On the separability of some Green biset functors

TL;DR

This paper investigates the separability properties of certain Green biset functors, revealing conditions under which they are or are not separable, especially focusing on the Burnside functor and complex characters.

Contribution

It establishes new criteria for the separability of Burnside biset functors and complex character functors, clarifying their algebraic structure over various rings.

Findings

The complex character Green biset functor $R_{\mathbb{C}}$ is not separable.

The Burnside biset functor $RB_G$ is separable iff $|G|$ is invertible in $R$.

The Burnside $R$-algebra $RB(G)$ is separable iff $|G|$ is invertible in $R$.

Abstract

We show that the Green biset functor $R_{\mathbb{C}}$ of complex characters over $\mathbb{Z}$, is not separable, i.e. it is not projective as a bimodule over itself. Also, we show that $RB_G$, the Burnside biset functor shifted by a finite group $G$, over a commutative ring $R$, is separable if and only if $|G|$ is invertible in $R$. Finally, to address the question of the relation between functors and their evaluations, we show that the Burnside $R$-algebra $RB(G)$ is separable if and only if $|G|$ is invertible in $R$.