= Symmetric cube of a group representation
{title2=$\operatorname{Sym}^3V$}
The symmetric cube of a <group representation> $V$ is the symmetric part of $V^{\otimes3}$. Over the <complex numbers>, its <character> is
$$
\chi_{\operatorname{Sym}^3V}(g)=\frac{\chi_V(g)^3+3\chi_V(g)\chi_V(g^2)+2\chi_V(g^3)}6.
$$
The projection onto symmetric <tensors> averages the six permutations of the three factors. The identity permutation contributes $\chi(g)^3$, the three transpositions each contribute $\chi(g)\chi(g^2)$, and the two three-cycles each contribute $\chi(g^3)$. Cubic <polynomial invariants> are fixed vectors in $\operatorname{Sym}^3(V^*)$, whose dimension follows by averaging this <character> for a <finite group>.
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