= Explicit weight basis of the exterior square of Sym4 for sl2
{title2=$\Lambda^2(\operatorname{Sym}^4\mathbb C^2)=L(6)\oplus L(2)$}
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> $L(6)$ is
$$
w_{01},\ w_{02},\ w_{03}+2w_{12},\ w_{04}+8w_{13},\ w_{14}+2w_{23},\ w_{24},\ w_{34}.
$$
A <weight basis> of the complementary $L(2)$ is
$$
w_{03}-3w_{12},\quad w_{04}-2w_{13},\quad w_{14}-3w_{23}.
$$
The 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
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