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On the orthogonal basis of symmetry classes of tensors. (English) Zbl 0989.20014
Let $$V$$ be an $$n$$-dimensional inner product vector space over the complex field $$\mathbb{C}$$. Let $$G$$ be a subgroup of the symmetric group $$S_m$$ on $$m$$ letters and $$\chi$$ be an irreducible complex character of $$G$$. For $$g\in G$$, the mapping $$P_g\colon\bigotimes^m_1V\to\bigotimes^m_1V$$ is defined by $$P_g(v_1\otimes\cdots\otimes v_m)=v_{g^{-1}(1)}\otimes\cdots\otimes v_{g^{-1}(m)}$$ when $$v_i\in V$$. The symmetry class of tensors associated with $$G$$ and $$\chi$$ is the image of $$T(G,\chi)={\chi(1)\over|G|}\sum_{g\in G}\chi(g)P_g$$ which is denoted by $$V_\chi(G)$$. The tensor $$T(G,\chi)(v_1\otimes\cdots\otimes v_m)$$, $$v_i\in V$$, is denoted by $$v_1*\cdots*v_m$$ and is called a decomposable symmetrized tensor. For $$\alpha$$ a mapping from the set $$\{1,2,\dots,m\}$$ into $$\{1,2,\dots,n\}$$ we set $$e^*_\alpha=e_{\alpha(1)}*\cdots*e_{\alpha(m)}$$, where $$\{e_1,\dots,e_n\}$$ is an orthogonal basis of $$V$$. A basis of $$V_\chi(G)$$ consisting of the decomposable symmetrized tensors $$e^*_\alpha$$ is called an orthogonal $$*$$-basis. In [R. R. Holmes and T.-Y. Tam, Linear Multilinear Algebra 32, No. 1, 21-31 (1992; Zbl 0762.15015)] necessary and sufficient conditions are found such that $$V_\chi(G)$$ has an orthogonal $$*$$-basis, where $$G$$ is the dihedral group $$D_{2k}$$ and $$\chi\in\text{Irr}(D_{2k})$$. Also in [M. R. Darafsheh and M. R. Pournaki, Linear Multilinear Algebra 47, No. 2, 137-149 (2000; Zbl 0964.20006)] necessary and sufficient conditions are found such that $$V\chi(G)$$ has an orthogonal $$*$$-basis, where $$G$$ is the dicyclic group $$T_{4k}$$ and $$\chi\in\text{Irr}(T_{4k})$$.
In these two articles the arguments depend on the cycle structures of elements of $$G$$ as a subgroup of $$S_m$$. In the paper under review the authors prove that the necessary conditions found in the above mentioned papers do not depend on the permutation structure of these groups.

##### MSC:
 20C30 Representations of finite symmetric groups 15A69 Multilinear algebra, tensor calculus 20B35 Subgroups of symmetric groups
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##### References:
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