(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Potencial as a representation of S3

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