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

1. Jan 19, 2012

### 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

2. Jan 19, 2012

### 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.

3. Jan 20, 2012

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

4. Jan 20, 2012

### naima

5. Jan 20, 2012

### 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)?

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