Potencial as a representation of S3

[Magister]

Homework Statement

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 $S_3$

Homework Equations

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.

 

[Jhero]

Thank you for your question. To write the most general real scalar potential with an irreducible representation of the symmetric group S_3, we can use the following steps:

i) For one singlet and one doublet of S_3:
First, we need to define the basis of the singlet and doublet irreps of S_3. The singlet irrep can be written as a single scalar field, let's call it \phi_S. The doublet irrep can be written as a two-component vector field, let's call it \phi_D. Now, to form a scalar field with these two irreps, we can write:

F=\phi_S+\phi_D^\dagger\phi_D

where \phi_D^\dagger is the Hermitian conjugate of \phi_D. This gives us a scalar field that transforms as a three-dimensional representation of S_3, which can be decomposed into a one-dimensional irrep and a two-dimensional irrep, as you correctly mentioned.

ii) For two doublets of S_3:
Similarly, we can write:

F=\phi_{D1}^\dagger\phi_{D1}+\phi_{D2}^\dagger\phi_{D2}

where \phi_{D1} and \phi_{D2} are two independent doublet fields. This gives us a scalar field that transforms as a four-dimensional representation of S_3, which can be decomposed into two two-dimensional irreps.

I hope this helps. Let me know if you have any further questions.