On the separability of some Green biset functors

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 $R{B}_G$, 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 $R$.