Decomposing a representation under a subgroup

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I am trying to remind myself how decomposing representations works by looking at an easy example.

Using the notation in Slansky, consider the representation of $SU(5)$ that has highest weight $(0\ 1\ 0\ 0)$. How does this rep decompose under $$SU(5)\rightarrow SU(3)\times SU(2)\times U(1)\ ?$$

Could someone please remind me how this works, and maybe mention the general steps along the way as well?

This post imported from StackExchange Physics at 2015-01-16 22:02 (UTC), posted by SE-user Heterotic

edited Jan 16, 2015
Related: physics.stackexchange.com/q/159597/2451

This post imported from StackExchange Physics at 2015-01-16 22:02 (UTC), posted by SE-user Qmechanic

After some digging, I found these notes useful.

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Decomposing representations is the process of breaking down a representation of a group into representations of subgroups. In the case of SU(5) decomposing to SU(3) x SU(2) x U(1), the representation with highest weight (0 1 0 0) will be decomposed into irreducible representations of the subgroups.

Here are the general steps to decompose a representation of a group G into irreducible representations of subgroups H:

1. Determine the weights of the representation of G you are interested in.

2. Determine the weights of the representations of H.

3. Find the highest weight of each irreducible representation of H that appears in the decomposition.

4. Determine the multiplicities of the irreducible representations of H by counting the number of times each highest weight appears in the decomposition.

5. Write down the decomposition of the representation of G into a direct sum of irreducible representations of H, including the multiplicities.

It is important to note that in general, the decomposition of a representation of a group into irreducible representations of subgroups is not unique, and the decomposition you obtain may depend on the method you use to perform the decomposition.

answered Feb 6, 2023 by anonymous

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