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

The concept of rigid spherical t -designs was introduced by Eiichi Bannai. We want to find examples of rigid but not tight spherical designs. Sali investigated the case when X is an orbit of a finite reflection group and proved that X is rigid if and only if tight for the groups A n , B n , C n , D n , E 6 , E 7 , F 4 , H 3 . There are two cases left open, namely the group E 8 and the isometry group H 4 of the four-dimensional regular polytope, the 600-cell. In this paper, we study the rigidity of spherical t -designs X that are orbits of a finite reflection groups E 8 and H 4 , and prove that X is rigid if and only if tight or the 600-cell.