On the rigidity of spherical tt-designs that are orbits of reflection groups E8E8 and H4H4

The concept of rigid spherical tt-designs was introduced by Eiichi Bannai. We want to find examples of rigid but not tight spherical designs. Sali investigated the case when XX is an orbit of a finite reflection group and proved that XX is rigid if and only if tight for the groups AnAn, BnBn, CnCn, DnDn, E6E6, E7E7, F4F4, H3H3. There are two cases left open, namely the group E8E8 and the isometry group H4H4 of the four-dimensional regular polytope, the 600-cell. In this paper, we study the rigidity of spherical tt-designs XX that are orbits of a finite reflection groups E8E8 and H4H4, and prove that XX is rigid if and only if tight or the 600-cell.