# Standard Model decompositions of larger group representations?

1. Sep 29, 2015

### Anchovy

When reading about GUTs you often come across the 'Standard Model decomposition' of the representations of a given gauge group. ie. you get the Standard Model gauge quantum numbers arranged between some brackets. For example, here are a few SM decompositions of the SU(5) representations $\textbf{5}, \textbf{10}, \textbf{15}$ and $\textbf{24}$.

So, for instance this is telling us that the representation $\textbf{5}$ will contain fields that are either

($SU(3)_{C}$ triplet, $SU(2)_{L}$ singlet, hypercharge $\tfrac{1}{2}Y = -\tfrac{1}{3}$) for the (3, 1, $-\tfrac{1}{3}$),

or

($SU(3)_{C}$ singlet, $SU(2)_{L}$ doublet, hypercharge $\tfrac{1}{2}Y = \tfrac{1}{2}$) for the (1, 2, $\tfrac{1}{2}$).

That's straightforward enough. However, I can't seem to find anything online explaining how these have been determined. I can find plenty about how you might go about constructing the $\textbf{10}, \textbf{15}$ and $\textbf{24}$ starting from combinations of the fundamental $\textbf{5}$ by the 'Young's Tableaux' method, but nothing about starting with one of these SU(5) representations and breaking them down. Can anyone explain or link to an explanation?

2. Sep 29, 2015

### Andrea M.

Try chapter 18 of "Lie algebras in particle physics" by Georgi.

3. Sep 29, 2015

### Anchovy

4. Sep 29, 2015

### Anchovy

I'm wondering something about what it says here. Specifically I am trying to understand the motivation for the part that's in the red box. I have also highlighted earlier parts in green boxes that I suspect might be related, but I don't quite understand.

So it wants to get the SM decomposition of the $\textbf{5}$ by choosing from constituents of the equation (18.13) that combine to form a 5-dimensional subset. Fair enough. What I don't understand is the $SU(2) \times U(1)$ part. Why must the $\textbf{5}$ incorporate that specifically? Why not require, say, $SU(3) \times SU(2) \times U(1)$?

5. Sep 30, 2015

### Andrea M.

The multiplets in $(18.15)$ aren't a subgroups of $SU(3)\times SU(2)\times U(1)$?