= Tensor cube of the defining unitary representation
{title2=$V^{\otimes3}=\operatorname{Sym}^3V\oplus\Lambda^3V\oplus V_{(2,1)}^{\oplus2}$}
For the <unitary group> $U(n)$ with $n\geq3$, the third <tensor power> of its <defining representation> decomposes as displayed. The first two summands have <highest weights> $(3,0,\ldots)$ and $(1,1,1,0,\ldots)$ and <dimensions> $\binom{n+2}3$ and $\binom n3$. The mixed summand has <highest weight> $(2,1,0,\ldots)$ and <dimension> $n(n^2-1)/3$. In the three-dimensional <weight space> spanned by <permutations> of $e_1\otimes e_1\otimes e_2$, the vectors killed by all positive <root vectors> have coefficient sum zero, giving multiplicity two. For $n=2$, the exterior cube vanishes and $V_{(2,1)}=\det\otimes\mathbb C^2$.
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