On the orthogonal dimension of orbital sets
type: Item
instance of: scholarly article
title: On the orthogonal dimension of orbital sets
zbMATH Open document ID: 1077.15025
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
review text: Let \(V\) be a complex inner product vector space and let \(e_1,\dots, e_n\) be an orthonormal basis of \(V\). Let \(\lambda\) be a complex irreducible representation of a subgroup \(G\) of the full symmetric group \(S_m\). Consider on \(V^{\otimes m}\) the linear operator \(T(G,\lambda):=\frac{\lambda (\text{id})}{| G| }\sum_{\sigma\in G}\lambda(\sigma)P(\sigma)\) where \(P(\sigma)\) is defined by \(P(\sigma)(v_1\otimes\dots\otimes v_m)= v_{\sigma^{-1}(1)}\otimes\dots\otimes v_{\sigma^{-1}(m)}\). The authors give a combinatorial necessary and sufficient condition for orthogonality of critical decomposable symmetric tensors \(e^*_\alpha:=T(G,\lambda)(e_{\alpha(1)}\otimes\dots\otimes e_{\alpha(m)})\) and \(e^*_\beta:=T(G,\lambda)(e_{\beta(1)}\otimes\dots\otimes e_{\beta(m)})\). The notion of sign-uniform partition is introduced and the set of such partitions is described. The characterization of the sign-uniform partitions is used for obtaining more manageable conditions of orthogonality of \(e^*_\alpha\) and \(e^*_\beta\). The concept of orthogonal dimension of a finite set of nonzero vectors is introduced. The orthogonal dimension of critical orbital sets is computed for a class of irreducible characters \(\lambda\).
reviewed by: Vladimir L. Popov
Mathematics Subject Classification ID: 15A72
zbMATH DE Number: 2173143
zbMATH Keywords: decomposable tensors
zbMATH Keywords: critical orbital sets
zbMATH Keywords: Schur polynomials
zbMATH Keywords: induced orthogonal bases
zbMATH Keywords: immanents
author: J. A. Dias Da Silva
author: Maria M. Torres
MaRDI profile type: Publication
full work available at URL: j.laa.2003.11.005
OpenAlex ID: W2137023635
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DOI: 10.1016/J.LAA.2003.11.005
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description: scientific article; zbMATH DE number 2173143