Complex spherical designs from group orbits

TL;DR

This paper investigates conditions under which group orbits form optimal complex spherical designs, developing a theoretical framework and providing explicit constructions of certain optimal designs.

Contribution

It introduces a general theory linking group actions to complex spherical designs and presents explicit constructions of optimal real and complex t-designs.

Findings

Group orbits with large stabilisers are good candidates for optimal designs

Developed harmonic Molien series for analyzing complex designs

Constructed explicit examples of optimal spherical t-designs

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.