Group shift operator
= Group shift operator
{title2=$U(h)$}
For a <finite abelian group> $G$, $U(h)|g\rangle=|g+h\rangle$ defines the <regular representation> on the basis labelled by $G$. The <linear characters> give the common <eigenbasis>
$$
|v_\chi\rangle=|G|^{-1/2}\sum_g\overline{\chi(g)}|g\rangle,
\qquad U(h)|v_\chi\rangle=\chi(h)|v_\chi\rangle.
$$
<Character orthogonality> proves orthonormality, and changing variables by the <translation in a group> proves the eigenvalue formula.