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.