Cubature formulae of the fifteenth degree of accuracy for a hypersphere (Q1311574)

Let \(S_{n-1}\) be the unit sphere in \(\mathbb{R}^ n\) and \(Q_ n\) be the hyperoctahedron with vertices at \(\{(p_ 1, \dots, p_ n)\): \(p_ j=0\) for \(j \neq i\), \(p_ i=1\) or \(p_ i=-1\}^ n_{i=1}\). The author considers the existence of a cubature formula on \(S_{n-1}\) which integrates exactly all polynomials of degree less than or equal to 15 and which are invariant under the group of all transformations of \(Q_ n\) into itself. The number of the free parmeters (nodes and coefficients) is equal to the number of the fundamental invariant polynomials (equal to 15). Tables with the nodes and coefficients are given for some small \(n\) \((n=8,9,10)\) for which the cubature formula exists.