Question

What is the exact relationship between principal bundles, representations, and vector bundles?

Answer 1

Let G be an algebraic group (or, since the question was tagged as differential geometry, a Lie group). Then if we're given a principal G-bundle $E_G$ and a representation V of G, we get a vector bundle out of this through the associated bundle construction: $(E_G \times V)/G$ is a vector bundle with generic fiber V. Here, G acts on $E_G \times V$ as $g(x,v) = (xg^{-1},gv).$

This shows that fixing a G-bundle determines an exact tensor functor from the category of representations of G to the category of vector bundles. There's a converse to this which says that giving an exact tensor functor from representations of G to vector bundles is equivalent to a G-bundle.

Answer 2

From a $G$-principal bundle $E\to B$ and a representation $V$ of $G$ you can construct a vector bundle $E\times_GV\to B$. A vector bundle $\mathcal E$ with fiber $V$, on the other hand, gives you a $GL(V)$-principal bundle by taking the bundle $F(\mathcal E)$ of frames in $\mathcal E$, and you can reconstruct $\mathcal E$ from $F(\mathcal E)$ and the tautological representation of $GL(V)$ on $V$.

Answer 3

Just for fun, I'm going to give a fancy reinterpretation of Mike's answer. There's nothing new here, but it's fun to say it this way:

A principal $G$-bundle $P$ on a space $X$ is the same thing as a map $f_P: X \to BG$ to the classifying stack $BG$. We can identify $BG$ with the quotient stack $[\operatorname{pt}/G]$, so a vector bundle $V$ on $BG$ is exactly the same thing as a $G$-equivariant vector bundle on the point, i.e., $V$ is a vector space with a $G$-action, i.e. $V$ is a representation of $G$. The vector bundle associated to $P$ and $V$ is precisely the pullback bundle $f_P^*V$.

Answer 4

Just to be pedantic, these notions are for finite dimensional groups and representations. In infinite dimensions (either group or representation) one has to be a little bit more careful, see this question.

Answer 5

To add some small point about the converse to the above answers for discrete groups: An n-dimensional vector bundle on X equipped with a flat connection is the same thing as a representation of the fundamental groupoid of X in GL_n. In this case, monodromy around paths is an invariant of homotopy class of path, and so any homotopy class of path produces an isomorphism between the vector spaces over start and end points.