The Dade group of Mackey functors for p-groups

TL;DR

This paper introduces the Mackey-Dade group of a p-group, exploring its properties and its relation to Dade groups of Weyl groups, revealing its role as a kernel of a linearization map for Mackey functors.

Contribution

It defines endo-permutation Mackey functors and the Mackey-Dade group, establishing their fundamental properties and connections to existing Dade groups and linearization maps.

Findings

Mackey-Dade group relates to Dade groups of Weyl groups

Tensoring with Q reveals the Mackey-Dade group as a kernel

Introduces endo-permutation Mackey functors

Abstract

We introduce endo-permutation Mackey functors and the Mackey-Dade group of a $p$-group and study their basic properties. We exhibit relations between the Mackey-Dade group of a finite $p$-group $P$ and the Dade groups of the Weyl groups of all subgroups of $P$. As a result, we show that the Mackey-Dade group tensored with $\mathbb Q$ over $\mathbb Z$ is the kernel of the linearization map for Mackey functors, introduced by the author.