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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.

Idempotent characters and equivariantly multiplicative splittings of K‐theory