On the cyclic symmetry classes
type: Item
instance of: scholarly article
title: On the cyclic symmetry classes
zbMATH Open document ID: 0858.20007
published in: Journal of Algebra
publication date: 1997-03-16T00:00:00Z
review text: Let \(\otimes^mV\) denote the \(m\)th tensor power of an \(n\)-dimensional complex vector space \(V\). For each permutation \(\pi\) in the symmetric group on \(m\) letters one can define a linear mapping \(P(\pi)\) on \({\otimes^m}V\) by defining \(P(\pi)(v_1\otimes\dots\otimes v_m)=v_{\pi^{-1}(1)}\otimes\dots\otimes v_{\pi^{-1}(m)}\) and extending it to the whole of \(\otimes^mV\) by linearity. Now for a subgroup \(G\) of \(S_m\) and an ordinary irreducible character \(\chi\) of \(G\) we can define the linear mapping \(T(G,\chi)\) on \({\otimes^m}V\) by setting \(T(G,\chi)={\chi(1)\over|G|}\sum_{\pi\in G}\chi(\pi)P(\pi)\). In this case the image of \(\otimes^mV\) under \(T(G,\chi)\) is called the symmetry class of tensors associated with \(G\) and \(\chi\) and is denoted by \(V_\chi(G)\). The purpose of the paper is to obtain a formula for the dimension of \(V_{\chi}(G)\) in the case when \(G\) is the direct product of cyclic subgroups. This result extends the result of \textit{L. J. Cummings} [in J. Algebra 40, 401-405 (1976; Zbl 0354.15012)] who considered only a cyclic subgroup \(G\) of \(S_m\) and where \(\chi\) is the trivial or a primitive linear character of \(G\).
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 922335
zbMATH Keywords: symmetric groups
zbMATH Keywords: ordinary irreducible characters
zbMATH Keywords: symmetry class of tensors
zbMATH Keywords: dimension
zbMATH Keywords: direct product of cyclic subgroups
author: Tin-Yau Tam
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1996.0189
OpenAlex ID: W2079432793
DOI: 10.1006/JABR.1996.0189
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description: scientific article; zbMATH DE number 922335