As we all know, for any homogeneous space $G/H$ we have that the equivariant vector bundles over $G/H$ are characterized by the representations of $H$. Thus, for the the complex projective line $CP^1 \simeq SU(2)/U(1)$, it must hold that all its line bundles are indexed by the integers $L\_k$, for $k \in Z$, and more generally, its rank-$k$ (*equivarian*) vector bundles are of the form
$$
L\_{\bf z} = L\_{z\_1} \oplus \cdots \oplus L\_{z\_k}, {\text ~~~ for ~~~ } {\bf z} \in Z^k.
$$
Does this then extend to all the total flag manifolds $F(n)$, ie the spaces of the form 
$$
F(n) := SU(n)/(U(1)^{\otimes n-1}).
$$


Edit: I omitted the word equivariant by mistake and have now entered it as (*equivariant*)



Are all Equivariant Bundles of a Total Flag Manifold Constructable from Line Bundles?
I think perhaps the confusion stems from the following.


It is true that the category of $G$-equivariant $G$-bundles on $G/H$ is equivalent to the category of representations of $H$. The flag variety $\mathcal F l\_n$ can be realized as either $U(n)/U(1)^n$, or as $GL\_n(\mathbb C)/B\_n$, where $B\_n$ is the group of upper triangular matrices.


Thus $GL\_n(\mathbb C)$-equivariant vector bundles on $\mathcal Fl\_n$ are equivalent to representations of $B\_n$. This group is not reductive, and not every representation is a direct sum of 1-dimensional representations. Thus, not every equivariant vector bundle is a direct sum of line bundles.


On the other hand, the category of $U(n)$-equivariant principal $U(n)$-bundles is equivalent to representations of $U(1)^n$. This category is semisimple.


