The $\mathbb{Z}_N^{\times 3}$ symmetry protected boundary modes in two-dimensional Potts paramagnets

We construct and analyze a class of one-dimensional boundary Hamiltonians arising from two-dimensional symmetry-protected topological phases with $\mathbb{Z}_N^{\times 3}$ symmetry on a triangular lattice. Using a cohomology-based transformation, the lattice models for the edge modes are explicitly obtained, and their structure is shown to be governed by the arithmetic properties of $N$. For prime $N$, the boundary theory admits a formulation in terms of mutually commuting Temperley-Lieb algebras. For the composite values of $N$, the models exhibit hierarchical or factorized structures. We demonstrate that all phases can be understood in terms of primary models augmented by local defect degrees of freedom that partition the system into independent segments. Finally, the global symmetry is realized on the boundary in a non-on-site and anomalous manner via a projective representation, directly realizing the corresponding 't Hooft anomaly.