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

The subrepresentation of $V\otimes V$ fixed by exchange of the two tensor factors. In characteristic zero the projector is $(I+\tau )/2$, and its character is $[\chi (g{)}^2+\chi (g^2)]/2$. Real coordinate-quadratic perturbations of a nonmagnetic electronic multiplet transform in this symmetric square.