Spherical tensors, in one paragraph

Jun 24, 2026

A rank-$n$ Cartesian tensor carries a reducible representation of the rotation group. Decomposing it into spherical tensors $T^{(k)}_q$ block-diagonalizes the action of rotations: each irreducible piece transforms only among its own $2k+1$ components, via the Wigner-D matrices,

A point-group constraint then reduces to asking which components survive averaging over the group — which, in this basis, is almost by inspection.

For example, the second-harmonic susceptibility $\chi^{(2)}_{ijk}$ — a rank-3 polar tensor — decomposes as $1 \otimes (1 \otimes 1)_{\mathrm{sym}}$, giving angular-momentum content $k \in \{1, 2, 3\}$. It is exactly this content that the calculator tabulates for any (magnetic) point group.