Reducing infinite representations (groups)

[Rory9]

Hi,

I am trying to work through some problems I have found in preparation for a test. I am running out of time, and getting somewhat confused, however...

Homework Statement

a) Show that every irreducible representation of SO(2) has the form

$$\Gamma\ \left( \begin{array}{ccc} cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \\ \end{array} \right) = ( exp(in\theta) )$$

where n = ... -2, -1, 0, 1, 2, ...

b) Determine which irreducible representations of SO(2) appear in the reduction of the representation of SO(2) in which each element of SO(2) is represented by itself.

The Attempt at a Solution

I can only think of a 'wordy' explanation for (a), and I'm just not sure how I should go about 'reducing' this infinite representation in (b).

U(1) is a compact linear lie group (right?), so the number of inequivalent reps. it has is infinite but countable. Since the form of U(1) is $$exp(i\theta)$$, I can define an infinite no. of reps with $$\Gamma^{n} = ( exp(in\theta) )$$, and since U(1) is isomorphic to SO(2), SO(2) has the same reps... something like that?

For part (b), I guess the rep. it wants me to reduce is

$$\Gamma(T) = \left( \begin{array}{ccc} cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \\ \end{array} \right) = ( exp(in\theta) )$$

Should I be thinking about determining

$$n_{p} = \int \chi(T)\chi^{p}(T) dT$$

or is that barking up the wrong tree? (So far, I've only been looking at reducing finite groups, so I'm a bit confused about what I should do with infinite groups like this...)

Cheers!

 

[Jhero]

Hello,

Thank you for your post. It looks like you are trying to understand the irreducible representations of SO(2) in preparation for a test. I can offer some guidance on how to approach this problem.

First, let's review the definition of an irreducible representation. An irreducible representation is a linear transformation that cannot be further reduced into smaller representations. In other words, it is a representation that cannot be written as a direct sum of two or more representations.

Now, for part (a), you are asked to show that every irreducible representation of SO(2) has the form (exp(inθ)). To do this, you can use the fact that SO(2) is isomorphic to U(1), as you mentioned. This means that the representations of SO(2) are essentially the same as the representations of U(1). Since U(1) is a compact linear Lie group, its representations are indexed by an integer n, as you correctly stated. This integer n corresponds to the exponent in (exp(inθ)). Therefore, every irreducible representation of SO(2) has the form (exp(inθ)).

For part (b), you are asked to determine which irreducible representations of SO(2) appear in the reduction of the representation of SO(2) in which each element of SO(2) is represented by itself. In other words, you need to find the irreducible representations that make up the representation (exp(inθ)). To do this, you can use the fact that the irreducible representations of SO(2) are indexed by an integer n. So, for each value of n, you can determine the corresponding irreducible representation and see if it appears in the reduction of (exp(inθ)).

To summarize, for part (a), you need to use the fact that SO(2) is isomorphic to U(1) to show that every irreducible representation of SO(2) has the form (exp(inθ)). For part (b), you need to use the fact that the irreducible representations of SO(2) are indexed by an integer n to determine which irreducible representations appear in the reduction of (exp(inθ)).

I hope this helps. Good luck with your test!