# Recipe to compute dimension and decompose product of $SO(N)$ group representations

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As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts:

1. There is a correspondence between irreps of $SU(N)$ and YT;

2. There is an easy way to compute dimension of a certain irrep of $SU(N)$ (using again YT);

3. There is an easy algorithm to multiply two or more irreps of $SU(N)$ in terms of YT;

Questions: Is there a way to do the same (points 1. 2. 3.) with $SO(N)$? What are the algorithms to do that?

This post imported from StackExchange Mathematics at 2015-04-29 18:23 (UTC), posted by SE-user MaPo

edited Apr 29, 2015

1) There is a correspondence, but there are more restrictions on YT than in the case of $SU(N)$ where the restriction is not to have more than N-1 boxes in a column. Due to an additional invariant tensor, metric $\delta^{ab}$, we can take traces and irreducible tensors need to be traceless. Combining it with the $\epsilon$-symbol you get that the total height of the first two columns has to be not greater than $N$, otherwise the tensor is identically zero.
2) There is a generalization of the Hook formula that works for $SU(N)$. The details can be found in this old paper.
3) Again there is an algorithm, which is a superposition of two $SU(N)$ tensor product rules. One has to take into account traces. The precise formulas are the very first in this paper