= Symmetric square of a group representation
{title2=$\operatorname{Sym}^2V$}
The symmetric square of a <group representation> on $V$ is the <quotient vector space>
$$
\operatorname{Sym}^2V=(V\otimes V)/\langle v\otimes w-w\otimes v\rangle,
$$
with the induced <tensor product representation>. If the <characteristic of a field> is different from two, $vw\mapsto(v\otimes w+w\otimes v)/2$ identifies it with the <invariant subspace> fixed by the factor-exchange map $\tau$. Over the <complex numbers>, the <projection operator> $(I+\tau)/2$ gives
$$
\chi_{\operatorname{Sym}^2V}(g)=\frac{\chi_V(g)^2+\chi_V(g^2)}2.
$$
In <characteristic two>, the quotient definition still works, but this averaging <projection operator> does not exist. Real coordinate-quadratic perturbations of a nonmagnetic electronic <multiplet> transform in this symmetric square.
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