Group: most general scalar potential out of 2 doublet irreps of S3.

[salparadise]

Group Theory: Most general scalar potential out of 2 doublet irreps of S3.

I'm taking a course on group theory in physics, but the teacher is really bad at making the bridge between the maths and the physics.

As homework I have to do the exercise below. I think I know how to do it but I'm also posting it to see if someone could please recommend a good reference book where this kind of questions are treated. A reference that clearly explains Young diagrams (not just Young tableau) is also something I can't find. I've consulted the following books: Georgi.H and Cornwell.

Homework Statement

Symmetry group S3. Taking into account the direct product of the 2D irreps as follows:
$$\psi_{i}\otimes\psi^{}_{j} = \Psi_{1}+\Psi_{1}+\Psi_{2}$$
where:
$$\Psi_{1} = \psi_1\psi^{'}_1 + \psi_2\psi^{'}_2\\$$

$$\Psi_{1'} = \psi_1\psi^{'}_2 + \psi_2\psi^{'}_1\\$$

$$\Psi_{2} = (\psi_1\psi^{'}_2 + \psi_2\psi^{'}_1 , \psi_1\psi^{'}_1 - \psi_2\psi^{'}_2 )^T\\$$

Write the most general scalar potential up to power four, made exclusivly with
two S3 doublets, namely $$\psi$$ and $$\chi$$.

The Attempt at a Solution

Knowing that the product of the 2D irreps of S3 is 2⊗2=1+1'+2, and knowing that a scalar invariant potential can only be formed by spaces of trivial representation. We only need to form all possible products of $$\psi$$ and $$\chi$$ up to power 4 and at the end only take the resulting 1 irrep (trivial one) terms.

Thanks in advance

PS - If this should be in another forum section, please let me know.

 

[salparadise]

Hi,

I noticed there's an error on my post. on the 1' irrep as direct product of the 2D irreps there should be minus sign.

Anybody can help with this question?

Thanks

 

[salparadise]

Hello,

Anybody could please help?

Thanks