Natural Morita equivalences of degree $n$

Abstract

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $b$ and $c$ block idempotents of $\mathcal{O}G$ and $\mathcal{O}H$ respectively. Under the assumption that $C_{H}(R)\subset O_{p',p}(H)$ for a Sylow $p$-subgroup $R$ of $O_{p',p}(H)$ and $c$ is also a block idempotent of $\mathcal{O}O_{p'}(H)$, we give two equivalent conditions about when $\mathcal{O}Gb$ and $\mathcal{O}Hc$ are natural Morita equivalent of degree $n$ (see Theorem 1.5).