Complex spherical designs from group orbits

Mozhgan Mohammadpour and Shayne Waldron


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