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Generalized symmetric tensors and related topics. (English) Zbl 0846.15012
For $$T= \sum_{\sigma\in G} M(\sigma) \otimes {\mathcal P} (\sigma)$$, where $$M$$ is a unitary matrix representation of the group $$G$$ as unitary linear operators on a space $$U$$, and $${\mathcal P} (\sigma)$$ the permutation operator on $$W= \otimes^n V$$, a generalized symmetric tensor is defined as a tensor of the form $$T(u \otimes w)$$, where $$u\in U$$ and $$w$$ is a decomposable tensor of $$W$$. The author discusses the properties of generalized symmetric tensors. The conditions for two generalized symmetric tensors to be equal are obtained. Finally, a new characterization of the set $$A$$ satisfying $${\mathcal M} (AX)= {\mathcal M} (X)$$ for arbitrary $$X$$ with $${\mathcal M} (A)= \sum_{\sigma\in G} M(\sigma) \prod^n_{i=1} a_{i\sigma (i)}$$ is presented.

##### MSC:
 15A72 Vector and tensor algebra, theory of invariants
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##### References:
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