Generalized symmetric tensors and related topics

Let T = Sigma(sigma is an element of G) M(sigma) x P(sigma), where M is a unitary matrix representation of the group G as unitary linear operators on a space, U, and P(sigma) the permutation operator on W = x(n) V. A generalized symmetric tensor is a tensor of the form T(u x w), where u is an element of U and w is a decomposable tensor of W. We discuss the properties of generalized symmetric tensors. The conditions when two generalized symmetric tensors are equal are also considered. We present a new characterization of the set of A satisfying M(AX) = M(X) for arbitrary X with M(A)= Sigma(sigma is an element of G) M(sigma) Pi(i = 1)(n) a(i sigma)(i).