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

摘要

We consider a finite subgroup Θ_n of the groupO(N) of orthogonal matrices, where N=2n, n=1, 2, ... . Thisgroup was defined in [4] and we use it to construct spherical designs inthe 2n-dimensional Euclidean space RN. We provethat representations ρ₁, ρ₂ andρ₃ of the group Θ_n on the spaces ofharmonic polynomials of degrees 1, 2 and 3 respectively are irreducible.This together with the earlier results [1, 3] imply that the orbitΘ_n,2x of any initial point x on the unit sphereS_N-1 is a 7-design in the Euclidean space of dimension2n