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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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##### References:
 [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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