Definition
Let be a principal G-bundle.
Let be a representation of .
The associated vector bundle with fiber is
Local trivializations
For any local trivialization of the vector bundle , then we can look at the associated vector bundle
The action on is trivial by the trivialization, so
Note that
since we have the map
Then we claim that the candidate inverse map
is well-defined.
This is since any element can be written as .
Thus, the local trivialization of the associated bundle is
Associated bundles with normal subgroups
Let be a group, and a normal subgroup.
Then is a natural principal bundle.
Thus, the associated bundle of a representation gives the vector bundle