Real and complex spherical designs and their Gramian

TL;DR

This paper characterizes real and complex spherical designs via Gramian matrices, introducing a potential function approach for their construction and analysis, especially when polynomial spaces are reducible under the unitary group.

Contribution

It provides a general method to characterize spherical designs through Gramian-based potentials, facilitating their numerical and analytical construction.

Findings

Designs are determined by their Gramian matrices.

Potential functions can be used to construct spherical designs.

Flexibility in potentials allows simple forms for certain designs.

Abstract

If a (weighted) spherical design is defined as an integration (cubature) rule for a unitarily invariant space P of polynomials (on the sphere), then any unitary image of it is also such a spherical design. It therefore follows that such spherical designs are determined by their Gramian (Gram matrix). We outline a general method to obtain such a characterisation as the minima of a function of the Gramian, which we call a potential. This characterisation can be used for the numerical and analytic construction of spherical designs. When the space P of polynomials is not irreducible under the action of the unitary group, then the potential is not unique. In several cases of interest, e.g., spherical t-designs and half-designs, we use this flexibility to provide potentials with a very simple form. We then use our results to develop certain aspects of the theory of real and complex spherical designs for unitarily invariant polynomial spaces.