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As the result of the high-temperature improper ferroelectric transitionoccurring in YMnO3 and other hexagonal manganites a striking domain pattern wasobserved: an emergence of six domains of alternating polarization from onepoint in a vortex-like pattern. We derive the formation and distribution of thedomain-vortex network from a combinatorial analysis based on order-parametersymmetry and lattice geometry. The analysis leads to stable vortex-likeconfigurations of the six domain states in the basal plane perpendicular to thepolarization and to fragmented vortices in planes parallel to the polarization.The predictions are debated in the light of existing experimental andtheoretical work on the vortex domain state. The relation of thesymmetry-determined geomet- rical defects to the topological defects proposedto describe the behavior of YMnO3 near the Curie temperature is discussed.

Combinatorial model for the ferroelectric domain-network formation in hexagonal manganites