= Solution
Take a basis $x,y$ of the <defining representation> of the <sl2 Lie algebra>, with $Hx=x$, $Hy=-y$, $Ey=x$, $Ex=0$, $Fx=y$ and $Fy=0$. A monomial basis of the <symmetric power> $\operatorname{Sym}^4V$ is
$$
v_i=x^{4-i}y^i\quad(0\leq i\leq4),\qquad Hv_i=(4-2i)v_i.
$$
For the <exterior power> $U=\bigwedge^2\operatorname{Sym}^4V$, put $w_{ij}=v_i\wedge v_j$ for $i<j$. These ten elements form a basis, and their <weights of a representation> add:
$$
Hw_{ij}=(8-2(i+j))w_{ij}.
$$
Thus the complete <weight-space decomposition>, with explicit bases, is
$$
\begin{array}{c|l}
\text{weight}&\text{basis}\\\hline
6&w_{01}\\
4&w_{02}\\
2&w_{03},\ w_{12}\\
0&w_{04},\ w_{13}\\
-2&w_{14},\ w_{23}\\
-4&w_{24}\\
-6&w_{34}
\end{array}
$$
The <weight diagram> therefore has \b[multiplicities $1,1,2,2,2,1,1$ at weights $6,4,2,0,-2,-4,-6$], respectively.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-3-sl2-weights.png]
{title=Weights and multiplicities of the exterior square of Sym4 C2}
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