= sl3 decomposition of the symmetric-square dual tensor product
{title2=$(S^2V)\otimes V^*\cong\Gamma_{2,1}\oplus\Gamma_{1,0}$}
For the defining three-dimensional <special linear Lie algebra> representation $V$, contraction $(uv)\otimes f\mapsto f(u)v+f(v)u$ maps $(S^2V)\otimes V^*$ onto $V$. The kernel contains the <highest-weight vector> $e_1^2\otimes e_3^*$ of <Dynkin labels> $(2,1)$. The <Weyl dimension formula> gives its irreducible module dimension $15$, the entire kernel dimension; the <Weyl complete reducibility theorem> supplies the displayed splitting.
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