Let $V\_\lambda$ be the irreducible representation of $sl\_{n}(\mathfrak{C})$ with highest weight $\lambda$. There are well known formulas for the decomposition of $V\_\lambda^{\otimes^k}= V\_\lambda\otimes V\_\lambda\otimes \cdots\otimes V\_\lambda$ into irreducble representations using Littelwood-Richardson, or Littelmann paths, among others. 


Is there any similar for the symmetric power $S^k(V\_\lambda)$ or for $\wedge^k(V\_\lambda)$, even for $sl\_2(\mathfrak{C})$ or$k=3, 4$, or just some results in some cases??


I know that small cases can be computed, for instance using LiE
 <http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/form.html>
but this formulas seems to be recursive. Is there any closed formula like Littelwood-Richardson, or Littelmann paths?


What is the best recent reference?


Thanks



Symmetric Powers for Lie Algebras
For $sl\_2$ one has formulas. If $V\_n=S^n(\mathbb{C}^2)$, then the multiplicity of the irreducible $V\_k$ inside $S^m(V\_n)$ is
$$
M\_{m,n,k}=\left[m,n,\frac{mn-k}{2}\right]-\left[m,n,\frac{mn-k}{2}-1\right]
$$
where $[m,n,w]$ is the number of integer partitions of $w$ with at most $n$ parts of length $\le m$.
This is the Cayley-Sylvester formula. It was discovered by Cayley in a letter to Sylvester around 1854/1855 but only proved in 1878 by Sylvester (see [this article](https://www.sciencedirect.com/science/article/pii/0315086086900911)).


For simple cases like $S^3(V\_n)\simeq S^n(V\_3)$ (by Hermite's reciprocity of 1854) one has even more explicit formulas, see Theorem 1.3 in [this article](https://arxiv.org/abs/1706.03166).


For $sl\_n$, as far as I know the state-of-the-art is summarized in [this article](https://link.springer.com/article/10.1007/s10208-015-9275-7) by Khale and Michałek.


