As can be guessed from some of my previous questions, I'm trying to understand, at the moment, the relationship between principal and their associated vector bundles. To this end I've been looking at $\mathbb{CP}^{n} = SU(n+1)/U(n)$ and trying to find the representation of $U(n)$ that gives $\Omega^{(1,0)}(\mathbb{CP}^n)$, for all $n$. For $n=1$, I worked it out using a transition function argument. But for $n>1$ this is proving very cumbersome. Can anyone point me in the direction of a more effective method.

Sketch of transition function method (as requested): So $\Omega^{(1,0)}(\mathbb{CP}^n)$ is the dual of $T^{(0,1)}(\mathbb{CP}^n)$, which is in turn, by definition, the dual of $T^{(1,0)}(\mathbb{CP}^n)$. Now $T^{(1,0)}(\mathbb{CP}^n)$ is defined to be the bundle whose transition functions are the Jacobian of the change-of-coordinate maps. Dual complex bundles have conjugate trans fns, and so, the Jacobian also provides the functions of $\Omega^{(1,0)}(\mathbb{CP}^n)$. These functions $$\phi_{ij}:U_i \cap U_j \to GL(n,\mathbb{C})$$ are elementary to calculate.

Now let $$ \psi:U_i \cap U_j \to U(n)$$ be the transition functions of the principal bundle $U(n) \to SU(n+1) \to \mathbb{CP}^n$. For the right representation $\pi:U(n) \to GL(n,\mathbb{C})$, we will have $$ \pi \circ \psi_{ij} = \phi_{ij}. $$ In the $n=1$, case both sets of transition functions are uncomplicated and it's easy to spot what $\pi$ must be. For $n>1$, it's proving to be more messy, and that's why I'm wondering if there's a smarter way of doing things.