I am wondering whether there is a systematic way to decompose a certain representation (say the completely symmetric representation for concreteness) of $SU(n_1n_2)$ in terms of the representations of $SU(n_1)\times SU(n_2)$.
For instance, let $n_1=2$ and $n_2=3$ i.e., $n_1n_2=6$. Consider a fully symmetric representation of $SU(6)$ with dimension 56. This can be decomposed into representations of $SU(2)\times SU(3)$ as:
56= (4,10) + (2,8)
The first entry in the parentheses is the dimension of the $SU(2)$ representation and the second one is the $SU(3)$ dimension. For more details of this, please have a look at 15th chapter of "Lie algebras of Particle Physics" by Georgi. There is a version online: http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf
In the above example, as far as I understand, Georgi obtains the decomposition just by trail and error and counting the dimensions. My question is whether one can do this decomposition more systematically?