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

Use $h=\mathrm{diag}(1,-1)$, $e=E_{12}$ and $f=E_{21}$ in the sl2 Lie algebra, so $hx=x$, $hy=-y$, $ey=x$ and $fx=y$. On the symmetric power, a monomial $x^{3-k}{y}^k$ has weight $3-2k$. The tensor product of Lie algebra representations adds weights, because $h$ acts as $h\otimes I+I\otimes h$.

Put $A_k=x\otimes x^{3-k}{y}^k$ and $B_k=y\otimes x^{3-k}{y}^k$ for $0\le k\le 3$. These eight vectors are a basis of $U$; their weights are $4-2k$ and $2-2k$. Thus the weight-space decomposition and a weight basis for each summand areIn particular $\boxed{U=U_4\oplus U_2\oplus U_0\oplus U_{-2}\oplus U_{-4}}$, with dimensions summing to eight. Here the subscripts are the eigenvalues of the chosen Cartan element $h$.