Check that the Boltzmann weights (7.153) for the chiral Potts model satisfy the Yang-Baxter equation (7.151) provided all the rapidities $p, q$ and $r$ lie on the same spectral manifold $\Gamma_\kappa$ (7.156). In order to do so, perform first the Fourier transform of the weights (7.153) with respect to $\sigma_3$, and introduce
$$
\begin{aligned}
V_{a b} & =\sum_{k=1}^N \omega^{b k} W_{p r}(a+k) \bar{W}_{q r}(k) \\
\bar{V}_{a b} & =\sum_{k=1}^N \omega^{a k} \bar{W}_{p r}(k) W_{q r}(b+k)
\end{aligned}
$$
The proof of (7.151) follows from the remark that the product formulae in (7.153) are equivalent to a pair of linear recurrence relations for both $V_{a b}$ and $\bar{V}_{a b}$.