 Symmetric powers of the defining sln representation

For $n\ge 2$ and $m\ge 0$, the representation ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of ${\mathfrak{sl}}_n(\mathbb{C})$ is irreducible with highest weight $m{\omega }_1$. Its monomial basis supplies weights by adding $m$ defining-representation weights, with repeated weights counted separately.