On a conjecture of Gow and Kleshchev concerning tensor products
type: Item
instance of: scholarly article
title: On a conjecture of Gow and Kleshchev concerning tensor products
zbMATH Open document ID: 0957.20009
published in: Journal of Algebra
publication date: 2000-09-20T00:00:00Z
review text: Let \(\Sigma_n\) denote the symmetric group of degree \(n\) and let \(F\) denote a field of characteristic \(2\). It is well known that the irreducible representations of \(\Sigma_n\) over \(F\) are labelled by \(2\)-regular partitions of \(n\), that is, partitions of \(n\) into distinct parts. Given a \(2\)-regular partition \(\lambda\) of \(n\), let \(D^\lambda\) denote the irreducible \(F\Sigma_n\)-module labelled by \(\lambda\). The trivial irreducible \(F\Sigma_n\)-module is the representation \(D^{(n)}\) corresponding to the partition \(\lambda=(n)\) of \(n\). When \(\lambda\) and \(\mu\) are partitions of \(n\) different from the 1-part partition \((n)\), an interesting question is to decide when the tensor product \(D^\lambda\otimes D^\mu\) is irreducible. \textit{C. Bessenrodt} and \textit{A. Kleshchev} have shown [in Bull. Lond. Math. Soc. 32, No. 3, 292-296 (2000)] that such an irreducible tensor product can occur only if \(n\) is even and all the parts of at least one of the partitions \(\lambda\) and \(\mu\) have the same parity. On the other hand, \textit{R. Gow} and \textit{A. Kleshchev} conjectured [in J. Algebra 221, No. 1, 60-89 (1999; Zbl 0943.20013)] that an irreducible tensor product of the kind considered above occurs if and only if \(n=4l+2\) for some positive integer \(l\) and, with a suitable choice of the partitions, \(\lambda=(2l+2,2l)\) and \(\mu\) is the 2-part partition \((4l+1-2j,2j+1)\), where \(0\leq j<l\). The irreducible \(F\Sigma_n\)-module \(D^{(2l+2,2l)}\) is in this case the spin module of dimension \(2^{2l}\), obtained from the basic spin representation in characteristic \(0\) of the double cover of \(\Sigma_n\). In the paper under review, the authors prove one part of the conjecture of Gow and Kleshchev, namely, that the tensor product \(D^{(2l+2,2l)}\otimes D^{(4l+1-2j,2j+1)}\) is irreducible and isomorphic to \(D^\nu\), where \(\nu\) is the 4-part partition \((2l+1-j,2l-j,j+1,j)\), known as the double of the partition \((4l…
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1471040
zbMATH Keywords: symmetric groups
zbMATH Keywords: irreducible tensor products
zbMATH Keywords: fields of characteristic \(2\)
zbMATH Keywords: 2-part partitions
zbMATH Keywords: irreducible representations
zbMATH Keywords: \(2\)-regular partitions
zbMATH Keywords: irreducible modules
author: John J. Graham
author: Gordon James
MaRDI profile type: Publication
full work available at URL: jabr.1999.8247
OpenAlex ID: W2092402401
Wikidata QID: Q122874985
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DOI: 10.1006/JABR.1999.8247
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description: scientific article; zbMATH DE number 1471040