Relation between spherical designs through a Hopf map

TL;DR

This paper proves a specific case of a known construction relating spherical designs on different spheres via a Hopf map, providing an explicit algorithm for constructing 2t-designs on S^3 from designs on S^2 and S^1.

Contribution

It establishes the case n=2 of a previous conjecture, offering an explicit construction algorithm for spherical designs on S^3 using Hopf maps.

Findings

Confirmed the construction for n=2 case.

Provided an explicit algorithm for 2t-designs on S^3.

Extended the understanding of spherical design relationships.

Abstract

Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for the case where $n=2$. That is, we give an algorithm to construct $2t$-designs on $S^{3}$ as products through a Hopf map $S^3 \rightarrow S^2$ of a $t$-design on $S^2$ and a $2t$-design on $S^1$.