What do $SU(N)$ Dynkin labels say about a multi-component bosonic wavefunction? Suppose you have a system of A bosons of $N$ different types, each of which can be in different angular momentum orbitals. We define the creation operator of a boson of type $a$ with angular momentum $k$ by $ b_{a,k}^\dagger $. 
A basis for the states of such bosons with angular momentum $ L $ can then be created by acting with a products of  $ b_{a,k}^\dagger $s on the vacuum state in such a way that the total angular momentum is correct. 
Suppose now that the Hamiltonian (including interaction) is $SU(N)$
 symmetric (i.e. homogeneous - same interaction between every pair of particles) and commutes with the total angular momentum operator. We then know that the eigenstates can be labelled by SU(N)-quantum numbers (called Dynkin labels, I think).
My question is: what are these numbers and how do they reflect the symmetries of the corresponding eigenstates?
 A: The Dynkin labels $(\lambda_1,\lambda_2,\ldots,\lambda_{N-1})$ label irreducible representations of $SU(N)$.  Using Young diagrams and Schur-Weyl duality is it possible to infer the permutation symmetry properties of the multiparticle states from the Dynkin labels.  
These, however, will not help you get angular momentum labels $L$.  For these you need branching rules from an irrep $(\lambda_1,\lambda_2,\ldots,\lambda_{N-1})$ to the $SO(3)$ subgroup of rotations.  The branching rules are usually clarified using a subgroup chain, i.e. something like
$$
SU(N)\supset SO(N)\supset SO(N-1)\ldots \supset SO(3)\tag{1}
$$
with each link in the chain supplying quantum numbers to properly label your states.
The problem is actually highly non-trivial.  For $SU(3)\supset SO(3)$ the branching rules are known from the work of Elliott on the nuclear $SU(3)$ model, and actually constructing basis states is extremely non-trivial as some $SO(3)$ irreps will usually occur more than once.  For example, in the irrep $(3,2)$, one can find the $SO(3)$ irreps $L=1,2,3,3,4,5$ with $L=3$ occurring twice.
Of course the subgroup chain of Eq.(1) is not the only way to get to $SO(3)$, i.e one could use $SU(N)\supset SU(N-1)\ldots SU(3)\supset SO(3)$.  This chain and the labels it supplies will have a different physical interpretation than the chain of (1).
The place to look are the tables of the review article by Slansky: Group theory for unified model building, Phys.Rep. vol. 79 (1981) p.1-128.  The physics of the simplest subgroup chains is best explained in nuclear physics texts, and a good modern one is the one by Rowe and Wood.  Otherwise, the actual first principle computation use modified tableaux and Schur functions, which are not for the faint-at-hearts.
