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RobyVonRintein
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Homework Statement
(a) For SU(N), we have:
N ⊗ N = A_A + S_S
where A corresponds to a field with two antisymetric fundamental SU(N) in- dices φij = −φji, and S corresponds to a field with two symmetric fundamental SU(N) indices φij = φji. By considering an SU(2) subgroup of SU(N), compute T(A) and T(S).
Homework Equations
Under the SU(2) subgroup, N transforms as:
[2 ⊕ (N − 2)1S] ⊗ [2 ⊕ (N − 2)1s] = A + S.
The Attempt at a Solution
Distributing, we get, among other terms, 2 ⊗ (N − 2)1S ⊕ 2 ⊗ (N − 2)1S. It's then just a matter of separating the symmetric and anti-symmetric terms.
What I don't understand is why one of these 2 ⊗ (N − 2)1S terms is symmetric and the otherone is anti-symmetric. They seem identical to me... I think part of the problem is I don't understand Srednicki's notation 1s (sometimes 1's) -- does this just mean the singlet is symmetrical?
I understand Young Tableaux, but not sure how to use them here, the only diagram I can work out is adjoint, and therefore neither symmetrical nor anti-symmetrical.
This is the only difficulty, if we can figure out this, I know how to solve the problem from there...
Thanks!
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