)
• Page Start: 2510
• Page End: 2533
• Publication Date: 2018-06-03
• Subjects: Discretization of SU(2) -- golden numbers -- icosahedral symmetry -- infinite reflection groups -- non-crystallographic root systems -- quaternions; 17B22 -- 20F55 -- 51F25 -- 49M25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1388815
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: The vertices of the four-dimensional 600-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 τ . The two are related by the conjugation τ ↦ τ ′ = −1∕ τ . 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' Σ which itself turns out to have a natural structure of the unitary group SU (2, ℛ ) over the ring ℛ = ℤ [ 1 2, τ ] (called here golden numbers). Acting upon it is the naturally associated infinite reflection group H ∞, which we prove is of index 2 in the orthogonal group O (4, ℛ ). The paper makes extensive use of the quaternions over ℛ and leads to a 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.