Inducing native Mackey functors to biset functors

TL;DR

This paper develops an induction functor from native Mackey functors to biset functors for finite groups over characteristic zero fields, and applies it to classify projective biset functors and describe indecomposables.

Contribution

It introduces an explicit induction functor between native Mackey and biset functors and applies it to classify projective and indecomposable biset functors.

Findings

Any projective biset functor is induced from a rational native Mackey functor.

Explicit description of projective indecomposable biset functors.

Parameterization of indecomposables by simple groups.

Abstract

In this paper, we describe the induction functor from the category of native Mackey functors to the category of biset functors for a finite group $G$ over an algebraically closed field $k$ of characteristic zero. We prove two applications of this description. As the first application, we exhibit that any projective biset functor over $k$ is induced from a (rational) virtual native Mackey functor. The second application is the explicit description of the projective indecomposable biset functors parameterized by simple groups.