On generalized Schur's lemma and its applications. (Q2883153)

Let \(G\) be a finite group acting transitively on sets \(\Omega\) and \(\Gamma\). For each orbit \(\mathcal O\) of \(G\) on \(\Omega\times\Gamma\), define a \(|\Omega|\times|\Gamma|\) matrix \(M^{\mathcal O}\) by \(M^{\mathcal O}_{\omega\gamma}=1\) if \((\omega,\gamma)\in\mathcal O\) and \(M^{\mathcal O}_{\omega\gamma}=0\) otherwise. Let \(F\) be a field and let \(F\Omega\) and \(F\Gamma\) be the permutation modules for \(FG\) corresponding to \(\Omega\) and \(\Gamma\), respectively. The relevant Schur's Lemma is the result that the matrices \(M^{\mathcal O}\) form an \(F\)-basis for \(\Hom_{FG}(F\Omega,F\Gamma)\).NEWLINENEWLINE In the paper under review, Schur's Lemma is generalized to give a basis for the space of \(FG\)-homomorphisms between any two modules induced from subgroups of \(G\). Applications to idempotents and relatively projective homomorphisms are discussed. In the case where \(F\) has prime characteristic \(p\), this discussion leads to a new proof of the Green correspondence between indecomposable \(FG\)-modules and indecomposable \(FN_G(P)\)-modules, both with vertex \(P\) for a given \(p\)-subgroup \(P\leq G\). Some further results are given for modules induced from a normal subgroup, and an application to permutation modules is given.