The Green correspondence and ordinary induction of blocks in finite group modular representation theory

Abstract

The first step in the fundamental Clifford theoretic approach to general block theory of finite groups reduces to: $H$ is a subgroup of the finite group $G$ and $e$ is a central idempotent of $H$ such that $e({}^{g}e)=0$ for all $g \in G-H$. Then $\Tr_{H}^{G}(e)$ is a central idempotent of $G$ and induction from $H$ to $G$, $\Ind_{H}^{G}$, is part of a Morita equivalence between the categories of $e$-modules and of $\Tr_{H}^{G}(e)$-modules. Let $W$ be an indecomposable $e$-module, so that $V=\Ind_{H}^{G}(W)$ is an indecomposable $\Tr_{H}^{G}(e)$-module. We present results that relate the Green correspondents of $W$ and $V$ via induction and restriction.