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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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