On the orthogonal bases of symmetry classes

Let V be an n-dimensional complex inner product space and let {e₁,...,e_n} be an orthonormal basis of V. Suppose G is a permutation group of degree m and χ is an irreducible character of G. We denote by V_χ(G) the symmetry class of tensors associated with G and χ. In this article, we discuss the problem of existing orthogonal bases for V_χ(G) consisting of symmetrized decomposable tensors e*_α.