# SO(10) 16 representation, decomposition & young tableau

## 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.

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## 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

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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

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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