I'm currently learning about Lie Group representations and I have two questions about them.
What is the inuitive difference between group actions and representations? Is the only real difference that in the former case, a group acts on a manifold, and the latter case, it acts on a vector space? And why are we so intensely focused on groups acting on vector spaces in particular?
What are intuitive, geometric examples of representations? For example, elements of the rotation group $\text{SO}(2)$ act on $\mathbb{R}^2$ by rotating it. Elements of $\text{SO}(3)$ act on $\mathbb{R}^3$ by rotating it. But we already studied both of these examples without developing the whole theory of representations. Are there any examples of representations that one can visualize (for example, not the adjoint representation) that are not trivial like the examples above?
Any clarification about these questions would be much appreciated. Thanks for the help!