Let $V(\pi\_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}\_n$, for $n > 2$. A basic fact is the tensor product $V(\pi\_1) \otimes V(\pi\_1)$ decomposes as
$$
V(\pi\_1) \otimes V(\pi\_1) \simeq V\_{2\pi\_1} \oplus V\_{\pi\_2}.
$$
For higher tensor powers
$$
V(\pi\_1)^{\otimes k}
$$
does there exist a formula to describe the decomposition into irreducibles? Thinking about this in Young diagram terms, it seems (more or less) clear that the $k^{\text{th}}$-tensor power will have summands whose diagrams contain $k$-boxes, and any Young diagram with $k$-boxes will appear. Hence the question reduces to the question of multiplicities.



Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
The multiplicity of the Young diagram of shape $\lambda$ is the number of standard Young tableaux of shape $\lambda$, which can be computed with the [hook length formula](https://en.wikipedia.org/wiki/Hook_length_formula). This can be deduced from [Schur-Weyl duality](https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality).


