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It is found that $\alpha$ cluster structure has the apparently opposing dualproperty of crystallinity and condensation simultaneously. The mathematicalequivalence of the spatially localized Brink $\alpha$ cluster model in thegenerator coordinate method (GCM) and the nonlocalized cluster model (NCM),which is also called the THSR (Tohsaki-Horiuchi-Schuck-R$\ddot{\rm o}$pke) wavefunction based on the condensation of $\alpha$ clusters, is shown. The latteris found to be an equivalent representation of the localized cluster model andit is a natural consequence that the many NCM (THSR) calculations reproduce theproceeding cluster model calculations using the GCM and the resonating groupmethod (RGM). Localized cluster models, which have been successfully used formore than half a century, will continue to be very powerful. The equivalence isa manifestation of the duality of incompatible aspects: crystallinity andcoherent wave nature due to condensation of $\alpha$ clusters, i.e. the dualproperties of a supersolid. The Pauli principle causes the duality. Theevidence for supersolidity, the emergence of a Nambu-Goldstone mode caused bythe spontaneous symmetry breaking of the global phase, is discussed

Equivalence of generator coordinate Brink cluster model and nonlocalized cluster model and supersolidity of $\alpha$ cluster structure in nuclei