On the orthogonal basis of symmetry classes of tensors.

*(English)*Zbl 0989.20014Let \(V\) 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.

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.

Reviewer: Mohammad-Reza Darafsheh (Tehran)

##### MSC:

20C30 | Representations of finite symmetric groups |

15A69 | Multilinear algebra, tensor calculus |

20B35 | Subgroups of symmetric groups |

##### Keywords:

inner product vector spaces; symmetric groups; irreducible complex characters; symmetry classes of tensors; symmetrized tensors; orthogonal bases
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\textit{M. A. Shahabi} et al., J. Algebra 237, No. 2, 637--646 (2001; Zbl 0989.20014)

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##### References:

[1] | Cummings, L.J., Cyclic symmetry classes, J. algebra, 40, 401-405, (1976) · Zbl 0354.15012 |

[2] | M. R. Darafsheh, and, M. R. Pournaki, On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group, Linear and Multilinear Algebra, in press. · Zbl 0964.20006 |

[3] | Freese, R., Inequalities for generalized matrix functions based on arbitrary characters, Linear algebra appl., 7, 337-395, (1973) · Zbl 0283.15004 |

[4] | Holmes, R.R.; Tam, T.Y., Symmetry classes of tensors associated with certain groups, Linear and multilinear algebra, 32, 21-31, (1992) · Zbl 0762.15015 |

[5] | James, G.; Liebeck, M., Representations and characters of groups, (1993), Cambridge Univ. Press Cambridge · Zbl 0792.20006 |

[6] | Merris, R., Multilinear algebra, (1997), Gordon & Breach New York |

[7] | M. Shahryari, On the orthogonal bases of symmetry classes, J. Algebra, in press. · Zbl 0935.15024 |

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