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Translation by one site multiplies a lattice wave by $e^{ika}$, so the coefficients of a finite periodic tight-binding ring arePeriodicity after three sites requiresChoosing representatives in the Brillouin zonegivesThe corresponding states areThey reproduce respectively the coefficient vectors $(1,1,1)$, $(1,\omega ,{\omega }^2)$ and $(1,{\omega }^2,\omega )$.