Projective superflows. II. $O(3)$ and the icosahedral group

TL;DR

This paper classifies 3D real superflows with specific symmetry groups, including the icosahedral group, and identifies all irreducible and reducible cases, highlighting their geometric and algebraic properties.

Contribution

It provides a complete classification of 3D real superflows with symmetries of Platonic and related solids, extending previous work on tetrahedral and octahedral symmetries.

Findings

Classified superflow with icosahedral symmetry.

Identified all irreducible 3D real superflows.

Found two reducible superflows with prism and antiprism symmetries.

Abstract

Let $X\in\mathbb{R}^{n}$. For $\phi:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $\phi^{t}=t^{-1}\phi(Xt)$. A projective flow is a solution to the projective translation equation $\phi^{t+s}=\phi^{t}\circ\phi^{s}$, $t,s\in\mathbb{R}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is in a sense unique and optimal. In this second part we classify $3$-dimensional real superflows. Apart from the superflow $\phi_{\hat{\mathbb{T}}}$ (with a group of symmetries being all symmetries of a tetrahedron) and the superflow $\phi_{\mathbb{O}}$ (with a group of symmetries being orientation preserving symmetries of an octahedron), both described in the first part of this study, here we investigate in detail the superflow $\phi_{\mathbb{I}}$ whose group of symmetries is the icosahedral group $\mathbb{I}$ of order $60$. This superflow is a flow on co-centric spheres, and is also solenoidal. These three superflows is the full (up to linear conjugation) list of $3$-dimensional irreducible real projective superflows. We also find all reducible $3$-dimensional real superflows. There are two of them: one with group of symmetries being all symmetries of a $3$-prism (group of order $12$), and the second with a group of symmetries being all symmetries of a $4$-antiprism (group of order $16$).