On the Orthogonal Bases of Symmetry Classes

Let V be an n-dimensional complex inner product space and let {e1,…,en} 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 Vχ(G) the symmetry class of tensors associated with G and χ. In this article, we discuss the problem of existing orthogonal bases for Vχ(G) consisting of symmetrized decomposable tensors e*α.