Connections between the representations of the symmetric groups and the symplectic group in characteristic \(2\)
type: Item
instance of: scholarly article
title: Connections between the representations of the symmetric groups and the symplectic group in characteristic \(2\)
zbMATH Open document ID: 0943.20013
author: Rod Gow
author: Alexander S. Kleshchev
published in: Journal of Algebra
publication date: 2000-02-01T00:00:00Z
review text: Let \(K\) be a field and let \(\text{SP}_{2l}(K)\) be the symplectic group defined over a non-degenerate symplectic space \(E\) of dimension \(2l\) over \(K\). The group \(\text{SP}_{2l}(K)\) acts on the exterior powers \(\bigwedge^rE\) and the composition factors of these modules are known. These modules are labelled by the fundamental domain weight of \(\text{SP}_{2l}(K)\). The irreducible \(\text{SP}_{2l}(K)\)-module with highest weight \(\omega_r\) is denoted by \(L(\omega_r)\), where \(\omega_r\) is a fundamental weight. The paper under review is divided into three sections and throughout it is assumed that \(K\) has characteristic 2. In section one the authors consider the symmetric group \(S_{2l+1}\) of degree \(2l+1\) as a subgroup of \(\text{SP}_{2l}(K)\) and prove that each composition factor of the action of \(\text{SP}_{2l}(K)\) on \(\bigwedge^rE\) remains irreducible when restricted to \(S_{2l+1}\). In section two they consider the symmetric group on \(2l+2\) letters as a subgroup of \(\text{SP}_{2l}(K)\) and show that if \(l\) is even then the irreducible module \(L(\omega_1+\omega_{2r})\) with highest weight \(\omega_1+\omega_{2r}\) is irreducible under the action of \(S_{2l+2}\). In section three the authors obtain general results about tensor products of irreducible modules of \(\text{SP}_{2l}(K)\) and the spin module.
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C33
zbMATH DE Number: 1384438
zbMATH Keywords: symplectic groups
zbMATH Keywords: exterior powers
zbMATH Keywords: composition factors
zbMATH Keywords: highest weights
zbMATH Keywords: symmetric groups
zbMATH Keywords: tensor products
zbMATH Keywords: irreducible modules
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1999.7943
OpenAlex ID: W2012998238
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DOI: 10.1006/JABR.1999.7943
description: scientific article; zbMATH DE number 1384438