= Tensor product of the standard representation with its exterior square
{title2=$V\otimes\Lambda^2V\cong\mathbb S_{(2,1)}V\oplus\Lambda^3V$}
For $V=\mathbb C^n$, $n\ge3$, the <tensor product of Lie algebra representations> for the <special linear Lie algebra> splits as displayed. The <exterior product> map $v\otimes(u\wedge w)\mapsto v\wedge u\wedge w$ has an equivariant right inverse obtained by dividing its three-term alternating insertion by three. Its <kernel> is the <Schur functor> of shape $(2,1)$, with <highest weight> $\omega_1+\omega_2$ and <dimension> $n(n^2-1)/3$. The other summand has <highest weight> $\omega_3$ for $n\ge4$ and is trivial for $n=3$.
The <weight space> of $2\varepsilon_i+\varepsilon_j$ has <dimension> one when $i\ne j$. The <weight space> of $\varepsilon_i+\varepsilon_j+\varepsilon_k$ has <dimension> three for distinct indices. Restricting these functionals to the traceless diagonal <Cartan subalgebra> creates no additional coincidences; for $n=3$ the latter weight is zero.
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