Induced operators on the generalized symmetry classes of tensors

Let V be a unitary space. Suppose G is a subgroup of the symmetric group of degree m and Λ is an irreducible unitary representation of G over a vector space U. Consider the generalized symmetrizer on the tensor space U⊗V⊗m, SΛ(u⊗v⊗)=1|G|∑σ∈GΛ(σ)u⊗vσ−1(1)⊗⋯⊗vσ−1(m) defined by G and Λ. The image of U⊗V⊗m under the map SΛ is called the generalized symmetry class of tensors associated with G and Λ and is denoted by VΛ(G). The elements in VΛ(G) of the form SΛ(u⊗v⊗) are called generalized decomposable tensors and are denoted by u⊛v⊛. For any linear operator T acting on V, there is a unique induced operator KΛ(T) acting on VΛ(G) satisfying KΛ(T)(u⊗v⊗)=u⊛Tv1⊛⋯⊛Tvm. If dimU=1, then KΛ(T) reduces to Kλ(T), induced operator on symmetry class of tensors Vλ(G). In this paper, the basic properties of the induced operator KΛ(T) are studied. Also some well-known results on the classical Schur functions will be extended to the case of generalized Schur functions. Keywords