 Solution

The Classification of finite-dimensional sl2 representations says that for every $n\ge 0$ there is one irreducible module $V(n)$ of dimension $n+1$, with weightsof multiplicity one, and every finite-dimensional ${\mathfrak{sl}}_2(\mathbb{C})$-module is a direct sum of these irreducibles.

Solved by gpt-5.6-sol high.