# Young tableaus for $SO(n)$

I know how to use young tableaus to find irreducible representations and their dimensions of $$SU(n)$$. Are there similar rules for $$SO(n)$$?

• Yes, see e.g. arxiv.org/abs/1411.7351. In particular footnote 1 and refs. therein. – MannyC Jan 26 '19 at 21:01
• See also doi.org/10.1063/1.1665778 – ZeroTheHero Jan 26 '19 at 21:11
• I recommend learning about roots, weights, and the Weyl dimension formula. These work for any simple Lie algebra. I think Young tableaux work for only some. – G. Smith Jan 27 '19 at 4:04
• You can first determine the GL irreps by projecting the tensor onto the relevant Young tableaux corresponding to the allowed partitions. Then, the SO irreps are the GL ones with the trace removed. Their dimension is given by King's rule. So in total, GL irreps=SO irreps + traces. – kospall Apr 25 '20 at 10:12
• Could you give some more information on King's rule? I googled it but I didn't find anything. Also I don't know much about GL irreps. I guess what you mean is the decomposition of a general tensor in symmetric, antisymmetric and trace parts? – toaster Apr 27 '20 at 13:08

This is not a complete answer to this question and is adapted from my answer to Irrep decompositions for $SO(N)$ tensors for $N>3$ as it also seems to apply here. Indeed you can do decompositions and find the dimension of irreps using young tableau also for SO(n). The young tableaux describe permutation of indices and thus are relevant for all lie algebra's coming from $$GL(N)$$. The application to $$SO(N)$$ is described well in Group theory: Birdtracks, Lie's, and exceptional groups (paywalled).
The Mathematica application lieART can also do the decompositions for and find the dimensions for you for all classical and exceptional Lie algebras including $$SO(N)$$. The documentation in https://arxiv.org/pdf/1206.6379.pdf also explains the algorithm used to do the decomposition. The only downside is that it does not work with Young Tableaus for SO(N). This means that you will have to convert your irrep in terms of young tableau (or number of boxes per row, i.e. {4} for four index symmetric, or {1,1} for the anti-symmetric) into their Dynkin label description of the same irrep (I did not find this easy. If you know an easy way please tell).