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*)