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A high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors. (English) Zbl 0735.15022
The authors present a high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors. This class provides a counter- example to the claim presented by M. Marcus and J. Chollet [ibid. 19, 133-140 (1985; Zbl 0591.15020)] that if \(e_ 1,e_ 2,\ldots,e_ n\) be an orthogonal basis of an \(n\)-dimensional unitary vector space \(V\), then there exists a subset \(S\) of \(\Gamma_{m,n}\), the set of all sequences of length \(m\) taken from the integers \(1,2,\dots,n\), such that the \(e^*_ \beta\) for \(\beta\) in \(S\) form an orthogonal basis of \(V_ \lambda(G)\), the symmetry class of tensors associated with \(G\), a subgroup of \(S_ m\), and \(\lambda\), an irreducible complex valued character of \(G\), if an only if \(\lambda(id)=1.\)
Moreover, an oversight on the proof of a theorem of Marcus and Chollet [loc. cit.] is pointed out which leads to their false claim.

MSC:
15A72 Vector and tensor algebra, theory of invariants
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[1] DOI: 10.1080/03081088608817710 · Zbl 0591.15020 · doi:10.1080/03081088608817710
[2] DOI: 10.1016/S0024-3795(73)80004-7 · Zbl 0283.15004 · doi:10.1016/S0024-3795(73)80004-7
[3] DOI: 10.1080/03081087308817022 · Zbl 0284.15025 · doi:10.1080/03081087308817022
[4] DOI: 10.1080/03081087808817251 · Zbl 0395.15013 · doi:10.1080/03081087808817251
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