The Roquette category of finite p-groups

TL;DR

This paper introduces the Roquette category R_p of finite p-groups, providing a new framework that simplifies the computation of rational p-biset functors and related algebraic structures.

Contribution

It defines the additive tensor category R_p, introduces the concept of edges of p-groups, and characterizes rational p-biset functors as additive functors from R_p to abelian groups.

Findings

Every p-group splits uniquely into edges of Roquette p-groups.

The tensor structure of R_p is described in terms of these edges.

Efficient computation methods for rational p-biset functors are derived.

Abstract

Let p be a prime number. This paper introduces the Roquette category R_p of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In R_p, every finite p-group P admits a canonical direct summand, called the edge of P. Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of R_p can be described in terms of such edges. The main motivation for considering this category is that the additive functors from R_p to abelian groups are exactly the rational p-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary p-groups : this applies to the representation functors R_K, where K is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.