K-theory, genotypes, and biset functors

TL;DR

This paper demonstrates that the genome of a finite p-group, derived from its rational irreducible representations, can be reconstructed from its first K-theory group, establishing it as a rational p-biset functor.

Contribution

It introduces the genome as a new invariant of p-groups, shows it can be recovered from K-theory, and proves it forms a rational p-biset functor with explicit biset action formulas.

Findings

The genome of a p-group is recoverable from K_1(QP).

The genome assignment is a p-biset functor.

The genome is a rational p-biset functor, factoring through the Roquette category.

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 $P \to \Gamma(P)$ 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.