Study Irreducible Representations of SU(2) Using Fourier Series

Introduction and Prerequisites

Representation theory is important in various branches of mathematics and physics. When studying representation of finite groups, we have quite some algebra and combinatorics. When differentiation (more precisely, smoothness) joins the party, we have Lie group, involving calculus, linear algebra, geometry and much more. Especially, theories around and are of great importance. On one hand, they are those simplest non-elementary and high-dimensional Lie groups. On the other hand, they describes rotations in and respectively, which is "physically realistic". I believe students in physics have more to say.

In this post we develop a way to study irreducible representations of these two Lie groups, in a mathematician's way. I try my best to make sure that everything is down-to-earth, and everything can be "reduced" to 19th (pre-modern) mathematics.

Nevertheless, the reader has to be assumed to be familiar with elementary languages of representation theory (and you know that, there are a lot of abuse of language), which I think is not a problem because otherwise you wouldn't be reading this post. You need to recall eigenvalue theories in linear algebra, as well as Fourier series. We need the fact that the trigonometric system is complete. In other words trigonometric polynomials are dense in the space of continuous functions.

We will first study and a first classification of irreducible representations of follows at once. This is because we have an isomorphism This is to say, is a "double cover" of . To see this, notice that and as Lie groups, meanwhile can be considered as the definition.

Of course, by representation we mean finite dimensional and unitary representations.

Irreducible Representations of the Special Unitary Group

Indeed it seems we have nowhere to start. Instead of trying to find all of them, we will try to work on seemingly immediate representations and it turns out that they are all we are looking for.

Let be the trivial representation on and be the standard representation on , which is given by ordinary matrix multiplication. These representations are irreducible. We want to extend this family to for . It is natural to think about generate representations of higher dimensions through . Here are several ways available.

• Direct sum: . The dimension is and unfortunately, the representation is determined by each component so essentially there is no "new thing".
• Tensor product: . The dimension is which is way too big.
• Wedge product: . It stops at and we have to deal with . This can be annoying.
• Symmetric product: . The dimension is and it doesn't stop. Besides, it can be understood as homogeneous polynomials of degree in two variables. This is a fantastic choice. Besides we have so nothing is abruptly excluded.

Spaces of Homogeneous Polynomials

Put , which can be understood as the space of homogeneous polynomials of degree in variables and . therefore has a canonical basis And we will make use of it later.

Definition of the representation

For each , we have a left action In other words, where and is matrix multiplication. Each has matrix representation Then When there is no confusion, we will write , viewing itself as an automorphism of . One can also replace with but we are not studying that bigger one.

Since is a homogeneous map of degree as it is linear and is non-degenerate, we have . In other words, are -invariant. We now have a well-defined representation. Note so the representation is trivial, and yields linear maps. Again, nothing is abruptly excluded. Even more satisfyingly, those are all irreducible.

Irreducibility

Proposition 1. The representations are irreducible.

Proof. By Schur's lemma, we need to show that each -equivariant automorphism of is a non-zero multiple of the identity, i.e. for some . By definition, for each , we have for all . And for simplicity we write , realising as a linear transform of , instead of an element of .

The group can be complicated, but is simple and can be considered as a subgroup of in two ways. We show that these two ways are just enough to expose the irreducibility of .

First of all we embed into by Call the matrix right hand side . Then for all . This is to say, is the eigenvector corresponding to eigenvalue . As , information on eigenvalues and eigenvectors can help a lot so we dig into it first.

Since are linearly independent, under this basis, we have a matrix representation but we don't know how eigenspaces are spanned because we may have for . However, the number can always be chosen that are pairwise distinct (for example, one can pick to be a primitive -th root of and is big enough). As a result, has distinct eigenvalues. Therefore, the -eigenspace can only be generated by .

On the other hand, by definition of , we have Hence lies in -eigenspace. Therefore we have for some . In other words, is the -eigenvector of . We obtain another matrix representation under the basis We want this matrix to be a scalar matrix. The result follows from another embedding of into . Note can be determined by , and we therefore have a matrix Still we have . As we can see, This follows from our observation on eigenvalues. Next, we immediately use the eigenvalue to obtain This is the definition of . Comparing coefficients of , we must have for all . Recall that is a basis so coefficients must be unique for a given vector. But we have already obtained what we want: .

Characters and Fourier Transform

So far we have used diagonalisation of representations of but the diagonalisation of itself is not touched yet. Neither have we made use of character functions. So now we invite them to the party.

Let's recall diagonalisation in . Pick . First of all it is diagonalisable. Let and be their two eigenvalues, then . Therefore we have where is one of the eigenvalues of . Since the diagonalised matrix is still in , we have , i.e., . We therefore write where We see, if and only if . By periodicity of function, we also see is in particular -periodic. If is a class function, then is an even -periodic function. Conversely, given an even -periodic function , we can recover it as a class function, and the process is as follows.

Define sending to the eigenvalue of with non-negative imaginary part (one can also pick non-positive one, because is even). Then given by is a well defined function sending into and is a class function. Besides we have and is the diagonalisation of . Therefore and as is expected.

With help of this and , we have this correspondence Recall that the space on the right hand side has a countable uniform basis In other words, spans a dense subspace. This is about the completeness of trigonometric system. Since there are only even functions, are excluded. For a reference to the completeness, one can check 4.25 Real and Complex Analysis by W. Rudin.

For class functions, we certainly want to know about characters. Let be the character of , then When , then . Otherwise, as a classic exercise in calculus, we have We have . For when , we have We see . By induction, every is a polynomial in variables . Therefore spans the same space as , which is dense in the space of even -periodic functions. Note the are linearly independent, because the leading term is .

The argument above shows that spans a dense subspace in the space of class functions. In other word, is the Fourier basis of class functions. As we all know, Fourier series is powerful. Let's see how powerful it is in the calculus of Lie group itself.

Proposition 2. For continuous class function , we have

Proof. On one hand, since the are irreducible, by fixed point theorem of representations, Here, for a group and a representation , is the fixed point set, i.e. the space of elements that are fixed by the action of on . Since is irreducible, fixed points can only be unless the representation itself is trivial. Now we move on and check the right hand side.

On the right hand side we are looking for even -periodic continuous functions, reflecting the denseness of . However we have so it does not vanish on . However, if we multiply it by , then it is transformed into the form and we are familiar with this orthonormality. More precisely, Since the functional is continuous in the uniform topology and spans a dense subspace, the result is now obtained.

Finally, surprisingly and satisfyingly enough, the denseness have actually axed out all other possibilities of irreducible representation. In other words, our search in symmetric products is optimal. We can see this through Parseval's identity. This is the heart of this blog post.

Proposition 3. Every irreducible representation of is isomorphic to one of the .

Proof. Suppose we have a character that is different from all of the . Then the orthonormality shows that for all and . Now let's see why this is absurd.

Since spans a dense subspace in the space of class functions, we actually have Therefore and It is impossible to have the sum of to be .

Irreducible Representations of the Special Orthonormal Group (First Classification)

Now we head to . In fact the result follows immediately from the surjection We have . Let be a representation of , i.e., we have a map Then by is an induced representation, and we write . If is irreducible, then is also irreducible. In particular, .

On the other hand, if is an irreducible representation where , then we have an associated representation given by . Let's denote it by . Again, if is irreducible, then is irreducible.

Therefore we have realised a correspondence So it remains to determine those of . Let be an irreducible representation, then because is homogeneous of degree . Therefore acts as an identity if and only if is even. We obtain

Proposition 4. Every irreducible representation of is of the form where is described in proposition 2.

This is, of course, just a first classification. But to introduce a classification as explicit as what we have done for , there has to be another post. As a quick overview, here is the result.

Let be the complex vector space of homogeneous polynomials in three variables of degree , which can be considered as functions on immediately. This setting makes sense immediately, just as what we have done for . Then, in fact, This is to say, can be understood as harmonic homogeneous polynomials in , which can also be considered to be uniquely determined on the unit sphere .

Reference

• Tendor Bröker and Tammo tom Dieck, Representations of Compact Lie Groups.
• Walter Rudin, Real and Complex Analysis, 3rd Edition.