On the Orthogonal Basis of Symmetry Classes of Tensors

A necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Pournaki (in press, Linear and Multilinear Algebra) and R. R. Holmes and T. Y. Tam (1992, Linear and Multilinear Algebra32, 21–31). These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups.