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∈g λ(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 ⊛ t v1 ⊛ · · · ⊛ t vm. if dim u = 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.