Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 1 / 1 / a / ii

Write $L(n)$ for the irreducible representation of ${\mathfrak{sl}}_2$ with highest weight $n$, dimension $n+1$ and weights $n,n-2,\dots ,-n$. The weight diagram below records multiplicities $1,2,2,2,1$ at weights $4,2,0,-2,-4$.

The unique top weight four forces one copy of $L(4)$, by complete reducibility of semisimple Lie algebra representations. Removing its five weights, each with multiplicity one, leaves weights $2,0,-2$ with multiplicity one, exactly the character of $L(2)$. HenceThe two lower rows draw their separate weight diagrams. The explicit invariant copies constructed next establish the Clebsch-Gordan splitting of a defining module and its cubic symmetric power within $U$ itself.