5 rep of SU(5) under SU(3)XSU(2)XU(1)

[LAHLH]

Hi,

What does is mean to say under SU(3)XSU(2)XU(1) the 5 representation of SU(5) transforms as

$$5\to(3,1,-1/3) \oplus (1,2,+1/2)$$

I can't work out what this (a,b,c) notation means exactly; could anyone point me to a link or chapter of a book that could explain what's going on here?

thanks

 

[Bill_K]

This decomposes the representation into an outer product of representations of SU(3), SU(2) and SU(1). For example (3,1,-1/3) means 3 ⊗ 1 ⊗ -1/3, the triplet of SU(3), the singlet of SU(2) and weak hypercharge -1/3.

 

[Simon_Tyler]

Bill_K gave the answer for what the notation means. As for references, you can start with
Wikipedia. The SU(5) model is known as the Georgi–Glashow model and is a simple example of a GUT.

Probably the best intro textbook for this topic is Georgi's Lie Algebras In Particle Physics, where both the SU(5) and SO(10) GUTs are discussed in detail.

 

[naima]

You have also Baez huerta
skip to SU(5) GUT

 

[LAHLH]

So I understand how to do a Clebsch-Gordon decomp and the rules of Young-Tableaux, for example I'm happy doing things like $8\otimes 8=\bar{10} \oplus 8 \oplus 8\oplus 1$ (well reasonably happy..)...so here we are just decomposing the outer product of two or more representations of a single group into the direct sum of other representations of the group.

I started having a look at p183 of Georgi where he discusses $SU(3)\to SU(2)\times U(1)$, as I think this is a simpler example of the mathematics I need to understand my original post? I'm not sure I follow what is going on in this section (12.3) however, despite feeling I reasonably well understand the previous Young Tableaux stuff. What is he doing to obtain the figure associated with (12.16) and (12.17)?