Half of an antipodal spherical design

TL;DR

This paper explores the properties of antipodal spherical designs, focusing on selecting half of the points to achieve balance at the origin, with applications to root systems, lattices, and harmonic designs.

Contribution

It introduces new insights into selecting balanced halves of antipodal spherical designs, connecting them with association schemes and harmonic index designs.

Findings

Balanced halves exist for certain root systems and lattices.

Connections established between antipodal designs and association schemes.

Insights into the structure of tight 7-designs on spheres.

Abstract

We investigate several antipodal spherical designs on whether we can choose half of the points, one from each antipodal pair, such that they are balanced at the origin. In particular, root systems of type A, D and E, minimal points of Leech lattice and the unique tight 7-design on $S^{22}$ are studied. We also study a half of an antipodal spherical design from the viewpoint of association schemes and spherical designs of harmonic index $T$.