Topology Seminar: Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

I provide a complete characterization of the equivariant commutative ring structures of all the idempotent summands of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. My results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, my 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, I obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.