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?


  • 2
    $\begingroup$ I think "plethysm" is the keyword you're looking for $\endgroup$ May 10, 2018 at 18:01
  • 1
    $\begingroup$ The character of the exterior power should be something like $e_k[s_\lambda]$ and the symmetric power $h_k[s_\lambda]$ $\endgroup$ May 10, 2018 at 18:02
  • 3
    $\begingroup$ 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. $\endgroup$ May 10, 2018 at 18:07
  • $\begingroup$ 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? $\endgroup$
    – user48969
    May 10, 2018 at 18:40
  • 2
    $\begingroup$ Giving a Littelmann path style description of this is a famous open problem. $\endgroup$
    – Ben Webster
    May 10, 2018 at 20:17

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.