Asymmetric Cubature Formulae with Few Points in High Dimension for Symmetric Measures

Let $\mu$ be a positive measure on $\mathbb{R}^d$ invariant under the group of reflections and permutations, and let m be a natural number. We describe a method to construct cubature formulae of degree m with respect to $\mu$, with n positive weights and n points in the support of $\mu$ and such that n grows at most like dm with the dimension d. We apply this method to classical measures to explicitly construct cubature formulae of degree 5 with the number of points growing at most like d3.