The splitting principle, representations of GL_n and cohomology of flag varieties

In these talks, we shall se how the theory of the algebraic representations of the general linear group $GL_n$ and the cohomology of flag varieties (like Grassmann varieties) can be understood by a method called `splitting principle'. Another incarnation of this method is the following: starting from a monic polynomial $P\in k[T]$, one may introduce  splitting field $K$
which is an extension of $k$ such that $P$ writes as $(X - x_1) (X - x_2) \dots (X - x_n) in $K[T].  In representation theory, the role of the roots  $x_i$ if $P$ shall be played by characters of the diagonal torus of $GL_n$ and in the cohomological study of flag varieties, these $x_i$ will be related to some line bundles (i,e., vector bundles of rank 1) on the complete flag variety.
An application of this method is the proof of some formulas for Chern classes by reducing to very simple cases. We shall see that .............(contd.) 
satisfy some `splitting principle'.