  At roots of unity the $N$-state integrable chiral Potts model and the
six-vertex model descend from each other with the $\tau_2$ model as the
intermediate. We shall discuss how different gauge choices in the six-vertex
model lead to two different quantum group constructions with different
$q$-Pochhammer symbols, one construction only working well for $N$ odd, the
other equally well for all $N$. We also address the generalization based on the
sl$(m,n)$ vertex model.
