= Clebsch-Gordan splitting of a defining module and its cubic symmetric power
{c}
{title2=$L(1)\otimes L(3)=L(4)\oplus L(2)$}
For the defining <sl2 Lie algebra> module with basis $x,y$, put $A_k=x\otimes x^{3-k}y^k$ and $B_k=y\otimes x^{3-k}y^k$. The raising operator has $eA_k=kA_{k-1}$ and $eB_k=A_k+kB_{k-1}$; the lowering operator has $fA_k=B_k+(3-k)A_{k+1}$ and $fB_k=(3-k)B_{k+1}$. The <highest-weight vectors> $A_0$ and $B_0-A_1$ generate irreducible submodules of highest weights four and two. Bases are respectively $(A_0,3A_1+B_0,A_2+B_1,A_3+3B_2,B_3)$ and $(B_0-A_1,B_1-A_2,B_2-A_3)$. Their vectors are independent in every <weight space>, proving the displayed <direct sum> without only comparing dimensions.
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