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Representation symmetric, antisymmetric or mixed 
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#1
Jun1414, 07:49 AM

P: 48

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, [itex](a,b) = \bigotimes_a (1,0)\bigotimes_b (0,1)[/itex] is symmetric if b = 0, antisymmetric if a = 0? Or is there more to it? Joris 


#2
Jun1414, 10:32 AM

Sci Advisor
Thanks
P: 4,160

In general, each part of a krank 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. 


#3
Jun1414, 12:37 PM

P: 48

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. 


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