= 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>.
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