# Representation symmetric, antisymmetric or mixed

1. Jun 14, 2014

### 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

2. Jun 14, 2014

### Bill_K

For SO(n), start with the vector representation, Va, a = 1,... n. Form higher rank tensors by taking tensor products, Vabc... 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, Vab may be reduced into a symmetric part (Vab + Vab)/2 and an antisymmetric part (Vab - Vab)/2.

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

In particular for SO(5) there is an invariant totally antisymmetric quantity εabcde 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 Sk this means that the Young's diagrams are restricted to having at most two rows.

Last edited: Jun 14, 2014
3. Jun 14, 2014

### 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.