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Potencial as a representation of S3

  1. Mar 3, 2008 #1
    1. The problem statement, all variables and given/known data
    I am asked to write the most general real scalar potencial (without SU(2)xU(1) structure and without spin) with a irreducible representation of the symmetric group [itex]S_3[/itex]. I am suppose to write it with:

    i) one singlet and one doublet of [itex]S_3[/itex]
    ii)two doublets of [itex]S_3[/itex]

    2. Relevant equations

    3. The attempt at a solution

    Well, I now already that the [itex]S_3[/itex] as 2 one-dimensional irreducible representations and 1 two-dimensional irreducible representations (irrep). I also know the basis of the invariant space which form each irrep. My question now, is how can a form a scalar field with this.
    For exemple, for i), I got,

    [tex]
    F=\phi_S+\phi_{D1}+\phi_{D2}
    [/tex]

    where the [itex]\phi_S[/itex] is the singlet and the other two terms forms the doublet. With this I get in fact a three-dimensional representation of [itex]S_3[/itex] which can be decomposed in [itex]D^{1}(S_3)\otimes D^{2}(S_3)[/itex], being the first a one-dimensional irrep and the second a two-dimensional irrep.
    Is that correct? I have no faith in this result...

    Thanks for any help.
     
  2. jcsd
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