Induced operators on symmetry classes of tensors

Let V be an n-dimensional Hilbert space. Suppose H is a subgroup of the symmetric group of degree m, and chi : H --> C is a character of degree 1 on H. Consider the symmetrizer on the tensor space circle times (m) V [GRAPHICS] defined by H and chi. The vector space V [GRAPHICS] is a subspace of circle times (m) V, called the symmetry class of tensors over V associated with H and chi. The elements in V-chi(m) (H) of the form S(v(1) circle times...circle timesv(m)) are called decomposable tensors and are denoted by v(1)*...*v(m). For any linear operator T acting on V, there is a (unique) induced operator K(T) acting on V-chi(m) (H) satisfying K(T)v(1)*...*v(m) = Tv(1)*...*Tv(m). In this paper, several basic problems on induced operators are studied.