The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of the general linear group. Using the Schur-Weyl duality, is there way to find the ring of representations of the general linear group in terms of the symmetric functions ?
There is an extensive literature on this subject. You're looking for the theory of Schur polynomials. These form a basis for the polynomials invariant under the symmetric group, and give the characters of the general linear group in characteristic zero. I suggest the following:
Fulton and Harris, Representation Theory.
Macdonald, Symmetric Functions and Hall Polynomials (2ed).
Sturmfels, Algorithms in Invariant Theory (esp. Ch 4).
Weyl, The Classical Groups.
On the other hand, if you're looking for the character theory of the finite groups $GL(n,q)$ in similar terms then I suggest
- Zelevinski, Representations of Finite Classical Groups.