Generalized symmetric tensors and related topics

Abstract

Let 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 = ⊗ n V. 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) П n i=1 a iσ(i) .