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Representation equivalent to a unitary one

  1. May 29, 2013 #1
    Hey guys,

    How come a representation [itex]\rho[/itex] of a group [itex]G[/itex] is always equivalent to a unitary representation of [itex]G[/itex] on some (inner product) space [itex]V[/itex] ?

    Can anyone provide a good source (book, preferrably) which states a proof?

    Thanks
     
  2. jcsd
  3. May 29, 2013 #2

    Office_Shredder

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    I think it's only true for finite groups in general. You keep your representation and just define an inner product:
    [tex] \left< u, v \right> = \frac{1}{|G|} \sum_{g\in G} \left< \rho(g) u, \rho(g) v \right> [/tex]

    And it's clear that this inner product is invariant under multiplying u and v by any [itex] \rho(h) [/itex] because the right hand side will still end up being the sum over all group elements of [itex] \rho(gh)[/itex] which still gives every [itex] \rho(g) [/itex] once.

    If you had a compact Lie group or something you could do the same thing with integrating over the group
     
  4. May 29, 2013 #3
    Ok, so what if I had [itex]SO(3)[/itex] ? It's supposed to hold for this group, but I can't seem to find a source (other than the lecture notes I'm using)
     
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