On enumeration in classical invariant theory

In this paper we study the Frobenius characters of the invariant subspaces of the tensor powers of a representation V. The main result is a formula for these characters for a polynomial functor of V involving the characters for V. The main application is to representations V for which these characters are known. The best understood case is for V the vector representation of a symplectic group or special linear group. Other cases where there are some related results are the defining representations of the symmetric groups and the adjoint representations of the general linear groups. This formula uses methods of enumerative combinatorics and in particular is similar to the cycle index series of regular graphs.