  We classify the primitive idempotents of the $p$-local complex representation
ring of a finite group $G$ in terms of the cyclic subgroups of order prime to
$p$ and show that they all come from idempotents of the Burnside ring. Our
results hold without adjoining roots of unity or inverting the order of $G$,
thus extending classical structure theorems. We then derive explicit
group-theoretic obstructions for tensor induction to be compatible with the
resulting idempotent splitting of the representation ring Mackey functor.
  Our main motivation is an application in homotopy theory: we conclude that
the idempotent summands of $G$-equivariant topological $K$-theory and the
corresponding summands of the $G$-equivariant sphere spectrum admit exactly the
same flavors of equivariant commutative ring structures, made precise in terms
of Hill-Hopkins-Ravenel norm maps.
  This paper is a sequel to the author's earlier work on multiplicative
induction for the Burnside ring and the sphere spectrum, see arXiv:1802.01938.
