On generalized Schur's lemma and its applications.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1266.20012
DOI: 10.1142/S1005386712000223
published in: Algebra Colloquium
publication date: 2012-05-11T00:00:00Z
full work available at URL: S1005386712000223.html
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 6033437
zbMATH Keywords: finite groups
zbMATH Keywords: Schur lemma
zbMATH Keywords: endomorphism rings
zbMATH Keywords: defect groups
zbMATH Keywords: permutation modules
zbMATH Keywords: induced modules
zbMATH Keywords: Green correspondence
author: Lizhong Wang
MaRDI profile type: Publication
Wikidata QID: Q124849089
cites work: Q4092167
cites work: Modules with a group of operators
cites work: On the vertices of irreducible modules
cites work: Q3826736
cites work: Generalized Foulkes' conjecture and tableaux construction
OpenAlex ID: W2011336892
title: On generalized Schur's lemma and its applications.
review text: 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.
reviewed by: Mark Wildon
description: scientific article; zbMATH DE number 6033437