Standard Model decompositions of larger group representations?

[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?

 

[Andrea M.]

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

 

[Anchovy]

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

OK, found a pdf version, will check it out very soon.

 

[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)?

 

[Andrea M.]

Why must the 5\textbf{5} incorporate that specifically? Why not require, say, SU(3)×SU(2)×U(1)SU(3) \times SU(2) \times U(1)?

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