# Representation Theory and Fourier Analysis

In Some Basics of Fourier Analysis we introduced some of the basic ideas in Fourier analysis, which is ubiquitous in many parts of both pure and applied math. In this post we look at these same ideas from a different point of view, that of representation theory.

Representation theory is a way of studying group theory by turning it into linear algebra, which in many cases is more familiar to us and easier to study.

A (linear) representation is just a group homomorphism from some group $G$ we’re interested in, to the group of linear transformations of some vector space. If the vector space has some finite dimension $n$, the group of its linear transformations can be expressed as the group of $n \times n$ matrices with nonzero determinant, also known as $\mathrm{GL}_n(k)$ ( $k$ here is the field of scalars of our vector space).

In this post, we will focus on infinite-dimensional representation theory. In other words, we will be looking at homomorphisms of a group $G$ to the group of linear transformations of an infinite-dimensional vector space.

“Infinite-dimensional vector spaces” shouldn’t scare us – in fact many of us encounter them in basic math. Functions are examples of such. After all, vectors are merely things we can scale and add to form linear combinations. Functions satisfy that too. That being said, if we are dealing with infinity we will often need to make use of the tools of analysis. Hence functional analysis is often referred to as “infinite-dimensional linear algebra” (see also Metric, Norm, and Inner Product).

Just as a vector $v$ has components $v_i$ indexed by $i$, a function $f$ has values $f(x)$ indexed by $x$. If we are working over uncountable things, instead of summation we may use integration.

We will also focus on unitary representations in this post. This means that the linear transformations are further required to preserve a complex inner product (which takes the form of an integral) on the vector space. To facilitate this, our functions must be square-integrable.

Consider the group of real numbers $\mathbb{R}$ (under addition). We want to use representation theory to study this group. For our purposes we want the square-integrable functions on some quotient of $\mathbb{R}$ as our vector space. It comes with an action of $\mathbb{R}$, by translation. In other words, an element $a$ of $\mathbb{R}$ acts on our function $f(x)$ by sending it to the new function $f(x+a)$.

So what is this quotient of $\mathbb{R}$ that our functions will live on? For now let us choose the integers $\mathbb{Z}$. The quotient $\mathbb{R}/\mathbb{Z}$ is the circle, and functions on it are periodic functions.

To recap: We have a representation of the group $\mathbb{R}$ (the real line under addition) as linear transformations (also called linear operators) of the vector space of square-integrable functions on the circle.

In representation theory, we will often decompose a representation into a direct sum of irreducible representations. Irreducible means it contains no “subrepresentation” on a smaller vector space. The irreducible representations are the “building blocks” of other representations, so it is quite helpful to study them.

How do we decompose our representation into irreducible representations? Consider the representation of $\mathbb{R}$ on the vector space $\mathbb{C}$ (the complex numbers) where a real number $a$ acts by multiplying a complex number $z$ by $e^{2\pi i k a}$, for $k$ an integer. This representation is irreducible.

If this looks familiar, this is just the Fourier series expansion for a periodic function. So a Fourier series expansion is just an expression of the decomposition of the representation of R into irreducible representations!

What if we chose a different vector space instead? It might have been the more straightforward choice to represent $\mathbb{R}$ via functions on $\mathbb{R}$ itself instead of on the circle $\mathbb{R}/\mathbb{Z}$. That may be true, but in this case our decomposition into irreducibles is not countable! The irreducible representations into which this other representation decomposes is the one where a real number $a$ acts on $\mathbb{C}$ by multiplication by $e^{2 \pi i k a}$ where $k$ is now a real number, not necessarily an integer. So it’s not indexed by a countable set.

This should also look familiar to those who know Fourier analysis: This is the Fourier transform of a square-integrable function on $\mathbb{R}$.

So now we can see that concepts in Fourier analysis can also be phrased in terms of representations. Important theorems like the Plancherel theorem, for example, also may be understood as an isomorphism between the representations we gave and other representations on functions of the indices. We also have the Poisson summation in Fourier analysis. In representation theory this is an equality obtained from calculating the trace in two ways, as a sum over representations and as a sum over conjugacy classes.

Now we see how Fourier analysis is related to the infinite-dimensional representation theory of the group $\mathbb{R}$ (one can also see this as the infinite-dimensional representation theory of the circle, i.e. the group $\mathbb{R}/\mathbb{Z}$ – the article “Harmonic Analysis and Group Representations” by James Arthur discusses this point of view). What if we consider other groups instead, like, say, $\mathrm{GL}_n(\mathbb{R})$ or $\mathrm{SL}_n(\mathbb{R})$ (or $\mathbb{R}$ can be replaced by other rings even)?

Things get more complicated, for example the group may not be abelian. Since we used integration so much, we also need an analogue for it. So we need to know much about group theory and analysis and everything in between for this.

These questions have been much explored for the kinds of groups called “reductive”, which are closely related to Lie groups. They include the examples of $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{SL}_n(\mathbb{R})$ earlier, as well as certain other groups we have discussed in previous posts such as the orthogonal and unitary (see also Rotations in Three Dimensions). There is a theory for these groups analogous to what I have discussed in this post, and hopefully this will be discussed more in future blog posts here.

References:

Representation theory on Wikipedia

Representation of a Lie group on Wikipedia

Fourier analysis on Wikipedia

Harmonic analysis on Wikipedia

Plancherel theorem on Wikipedia

Poisson summation formula on Wikipedia

An Introduction to the Trace Formula by James Arthur

Harmonic Analysis and Group Representations by James Arthur