Morita equivalence for blocks of Hecke algebras of symmetric groups
type: Item
instance of: scholarly article
title: Morita equivalence for blocks of Hecke algebras of symmetric groups
zbMATH Open document ID: 0883.20009
author: Thomas Jost
published in: Journal of Algebra
publication date: 1998-03-22T00:00:00Z
review text: This paper deals with the Hecke algebras which arise as endomorphism rings of the permutation representation of the finite general linear group on the cosets of the Borel subgroup which can also be viewed as specialization of the generic Hecke algebra \({\mathcal H}_q(S_n)\) of the symmetric group \(S_n\). Connections between representations of \({\mathcal H}_q(S_n)\) and \(S_n\) have been developed by \textit{R. Dipper} and \textit{G. James} [Proc. Lond. Math. Soc., III. Ser. 52, 20-52 (1986; Zbl 0587.20007) and ibid. 54, 57-82 (1987; Zbl 0615.20009)] and this paper reveals more of those connections. In particular it was shown that the bases of \(q\)-Specht modules and \(q\)-permutation modules for \({\mathcal H}_q(S_n)\) are \(q\)-analogues of the branching of the bases of the corresponding Specht and permutation modules for \(S_n\). Also a \(q\)-analogue of the branching theorem of Specht modules is proved for \(q\)-Specht modules over an arbitrary field. The main theme of this paper is to generalize a result proved by \textit{J. Scopes} [J. Algebra 142, No. 2, 441-455 (1991; Zbl 0736.20008)] on Morita equivalence of blocks of symmetric groups. The notion of \((w,k)\)-pair of blocks of \(S_n\) is generalized for blocks of \({\mathcal H}_{R,q}(S_n)\) and it is proved that blocks of \({\mathcal H}_{F,q}(S_n)\) forming a \((w,k)\)-pair have the same decomposition matrix.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1053881
zbMATH Keywords: Specht modules
zbMATH Keywords: permutation modules
zbMATH Keywords: Hecke algebras
zbMATH Keywords: endomorphism rings
zbMATH Keywords: representations
zbMATH Keywords: branching theorem
zbMATH Keywords: Morita equivalences
zbMATH Keywords: blocks of symmetric groups
zbMATH Keywords: decomposition matrices
reviewed by: Ahmed A. Khammash
MaRDI profile type: Publication
full work available at URL: jabr.1996.6988
OpenAlex ID: W2006558862
cites work: Q4056148
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cites work: On the representation theory of the symmetric groups and associated Hecke algebras
cites work: Specht modules and symmetric groups
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DOI: 10.1006/JABR.1996.6988
description: scientific article; zbMATH DE number 1053881