Can someone explain the representation-theory tag? I like to think of myself as pretty interested in representation theory, but I have no idea what last line of the tag description for representation-theory is talking about. It reads,

Representation theory is a broad field that studies the symmetries of mathematical objects. A representation of an object is a way to "linearize" that object as a group of matrices. It's the non-commutative analog of classical Fourier transforms.

I have a couple of nits to pick about this description overall, but that's probably a topic for the meta. My question here is, how is representation theory a non-commutative analog of classical Fourier transforms? I'm not very familiar with the classical theory of Fourier transforms, and I don't normally think of them as having anything to do with representation theory or its most basic object: a homomorphism from an algebraic object to the endomorphisms of a linear object.
My understanding is that the classical Fourier transform is an isomorphism of one familiar function space into another one with a convenient basis, but I don't see how this is a representation, or how representation theory is an analog of it. Can someone explain the connection to me?
 A: Let $G$ be the circle, thought of as complex numbers of norm one. The Hilbert space $L^2(G)$ of complex valued functions on $G$ carries a natural unitary representation of $G$ by the rule
$$
g \cdot f(x) = f(g^{-1}x).
$$
But $G$ is compact, so representations are (Hilbert space) direct sums of irreducible representations, and $G$ is abelian, so its irreducible representations are one-dimensional -- in this case, they are in bijection with $\mathbb{Z}$, the map being $z \mapsto z^n$. Let's call this representation $\mathbb{C}(n)$.
So we have
$$
L^2(G) = \widehat{\bigoplus_{n \in \mathbb{Z}}} \mathbb{C}(n). 
$$
So we can write functions on $G$ as convergent sums of functions of the form $z \mapsto z^n$,  which is to say, we can write periodic functions on $\mathbb{R}$ with period one as convergent sums of functions $x \mapsto \exp(2 \pi i n x)$, which is of course the classical Fourier series.
Similar story with $\mathbb{R}$ and the Fourier transform, except you can't hope for something as nice as breaking a function into discrete pieces because the representations of $\mathbb{R}$ are indexed by the continuous set $\mathbb{R}$ . You instead get that $L^2(\mathbb{R})$ is a "direct integral" of one-dimensional spaces, which really is just language for stating the Fourier inversion formula.
