Box Product of $C_p$-Mackey Functors

TL;DR

This paper explores the structure of $G$-Mackey functors with a focus on the box product, providing examples for cyclic groups and classifying invertible Mackey functors for $C_p$, a cyclic group of prime order.

Contribution

It introduces and analyzes the box product on Mackey functors and classifies all invertible $C_p$-Mackey functors under this product.

Findings

Computed examples of box products for $G=C_p$

Classified all invertible $C_p$-Mackey functors

Provided an exposition of $G$-Mackey functors

Abstract

Let $G$ be a finite group. In this paper, we begin by providing an exposition of $G$-Mackey functors and a symmetric monoidal product on the category of Mackey functors called the box product. After computing several examples of box products for the case of $G=C_p$, the cyclic group of order $p$, we move to the heart of the paper, which is to find and classify all $C_p$-Mackey functors invertible for the box product.