Symmetric square of a group representation (${\mathrm{Sym}}^{2}V$)

The symmetric square of a group representation on $V$ is the quotient vector spacewith 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 $(I+\tau )/2$ givesIn 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.