Biset transformations of Tambara functors

TL;DR

This paper extends the biset transformation framework from Mackey functors to Tambara functors for finite groups, establishing functoriality and algebraic compatibility, and constructs its left adjoint.

Contribution

It introduces a biset transformation for Tambara functors applicable to right-free bisets, and constructs the left adjoint of this transformation.

Findings

Biset transformation applies to Tambara functors when bisets are right-free.

The transformation is compatible with algebraic operations like ideal quotients.

Constructs the left adjoint of the biset transformation.

Abstract

If we are given an $H$-$G$-biset $U$ for finite groups $G$ and $H$, then any Mackey functor on $G$ can be transformed by $U$ into a Mackey functor on $H$. In this article, we show that the biset transformation is also applicable to Tambara functors when $U$ is right-free, and in fact forms a functor between the category of Tambara functors on $G$ and $H$. This biset transformation functor is compatible with some algebraic operations on Tambara functors, such as ideal quotients or fractions. In the latter part, we also construct the left adjoint of the biset transformation.