e^{2\pi i / a} \in {\mathbb{C}}^*$.

2.  For a summand of loop type $W_{\text{loop}} = x_1^{a_1}x_2 + x_2^{a_2}x_3 + \ldots +x_{m-1}^{a_{m-1}}x_m + x_m^{a_m}x_1$, the group $\emph{Aut}(W_{\text{loop}})$ is isomorphic to $ {\mathbb{Z}}/ \Gamma{\mathbb{Z}}$ where $\Gamma =a_1\cdots a_m + (-1)^{m+1}$ and generated by $(\varphi_1, \ldots, \varphi_m)\in ({\mathbb{C}}^*)^m$, where $$\varphi_1 : = e^{2\pi i (-1)^m/\Gamma}, \text{ and } \varphi_i : = e^{2\pi i (-1)^{m+1-i} a_1\cdots a_{i-1}/ \Gamma}, i \geq 2.$$

3.  For a summand of chain type, $W_{\text{chain}} = x_1^{a_1}x_2 + x_2^{a_2}x_3 + \ldots x_{m-1}^{a_{m-1}}x_m + x_m^{a_m}$, the group $\emph{Aut}(W_{\text{chain}} )$ is isomorphic to $ {\mathbb{Z}}/ (a_1 \cdots a_m) {\mathbb{Z}}$, and generated by $(\varphi_1, \ldots, \varphi_m) \in ({\mathbb{C}}^*)^m$, where $$\varphi_i = e^{2\pi i (-1)^{m+i}/a_i\cdots a_m}.$$

Note that there is some overlap in the subgroups of $({\mathbb{C}}^*)^{n+1}$. Let $J_{F_A}:= {\text{Aut}}(F_A) \cap {\mathbb{C}}^*$. The group $J_{F_A}$ is generated by $(e^{2\pi i q_0/d}, \ldots, e^{2\pi i q_n/d})$, which is clearly in ${\text{Aut}}(F_A)$ because $\sum_{j=0}^n a_{ij}q_j = d$ and the alternate description provided by the isomorphism above in Equation \[autisom\] (moreover, $(e^{2\pi i q_0/d}, \ldots, e^{2\pi i q_n/d}) = \prod_{i=0}^n \rho_i \in {\text{Aut}}(F_A)$).

We now introduce the group $$SL(F_A) := \left\{ (\lambda_0, \ldots, \lambda_n) \in {\text{Aut}}(F_A) \middle| \prod_{j=0}^n \lambda_j = 1\right\}.$$ The group $J_{F_A}$ is a subgroup 