Character of a localized orbital basis

A symmetry transformation acts on localized atomic orbitals both by moving their centres and by transforming their angular shapes. Only centres left in place can contribute to the trace of the corresponding group representation. At each fixed centre, sum the diagonal coefficients of the orbital transformation: a reversed p orbital contributes $-1$, whereas a fixed s orbital contributes $+1$. For Cartesian displacement vectors at a fixed atom, use the trace of the full polar-vector matrix, not merely the number of fixed components. This rule generates characters of a representation without writing its entire matrix.