# Homework Help: Reducing infinite representations (groups)

1. Jan 1, 2010

### 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...

1. The problem statement, all variables and given/known data

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.

3. 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!