# Complex spherical designs from group orbits

## Mozhgan Mohammadpour and Shayne Waldron

## Abstract:

We consider the general question of when all orbits under the unitary action
of a finite group give a complex spherical design. Those orbits which have large
stabilisers are then good candidates for being optimal complex spherical designs.
This is done by developing the general theory of complex designs and associated
(harmonic) Molien series for group actions. As an application, we give explicit
constructions of some putatively optimal real and complex spherical t-designs.

**Keywords:**
complex spherical design, unitary group action, complex reflection group,
harmonic Molien series, spherical t-designs, projective designs, complex τ -designs, tight
spherical designs, finite tight frames, integration rules, cubature rules, cubature rules for
the sphere, Weyl-Heisenberg SIC

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

**Length:** 26 Pages

**Last Updated:** 21 July 2023

## Availability: