Generalized symmetric tensors and related topics

AbstractLet T = ∑σ∈G M(σ) ⊗ P(σ), where M is a unitary matrix representation of the group G as unitary linear operators on a space U, and P(σ) the permutation operator on W = ⊗nV. A generalized symmetric tensor is a tensor of the form T(u ⊗ w), where u ∈ 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) = ∑σ∈G M(σ) Пni=1 aiσ(i)