  We provide a complete characterization of the equivariant commutative ring
structures of all the factors in the idempotent splitting of the G-equivariant
sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any
finite group. Our results describe explicitly how these structures depend on
the subgroup lattice and conjugation in G. Algebraically, our analysis
characterizes the multiplicative transfers on the localization of the Burnside
ring of G at any idempotent element, which is of independent interest to group
theorists. As an application, we obtain an explicit description of the
incomplete sets of norm functors which are present in the idempotent splitting
of the equivariant stable homotopy category.
