Let $G$ be a semisimple Lie group with the set of positive simple roots $\prod$. Let $\prod^{-}$ be the set of negative simple roots and let $\mathfrak{M}$ be the semigroup freely generated by $\prod$ and $\prod^{-}$. Then the braid semigroup is the quotient of the semigroup $\mathfrak{M}$ by some braid relations which related to the Cartan matrix $C$.
In the paper 'Cluster $\mathcal{X}$ -varieties, amalgamation and Poisson-Lie groups' https://arxiv.org/pdf/math/0508408.pdf written by V. V. Fock and A. B. Goncharov. On page 16, in order to explain the fact a mutation of a cluster seed $J(D)$ is not always a cluster seed corresponding to another word of the semigroup.I am at a loss for the following example.
Example Let $\prod=\{\gamma,\Delta,\eta\}$ be the root system of type $A_3$ with $C_{\eta\gamma}=0$. Then $\mu_{\binom{\gamma}{1}}J(\gamma \Delta \eta\gamma\Delta\gamma)=J(\Delta\gamma\Delta\eta\Delta\gamma)$, but $\mu_{\binom{\Delta}{1}}J(\gamma \Delta \eta\gamma\Delta\gamma)$ is a seed which does not correspond to any word.
Since $C_{\eta\gamma}=0$, then $\eta\gamma=\gamma\eta$ by the braid relation, which implies that $\gamma \Delta \eta\gamma\Delta\gamma=\gamma \Delta \gamma\eta\Delta\gamma$. Thus $\mu_{\binom{\gamma}{1}}J(\gamma \Delta \gamma\eta\Delta\gamma)=J(\Delta\gamma\Delta\eta\Delta\gamma)$. But I don't know how to mutate $\mu_{\binom{\Delta}{1}}J(\gamma \Delta \eta\gamma\Delta\gamma)$? Who can give me some mutation rule? Any help would be appreciated.
