Spherical designs and finite group representations (some results of E. Bannai)

We reprove several results of Bannai concerning spherical t-designs and finite subgroups of orthogonal groups. These include criteria in terms of harmonic representations of subgroups of O(n) for the corresponding orbits to be t-designs (t=0,1,2,3,…) in Sn−1. We also discuss a conjecture of Bannai, dating from 1984, according to which t is bounded independently of the dimension n (for n≥3) for such designs.