Idempotent characters and equivariantly multiplicative splittings of K-theory

TL;DR

This paper classifies primitive idempotents in the p-local complex representation ring of finite groups, linking them to cyclic subgroups and Burnside ring idempotents, with implications for equivariant K-theory and homotopy theory.

Contribution

It extends classical theorems by classifying idempotents without roots of unity, and connects these to equivariant K-theory and sphere spectra structures.

Findings

Primitive idempotents correspond to cyclic subgroups of order prime to p

Idempotents originate from the Burnside ring without adjoining roots of unity

Equivariant K-theory summands share the same commutative ring structures as sphere spectra

Abstract

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.