Symmetric Powers for Lie Algebras

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

• I think "plethysm" is the keyword you're looking for – Sam Hopkins May 10 '18 at 18:01
• The character of the exterior power should be something like $e_k[s_\lambda]$ and the symmetric power $h_k[s_\lambda]$ – Sam Hopkins May 10 '18 at 18:02
• This doesn't necessarily mean there's a simple formula- plethysm in general is very hard to understand. But anyways, Richard Stanley should probably be here soon to tell you the answer. – Sam Hopkins May 10 '18 at 18:07
• even for $sl_2$, something general like $S^3(V_n)$ there is not a simple closed formula, isn't it? Of course you can compute some cases, but even these case is hard. Am I right? – user48969 May 10 '18 at 18:40
• Giving a Littelmann path style description of this is a famous open problem. – Ben Webster May 10 '18 at 20:17

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.