By the Baxter's $Q_{ 72}$-operator method, we demonstrate the equivalent theory between the generalized $\tau^{ (2)}$-model (other than two special cases with a pseudovacuum state) and the $N$-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter $\varsigma$) of $U_{ \sf q}(sl_2)$ with ${ \sf q}^N=1$ for odd $N$ is identified with either (for $\varsigma^N=1$) the chiral Potts model with two superintegrable vertical rapidities, or (for $\varsigma^N \neq 1$) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices $T, \hat{ T}$ of the chiral Potts model (including the degenerate ones) serve as the $Q_R, Q_L$-operators of the corresponding $\tau^{ (2)}$-model, so that the functional relations hold as in the solvable $N$-state chiral Potts model.
Categories: cond-mat.stat-mech hep-th math-ph math.AG math.MP nlin.SI