Representation symmetric, antisymmetric or mixed

[JorisL]

Hi,

While studying Lie groups and in particular the representations I have some trouble with determining whether a representation is a symmetric, antisymmetric or mixed tensor product of the fundamental representations. I'm working with SO(5) to get an understanding about the actual mechanisms.

I used Georgi's book Lie Algebras in Particle Physics. Do I understand correctly when saying that a representation, $(a,b) = \bigotimes_a (1,0)\bigotimes_b (0,1)$ is symmetric if b = 0, antisymmetric if a = 0? Or is there more to it?


Joris

 

[Bill_K]

While studying Lie groups and in particular the representations I have some trouble with determining whether a representation is a symmetric, antisymmetric or mixed tensor product of the fundamental representations. I'm working with SO(5) to get an understanding about the actual mechanisms.

For SO(n), start with the vector representation, V_a, a = 1,... n. Form higher rank tensors by taking tensor products, V_{abc... k}. These representations are reducible, and may be reduced by taking traces and by symmetrizing the indices in all possible ways. For example for two indices, V_ab may be reduced into a symmetric part (V_ab + V_ab)/2 and an antisymmetric part (V_ab - V_ab)/2.

In general, each part of a k-rank tensor corresponds to a representation of the symmetric group S_k. For more than two indices, some of these representations have "mixed" symmetry.

In particular for SO(5) there is an invariant totally antisymmetric quantity ε_abcde that may be used for further reduction. Multiplication by this quantity will convert a totally antisymmetric set of three indices to an antisymmetric set of only two. In terms of S_k this means that the Young's diagrams are restricted to having at most two rows.

 

[JorisL]

So if I can show that the contractition of the totally antisymmetric tensor with the rep vanishes I'd know I'm dealing with a totally symmetric rep? Or is there a more suitable way?

Also, can you recommend an extra text? Because the Georgi text is suitable for examples, the generalities I find less obvious.