TITLE: Symmetric Powers for Lie Algebras
QUESTION [9 upvotes]: 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

REPLY [7 votes]: 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).
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.
For $sl_n$, as far as I know the state-of-the-art is summarized in this article by Khale and Michałek.