Spherical t-designs which are orbits of finite groups

A spherical t-design in Rᵈ is a finite nonempty subset X in the unit sphere Ωd=\(x₁, --, xd)∈ Rᵈ|x₁²+⋯+xd²=1\ such that (0.1)for all polynomials f(x)=f(x₁, x₂, ⋯ , xd) of degree r .The condition (0.1) is equivalent to the following condition:for all homogeneous harmonic polynomials f(x)=f(x₁, ⋯ , xd) of degrees 1, 2, t .The reader is referred to Delsarte-Goethals-Seidel [6] for the basic properties of spherical t-designs.In this paper, we study spherical t-designs X which are obtained from finite subgroups G of the real orthogonal group $O(d)$ in such a way that X:=xG :=ᵍ|g∈ G\⊂Ωd for some x∈Ωd . (Namely, X is a spherical t-design which is obtained as an orbit of a finite group G in $0(d).)$Let G be a finite subgroup of the real orthogonal group $0(d)$ acting on Rᵈ and on Ωd .Let ρᵢ(i=0,1, 2, ) be the i-th spherical representation of $O(d)$, $i.e.$ , the representation of $O(d)$ on the space of homogeneous harmonic poly- nomials of degree i .So dim ρᵢ=(arrayld+i-1iarray)-(arrayld+i-3i-2array) .In [1] the following theorem was proved: THEOREM A (Bannai [1, Theorem 1]). (i) Let G be a finite subgroup of $*)$