Constructing high order spherical designs as a union of two of lower order

TL;DR

This paper presents a novel method for constructing higher-order spherical designs by union of lower-order designs, leveraging group actions and variational characterizations, resulting in highly symmetric and efficient designs.

Contribution

It introduces a new approach to build high-order spherical designs from lower-order ones using unions of orbits under reflection groups, many of which are first known constructions.

Findings

Constructed new high-order spherical designs with fewer points.

Achieved designs with high symmetry and novel configurations.

Provided explicit examples for complex and real vector spaces.

Abstract

We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical $(t,t)$-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point $(9,9)$-design for $\mathbb{C}^2$, a $48$ point $(4,4)$-design for $\mathbb{C}^3$, and a $400$ point $(5,5)$-design for $\mathbb{C}^4$.From a real reflection group, we construct a $360$ point $(9,9)$-design for $\mathbb{R}^4$ (spherical half-design of order $18$), i.e., a $720$ point spherical $19$-design for $\mathbb{R}^4$.