K-theory, genotypes, and biset functors

Abstract

Let p be an odd prime number. In this paper, we show that the genome\(\Gamma (P)\) of a finite p-group P, defined as the direct product of the genotypes of all rational irreducible representations of P, can be recovered from the first group of K-theory \(K_1(\mathbb {Q}P)\). It follows that the assignment \(P\mapsto \Gamma (P)\) is a p-biset functor. We give an explicit formula for the action of bisets on \(\Gamma \), in terms of generalized transfers associated to left free bisets. Finally, we show that \(\Gamma \) is a rational p-biset functor, i.e. that \(\Gamma \) factors through the Roquette category of finite p-groups.

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Correspondence to Serge Bouc.

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Bouc, S. K-theory, genotypes, and biset functors. Bol. Soc. Mat. Mex. 26, 57–74 (2020). https://doi.org/10.1007/s40590-019-00234-6

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Keywords

  • K-theory
  • Genotype
  • Whitehead group
  • Biset functor
  • Roquette category
  • transfer

Mathematics Subject Classification

  • 19B28
  • 20C05
  • 18A99