Irreducible representation intuition

Question

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.

Thanks

Answer 1

A representation of a group is a homomorphism from the group $G$ to the group of transformations (the “automorphism group”) of some mathematical object $A$.

If $A$ is a vector space then its automorphism group is the group of linear transformations of $A$, and the representation is called a linear representation. If $A$ is a finite $n$ dimensional vector space over a field $K$ then the group representation is a homomorphism from $G$ to $GL(n,K)$, the group of $n$ dimensional invertible matrices over $K$.

If two representations $a$ and $b$ are related by an automorphism $c$ such that $a=cbc^{-1}$ then $a$ and $b$ are equivalent representations. For linear representations, this just means that we have chosen a different basis for the underlying vector space.

Representations can be combined to form more complex representations. For example, two $2$ dimensional linear representations can be combined to create a $4$ dimensional linear representation, with each component acting on a $2$ dimensional sub-space. Going in the opposite direction, some representations can be broken down into simpler representations. These are called reducible representations. Representations that cannot be broken down into simpler representations are called irreducible representations.

A set of inequivalent and irreducible representations of a group is an interesting object which can be used to investigate properties of the group itself.