By using the action of a Galois group on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, that each Galois character is a positive linear combination of the Galois irreducible characters. It is shown that whenever the complex characters of the groups of a tower produce a positive self-dual Hopf algebra (PSH), Galois characters of the groups of the tower also produce a PSH. Then we will classify the Galois characters and Galois classes of the general linear groups over finite fields. In the end, we will precisely indicate the isomorphism between the PSH of Galois characters and a certain tensor product of Hopf algebras isomorphic to symmetric functions. 查看全文>>