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salparadise
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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.
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.
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.
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.
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