Decomposing a representation under a subgroup

Question

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

Comments

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

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After some digging, I found these notes useful.

Answer 1

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.