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Reducing infinite representations (groups)

  1. Jan 1, 2010 #1
    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

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

    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 [tex]exp(i\theta)[/tex], I can define an infinite no. of reps with [tex]\Gamma^{n} = ( exp(in\theta) ) [/tex], 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

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

    Should I be thinking about determining

    [tex]n_{p} = \int \chi(T)\chi^{p}(T) dT[/tex]

    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!
     
  2. jcsd
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