Explicit weight basis of the exterior square of Sym4 for sl2 (${\Lambda }^2({\mathrm{Sym}}^4{\mathbb{C}}^2)=L(6)\oplus L(2)$)

Let $e_i=x^{4-i}{y}^i$ and $w_{ij}=e_i\wedge e_j$ for $i<j$. For the sl2 Lie algebra, the raising and lowering operators are $E=x{\partial }_y$, $F=y{\partial }_x$, and $w_{ij}$ has weight $8-2(i+j)$. A weight basis of the irreducible representation $L(6)$ isA weight basis of the complementary $L(2)$ isThe first vector in each list is a highest-weight vector. Applying $F$ along the first list gives successive coefficients $3,2,1,12,5,2$; along the second it gives $1,2$. The final vectors are killed by $F$. At the overlapping weights $2,0,-2$, the two listed vectors are independent, proving the direct sum.

// Target: semisimple-lie-algebra.bigb