Associated vector bundles and Characteristic Classes Assume that $P\to M$ is a principal $G$-bundle where $G$ is some (compact) Matrix group. Let $\rho\colon G \to \operatorname{Gl}(\mathbb{R}^n)$ be the tautological representation and $\rho^\prime\colon G\to \operatorname{Gl}(V)$ some other representation.
Let
$$ E = P \times_{\rho}\mathbb{R}^n, \qquad \text{and} \qquad F = P\times_{\rho^{\prime}}V
$$
be the associated vector bundles.
Using the Chern-Weyl theory and the usual definitions can associate characteristic classes to $P$, $E$ and $F$.
Are they equal?
To make this less vague, assume $G$ is a complex Matrix group and let`s look at $c_1(P)$, $c_1(E)$ and $c_1(F)$. I assume that $c_1(P) = c_1(E)$. Is this also true for $c_1(F)$?
To make it even less vague, consider the Hopf bundle $\mathbb{S}^3 \to \mathbb{S}^2$, and let $\rho_n\colon U(1)\to \operatorname{Gl}(\mathbb{C})$ be the usual irreducible representations for $n\in\mathbb{Z}$. Then $c_1$ of these vector bundles are a complete invariant. Assuming $c_1(\operatorname{Hopf}) = -1$, what is
$$c_1\left(\operatorname{Hopf}\times_{\rho_n}\mathbb{C}\right)?
$$
I apologize if this is too trivial.
 A: The first Chern class of the dual $-L$ of a line bundle $L$ is the negative of the first Chern class of $L$. In general Chern classes of different associated vector bundles are unrelated, and when they are related the story is complicated. In your example, when you tensor line bundles, the first Chern class scales.
A: Where you seem to be going wrong here is assuming you can talk about the Chern classes of $P$.  That's not a well-defined concept (unless $G$ is $GL_n(\mathbb{C})$); at best, you're just fixing $\rho$ and defining the Chern classes of $P$ to be those of $E$.  
A principal $G$-bundle is always the pullback of the tautological bundle on $BG$.  The Chern classes of the associated bundles $E$ and $F$ are just the pullbacks of the associated bundles to $\rho$ and $\rho'$ on $BG$.  So, the Chern classes will always be the same if and only if the associated bundles on $\rho$ and $\rho'$ have the same Chern classes on $BG$.  As Ben McKay notes, there are some cases where you can write a formula for one in the terms of the other, but if you don't have a lot of control over the representation theory of $G$, there's no hope.
