SO(10) 16 representation, decomposition & young tableau

[Porrum_doctum]

Homework Statement

I've been looking at Paul Langacker's "Grand unified theories and proton decay" for a course on GUTs. I'm stuck with the 16 irrep of SO(10), particularly I don't understand how to prove the statement that the 16 decomposes as 5* + 10 + 1. I can see why it's useful physics wise, but I would like to understand where it comes from mathematics wise. My lack of a decent group theory basics knowledge doesn't really help here.

Continuing from that if you look at 16 x 16 to get to a 126 for a Maiorana mass term, it would be my first idea to use Young tableaux and do it like I've done 5x5, 5x10 etc. in SU(5). However, I don't know what the young tableau of a 16 would be in SO(10). Maybe that would get me started?

Thanks for help towards a solution, or any decent references where I can look.

Homework Equations

--

The Attempt at a Solution

- I don't know how to do decompositions
- I tried working with Young tableaux, but with 16 that's a problem for me

 

[sgd37]

Well first of all the SO(10) has 20 elements 1 of which you can remove because det O = 1 where O $$\in$$ SO(10) similarly you remove another 3 elements from the fact that O$$^{2}$$=I i.e. 1 condition for 0 above the diagonal another for 1 along the diagonal and another for 0 below the diagonal. The decomposition comes from the fact that you can decompose representations into symmetric and anti-symmetric parts plus ( I'm not sure for the general case) into a 1 parameter subgroup. My guess would be the 5* in your case is the anti-symmetric part since that is more restrictive

Note by Moderators -- sdg37 sent us a note that his post is incorrect, but it is past the time limit for him to edit/delete it. Please disregard it. Thanks.

 

[Porrum_doctum]

Thanks, but that doesn't help me in proving how you would get to a 5* , a 10 and a 1 starting from a 16. Why not say, a 4 and a 12 (just to name two random numbers which add to 16)?

I understand now where the 16 comes from, it's one of two spinor representations of dimension 2^m for a group SO(2m). However, I would really like to know how to make/get a decomposition properly. I know you can find them in tables, but where do they come from?

The 5* + 10 + 1 decomposition is with respect to SU(5), but then you can also ask the same question for 16 = (3,2,1) + (3*,1,2) + (1,2,1) + (1,1,2) with respect to SU(3)_CxSU(2)_LxSU(2)_R (which so nicely gives two quark-doublets and two lepton SU(2) doublets).

 

[fzero]

You'll want to look at Slansky, "Group Theory for Unified Model Building", Phys.Rept. 79 (1981) 1-128 (http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=&TI=&AU=&AF=&CL=&RP=LA-UR-80-3495&YR= available), specifically section 6. The branching rules you're looking for could either be obtained by projecting weight vectors or by comparing various products of representations in the algebra and subalgebra.

 

[paranormal]

Dear student,

Thank you for your question. The 16-dimensional representation of SO(10) is a fundamental representation, meaning it is the smallest representation that can fully describe the symmetry group. The decomposition of this representation into smaller representations is a fundamental concept in group theory, and it has important implications in physics, such as in the study of grand unified theories.

To understand the decomposition of the 16 representation, we need to first understand the structure of SO(10) and its representations. SO(10) is a Lie group, which means it is a continuous group that can be described by a set of generators and their algebraic relations. The 16 representation is a spinor representation, meaning it describes the spin of particles in three-dimensional space.

The decomposition of the 16 representation into 5*, 10, and 1 representations is a consequence of the structure of SO(10) and its irreducible representations. In group theory, every representation can be decomposed into a direct sum of irreducible representations, which are the building blocks of the group. These irreducible representations cannot be further decomposed into smaller representations.

In the case of SO(10), the 16 representation can be decomposed into a direct sum of the 5*, 10, and 1 representations, as you have mentioned. This means that the 16 representation can be written as a linear combination of these three representations, and each component of the 16 representation transforms independently under the symmetry of SO(10).

To understand the decomposition in more detail, we can use the concept of Young tableaux. Young tableaux are a graphical tool used to represent the decomposition of a representation into its irreducible components. In the case of SO(10), the 16 representation can be represented by a Young tableau with 16 boxes. The decomposition into 5*, 10, and 1 representations can then be visualized by rearranging the boxes in a specific way, according to the rules of Young tableaux.

I would recommend studying the basics of group theory and the use of Young tableaux in the context of SO(10) to gain a better understanding of the decomposition of the 16 representation. There are many resources available, such as textbooks and online lectures, that can help you in this regard. Additionally, consulting with your course instructor or a group theory expert can also provide valuable insights.

I hope this helps in your understanding of the 16 representation, its decomposition, and the use of