On Young modules of defect \(2\) blocks of symmetric group algebras
type: Item
instance of: scholarly article
title: On Young modules of defect \(2\) blocks of symmetric group algebras
zbMATH Open document ID: 0957.20008
author: Joseph Chuang
author: Kai Meng Tan
published in: Journal of Algebra
publication date: 2000-04-03T00:00:00Z
review text: Let \(\Sigma_n\) denote the symmetric group of degree \(n\) and let \(k\) denote a field of prime characteristic \(p\). Given a partition \(\lambda\) of \(n\), let \(\Sigma_\lambda\) denote the Young subgroup of \(\Sigma_n\) corresponding to \(\lambda\). It is a direct product of symmetric groups that correspond to the parts of \(\lambda\). Let \(M^\lambda\) be the permutation module for \(\Sigma_n\) over \(k\) that arises from the permutation action on the cosets of \(\Sigma_\lambda\). The Specht module \(S^\lambda\) is a submodule of \(M^\lambda\) whose head \(D^\lambda\) is simple if \(\lambda\) is \(p\)-regular, that is, if no part of \(\lambda\) is repeated more than \(p-1\) times. As \(\lambda\) runs over the \(p\)-regular partitions of \(n\), the corresponding \(D^\lambda\) run over all the non-isomorphic simple \(k\Sigma_n\)-modules. The Young submodule \(Y^\lambda\) is the unique indecomposable direct summand of \(M^\lambda\) that contains \(S^\lambda\) as a submodule. \(Y^\lambda\) may be lifted to a characteristic zero module, unique up to isomorphism, and we may speak of the (ordinary) character \(\text{ch}(Y^\lambda)\), which contains the irreducible character \(\chi^\lambda\) according to the usual parametrization of the irreducible characters of \(\Sigma_n\). In the paper under review, the authors investigate \(Y^\lambda\) when it belongs to a \(p\)-block of \(k\Sigma_n\) of defect 2. Such blocks only occur when \(p\) is odd. Much is known about these blocks following work of K.~Erdmann, S.~Martin, M.~J.~Richards and J.~Scopes. For example, there are bounds for the entries of the Cartan matrix of such a block. Under this defect 2 hypothesis, the authors determine the irreducible constituents of \(\text{ch}(Y^\lambda)\), which occur with multiplicity at most 1. They also describe the Loewy and socle layers of \(Y^\lambda\). The third section of the paper contains additional results relating to Young modules.
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 1400114
zbMATH Keywords: symmetric groups
zbMATH Keywords: blocks of defect \(2\)
zbMATH Keywords: Young modules
zbMATH Keywords: Young subgroups
zbMATH Keywords: permutation modules
zbMATH Keywords: Specht modules
zbMATH Keywords: irreducible characters
zbMATH Keywords: Cartan matrices
MaRDI profile type: Publication
full work available at URL: jabr.1999.8022
OpenAlex ID: W2007392242
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DOI: 10.1006/JABR.1999.8022
description: scientific article; zbMATH DE number 1400114