Tensor products and dimensions of simple modules for symmetric groups
type: Item
instance of: scholarly article
title: Tensor products and dimensions of simple modules for symmetric groups
zbMATH Open document ID: 0849.20008
DOI: 10.1007/BF02567828
author: Karin Erdmann
published in: Manuscripta Mathematica
publication date: 1996-11-03T00:00:00Z
full work available at URL: https://eudml.org/doc/156133
review text: Let \(K\) be an algebraically closed field of characteristic \(p>0\) and let \(D^\lambda\) be the simple module of the symmetric group \(S_n\) over \(K\), where \(\lambda\) is a \(p\)-regular partition of \(r\). Then, the dimension of \(D^\lambda\) for \(\lambda\) with at most \(n\) parts are the same as the multiplicities of direct summands of \(E^{\otimes r}\) where \(E\) is the natural \(n\)-dimensional module for \(\text{GL}_n(K)\). The problem of finding the dimension of \(D^\lambda\) is a major open problem; thus that the author has been able to calculate recursive formulae for these multiplicities in the case \(n=2\) leading to a determination of their generating functions is of some interest. She shows that these are rational functions and remarkably, explicit expressions in terms of Chebyshev polynomials are obtained. These in turn lead to explicit formulae for the dimensions of the \(D^\lambda\).
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 871773
zbMATH Keywords: simple modules
zbMATH Keywords: symmetric groups
zbMATH Keywords: \(p\)-regular partitions
zbMATH Keywords: multiplicities of direct summands
zbMATH Keywords: dimensions
zbMATH Keywords: recursive formulae
zbMATH Keywords: generating functions
zbMATH Keywords: rational functions
zbMATH Keywords: explicit expressions
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: explicit formulae
MaRDI profile type: Publication
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OpenAlex ID: W2008713774
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description: scientific article; zbMATH DE number 871773