Resultant of an equivariant polynomial system with respect to the reflection group $G(r,n)$

We consider systems of homogeneous multivariate polynomials equivariant under the complex reflection group $G(r,n) = (\mathbb{Z}/r\mathbb{Z})^n \rtimes S_n$. Using divided differences indexed by partitions of $n$, we establish a decomposition formula expressing the resultant of such a system as a pr