Chiral Potts rapidity curve descended from six-vertex model and symmetry group of rapidities

In this paper, we present a systematical account of the descending procedure from the six-vertex model to the N-state chiral Potts model through fusion relations of τ(j)-operators, following the works of Bazhanov-Stroganov and Baxter-Bazhanov-Perk. A careful analysis of the descending process leads to the appearance of the high genus curve as the rapidity constraint for the chiral Potts models. Full symmetries of the rapidity curve are identified, as is its symmetry group structure. By normalized transfer matrices of the chiral Potts model, the τ(2)T relation can be reduced to functional equations over a hyperelliptic curve associated with rapidities, by which the degeneracy of τ(2)-eigenvalues is revealed in the case of the superintegrable chiral Potts model