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Idempotent characters and equivariantly multiplicative splittings of K-theory

MPG-Autoren
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Böhme,  Benjamin
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1808.09832.pdf
(Preprint), 215KB

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Zitation

Böhme, B. (2020). Idempotent characters and equivariantly multiplicative splittings of K-theory. Bulletin of the London Mathematical Society, 52(4), 730-745. doi:10.1112/blms.12362.


Zitierlink: https://hdl.handle.net/21.11116/0000-0006-BD63-6
Zusammenfassung
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.