# Potencial as a representation of S3

1. Mar 3, 2008

### Magister

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 $S_3$. I am suppose to write it with:

i) one singlet and one doublet of $S_3$
ii)two doublets of $S_3$

2. Relevant equations

3. The attempt at a solution

Well, I now already that the $S_3$ 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,

$$F=\phi_S+\phi_{D1}+\phi_{D2}$$

where the $\phi_S$ is the singlet and the other two terms forms the doublet. With this I get in fact a three-dimensional representation of $S_3$ which can be decomposed in $D^{1}(S_3)\otimes D^{2}(S_3)$, 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.