Young Tableaux of $SU(N)$: Group or Algebra? I am slightly confused about the use of Young Tableaux in the context of the Lie group and Lie algebras of $SU(N)$. A Young Tableaux has an associated representation which acts on a tensor e.g. $\psi_{[\mu,\nu]}$. From what I can tell this rep is of the Lie group $SU(N)$ rather then the Lie algebra $su(N)$. And as such Young Tableaux specify reps of Lie groups rather then Lie algebras. Is this correct?
That being said, with each Young Tableaux we can associate a Dynkin label - something that is defined on Lie algebras rather then Lie groups. Does this Dynkin diagram therefore correspond to Lie algebra rep associated with the Lie group rep defined above? If not how?
Lastly If this is the case then the issue of the complex rep arises. For a Lie group the complex rep seems to be defined as:
$$R(g)^*$$
whilst for the Lie algebra it is defined as:
$$-r(\zeta)^T$$
thus if we have a Young table for $R(g)^*$ is the corresponding Dynkin diagram for $-r(\zeta)^T$ or something else?
 A: Representations of the su$(n)$ algebra will exponentiate to representations of the SU$(n)$ group, preserving irreducibility so irreps of the algebra and the group share the same labelling scheme.  
The Young diagrams (the tableaux are diagrams filled with entries) are partitions $\{\lambda\}=(\lambda_1,\lambda_2,\ldots,\lambda_p)$ with $\sum_p\lambda_p=\lambda$.  For su$(n)$ they will contain at most $n$ parts.  The Dynkin labels are differences of consecutive parts in the partition so that to the partition $\{\lambda\}$ corresponds the Dynkin label $(\lambda_1-\lambda_2,\lambda_2-\lambda_3\ldots)$.
As to the last part: it depends a bit if you consider the generators to be hermitian or anti hermitian.  In general complex conjugation will transform $e^{i \theta h}$ with $h$ hermitian in the algebra to $e^{-i\theta h^\dagger}$; in the simplest case where $h$ is in the Cartan subalgebra (thus diagonal with real eigenvalues), it will reverse the sign of the eigenvalues, so the conjugate irrep has weights which are the negatives of the original irrep.
In terms of Young diagrams conjugate irreps have conjugate diagrams, i.e. diagrams are reflected along the diagonal so that the rows and columns get swapped.  For instance $\{2,1,1\}$ for instance becomes $\{3,1\}$.
