Correspondences of isomorphism types of irreducible modules in finite group block theory.
type: Item
instance of: scholarly article
title: Correspondences of isomorphism types of irreducible modules in finite group block theory.
zbMATH Open document ID: 1273.20010
published in: Journal of Algebra
publication date: 2013-04-17T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic \(p>0\), let \(M\) be a normal subgroup of a finite group \(G\), and let \(B\) be a block of \(kM\) with defect group \(P\). The Harris-Knörr correspondence gives a bijection between blocks of \(kG\) covering \(B\) and blocks of \(kN_G(P)\) covering the Brauer correspondent \(B^*\) of \(B\), a block of \(kN_M(P)\) with defect group \(P\). Now suppose that \(M\) is \(p\)-solvable. Then a result by \textit{T. Okuyama} [Hokkaido Math. J. 10, 299-318 (1981; Zbl 0488.20013)] not mentioned in the paper under review gives a bijection between isomorphism classes of simple \(B\)-modules with vertex \(P\) and isomorphism classes of simple \(B^*\)-modules with vertex \(P\). The main result of the present paper shows that whenever \(A\) is a block of \(kG\) covering \(B\) and whenever \(A^*\) is the Harris-Knörr correspondent of \(A\) then there is a vertex-preserving bijection between the isomorphism classes of simple \(A\)-modules lying over a simple \(B\)-module \(S\) with vertex \(P\) and the isomorphism classes of simple \(A^*\)-modules lying over the simple \(B^*\)-module \(S^*\) corresponding to \(S\) in Okuyama's theorem. This result extends a theorem by \textit{A. Laradji} [J. Algebra 324, No. 4, 749-757 (2010; Zbl 1232.20015)].
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 6154897
zbMATH Keywords: finite groups
zbMATH Keywords: blocks
zbMATH Keywords: defect groups
zbMATH Keywords: vertices
zbMATH Keywords: sources
zbMATH Keywords: Morita equivalences
zbMATH Keywords: Brauer correspondence
zbMATH Keywords: Harris-Knörr correspondence
zbMATH Keywords: simple modules
author: Morton E. Harris
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.05.023
OpenAlex ID: W1977918132
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DOI: 10.1016/J.JALGEBRA.2012.05.023
description: scientific article; zbMATH DE number 6154897