Spherical t-designs which are orbits of finite groups (Q797591)

A spherical t-design in \({\mathbb{R}}^ d\) is a finite nonempty subset X of the unit sphere \(\Omega_ d\) such that \(\sum_{x\in X}f(x)=0\) for all homogeneous harmonic polynomials f of degree \(\leq t\). In an earlier paper [J. Comb. Theory, Ser. A 26, 157-161 (1979; Zbl 0464.05016)] the author proved that if G is a finite subgroup of the real orthogonal group O(d), and the restriction of the i-th spherical representation \(\rho_ i\) of O(d) is irreducible for \(0\leq i\leq s,\) then for any \(x\in \Omega_ d\) the orbit of x under G is a spherical 2s-design. The theorems in this paper also concern t-designs obtained as orbits of subgroups of O(d). The main result states that if the orbit of \(x_ i\) is a \(t_ i\)-design but not a \((t_ i+1)\)-design \((i=1,2)\), then \(t_ 1\leq 2t_ 2+1\) and \(t_ 2\leq 2t_ 1+1.\) \textit{J. M. Goethals} and \textit{J. J. Seidel} [Proc. Symp. Pure Math. 34, 255-272 (1979; Zbl 0404.05015)] proved the converse of the earlier theorem stated above. In the present paper it is pointed out that there is an error in the proof and in fact a counterexample is given.