Spherical 7-design in the 2/sup n/-dimensional Euclidean space

We consider a finite subgroup /spl Theta//sub n/ of the group O(N) of orthogonal matrices, where N=2/sup n/, n=1, 2, ... . This group was defined in [4] and we use it to construct spherical designs in the 2/sup n/-dimensional Euclidean space R/sup N/. We prove that representations /spl rho//sub 1/, /spl rho//sub 2/ and /spl rho//sub 3/ of the group /spl Theta//sub n/ on the spaces of harmonic polynomials of degrees 1, 2 and 3 respectively are irreducible. This together with the earlier results [1, 3] imply that the orbit /spl Theta//sub n,2/x of any initial point x on the unit sphere S/sub N-1/ is a 7-design in the Euclidean space of dimension 2/sup n/.