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

## Mozhgan Mohammadpour and Shayne Waldron

## 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 $\CC^2$,
a $48$ point $(4,4)$-design for $\CC^3$,
and a $400$ point $(5,5)$-design for $\CC^4$.
From a real reflection group, we construct a $360$ point $(9,9)$-design
for $\RR^4$ (spherical half-design of order $18$),
i.e., a $720$ point spherical $19$-design for $\RR^4$.

**Keywords:**
complex spherical design,
harmonic Molien-Poincar\'e series,
spherical $t$-designs,
spherical half-designs,
tight spherical designs,
finite tight frames,
signed frame,
integration rules,
cubature rules,
cubature rules for the sphere,

**Math Review Classification:**
Primary 05B30, 42C15, 65D30;
Secondary 94A12.

**Length:** 21 pages

**Last Updated:** 16 December 2019

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