  The Ising model is the simplest to describe many-body effects in classical
statistical mechanics. Duality analysis leads to a critical point under several
assumptions. The Ising model itself has $Z(2)$ symmetry. The basis of the
duality analysis is a nontrivial relationship between low and high-temperature
expansions. However, the discrete Fourier transformation finds the hidden
relationship automatically. The duality analysis can be naturally generalized
into the case with the degrees of freedom with $Z(q)$ symmetry and random spin
systems. We further obtain the duality in a series of permutation models in the
present study by considering the symmetric group $S_q$ and its Fourier
transformation. The permutation model in the symmetric group is closely related
to the random quantum circuits and random tensor network model, often discussed
in the context of quantum computing and the holographic principle, a property
of string theories and quantum gravity. We provide a systematic way by our
duality analysis to analyze the phase transition in these models.
