Young diagrammatic methods in the homology of groups
type: Item
instance of: scholarly article
title: Young diagrammatic methods in the homology of groups
zbMATH Open document ID: 0697.20027
DOI: 10.1016/0022-4049(90)90006-4
author: Steven Schlicker
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: It was shown by H. Weyl that the irreducible representations of the classical groups GL(n,\({\mathbb{C}})\), Sp(n,\({\mathbb{C}})\), Sp(2n,\({\mathbb{C}})\) and \(SO(2n+1,{\mathbb{C}})\) are essentially in one to one correspondence with Young diagrams of depth at most n. Then K. Koike and I. Terada generalized the Littlewood-Richardson Rule for decomposing tensor products of representations for linear groups into sums of irreducible components to the symplectic and orthogonal groups. In the paper under review, the correspondence between representations and diagrams and the work of Koike and Terada are exploited to determine the homology of the groups \(SL(n,p^ 2)\), \(Sp(2n,p^ 2)\), \(SO(2n+1,p^ 2)\) with \({\mathbb{Z}}/p{\mathbb{Z}}\) coefficients in dimensions 1 through 5, with the exception of the group \(H_ 5(SL(n,p^ 2),{\mathbb{Z}}/p{\mathbb{Z}})\), for which only bounds are determined. In addition, the p primary component of the homology of the groups \(SL(n,p^ 2)\), \(Sp(2n,p^ 2)\), \(SO(2n+1,p^ 2)\) with \({\mathbb{Z}}\) coefficients is found in dimensions 1 through 5, with the exception of the group \(H_ 5(SL(n,p^ 2),{\mathbb{Z}})\), for which only bounds are determined. These groups are found by computing the relevant entries and differentials in certain spectral sequences involving the corresponding classical groups and the associated Lie algebras.
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 4142310
zbMATH Keywords: irreducible representations
zbMATH Keywords: classical groups
zbMATH Keywords: Young diagrams
zbMATH Keywords: Littlewood-Richardson Rule
zbMATH Keywords: tensor products of representations
zbMATH Keywords: symplectic and orthogonal groups
zbMATH Keywords: homology
zbMATH Keywords: spectral sequences
MaRDI profile type: Publication
full work available at URL: 0022-4049(90)90006-4
OpenAlex ID: W2082676854
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Wikidata QID: Q126629792
description: scientific article; zbMATH DE number 4142310