An Equivariant Tensor Product on Mackey Functors

TL;DR

This paper introduces explicit norm functors for Mackey functors over cyclic p-groups, enabling a monoidal structure where Tambara functors serve as commutative ring objects, advancing algebraic understanding in equivariant settings.

Contribution

It provides an explicit algebraic construction of norm functors for Mackey functors over cyclic p-groups, establishing a new monoidal structure with Tambara functors as ring objects.

Findings

Defined norm functors for all subgroups of cyclic p-groups.

Constructed a monoidal category of Mackey functors.

Identified Tambara functors as commutative ring objects.

Abstract

For all subgroups $H$ of a cyclic $p$-group $G$ we define norm functors that build a $G$-Mackey functor from an $H$-Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects.