Please explain this concept: Reducible and irreducible representations of a group, invariant subspaces of a group.
I suspect it will be easiest to understand with a finite group and a linear representation, but if there's extra insight to be gained by considering infinite, continuous, or non-linear groups, have at it.
I hope to understand things about the structure of the group itself, and I see linear algebra as merely a means to that end.
If possible, please give references to important related concepts.
In your answers, feel free to assume knowledge of the basics of group theory: Group axioms, Abelian/Nonabelian, invariant subgroups, homo/isomorphisms, related algebraic structures, etc.