Blocks of endomorphism algebras
type: Item
instance of: scholarly article
title: Blocks of endomorphism algebras
zbMATH Open document ID: 0837.20009
author: Laurence Barker
published in: Journal of Algebra
publication date: 1994-09-28T00:00:00Z
review text: Let \(G\) be a finite group, let \(\mathcal O\) be a suitable ring of coefficients, let \(A\) be an \({\mathcal O} G\)-lattice, let \({\mathcal H} (A) := \text{End}_{{\mathcal O} G} (A)\), and let \({\mathcal F} (A) := \text{End}_{{\mathcal H} (A)} (A)\). Then \({\mathcal F} (A)\) is a \(G\)-algebra, and \({\mathcal F} (A)^G = Z({\mathcal H} (A))\). Thus every block (idempotent) \(\alpha\) of \({\mathcal H} (A)\) has a defect group \(P\). The author shows that \(P\) is contained in a defect group of the block of \({\mathcal O} G\) containing \(A_\alpha\), and that \(P\) contains a vertex of every indecomposable direct summand of \(A_\alpha\). Let \(N_G(P) \leq H \leq G\). Then it is possible to associate to \(\alpha\) a unique block \(\text{ass}_H(\alpha)\) of \({\mathcal H} (\text{Res}^G_H (A))\). Conversely, given a block \(\beta\) of \({\mathcal H} (B)\) with defect group \(P\), for an \({\mathcal O} H\)-lattice \(B\), it is possible to associate to \(\beta\) a block \(\text{ass}_G (\beta)\) of \({\mathcal H} (\text{Ind}^G_H (B))\). The author studies the behaviour of these maps on blocks with respect to defect groups and the Brauer and the Green correspondence.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 645102
zbMATH Keywords: Brauer correspondence
zbMATH Keywords: block idempotents
zbMATH Keywords: finite groups
zbMATH Keywords: vertex
zbMATH Keywords: indecomposable direct summands
zbMATH Keywords: blocks
zbMATH Keywords: defect groups
zbMATH Keywords: Green correspondence
MaRDI profile type: Publication
full work available at URL: jabr.1994.1250
OpenAlex ID: W2014500855
DOI: 10.1006/JABR.1994.1250
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description: scientific article; zbMATH DE number 645102