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Orthogonal basis of the symmetry classes of tensors associated with the direct product of permutation groups. (English) Zbl 0954.20011

The authors consider an \(m\)-dimensional vectorspace \(V\) over the field \(\mathbb{C}\) of complex numbers with \(m\geq 2\). For a subspace \(W\) of the tensor product \(\otimes^n V\) they define the notion of an orthogonal basis of decomposable symmetrized tensors, called 0-basis. Therefore they use the well-known representation of \(S_n\) on \(\otimes^n V\). They prove their “main theorem”: Assume \(G_i\) is a subgroup of \(S_{n_i}\), \(i=1,\dots,k\), \(n=n_1+\cdots+n_k\), \(G=G_1\times\cdots\times G_k\), then there exists a 0-basis for \(\otimes^n V\) if and only if each \(\otimes^{n_i}V\) admits a 0-basis for \(i=1,\dots,k\).
Further necessary and sufficient conditions for the existence of a 0-basis are given associated to direct and central products of certain permutation groups.

MSC:

20C30 Representations of finite symmetric groups
15A69 Multilinear algebra, tensor calculus
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