\text{ and }
                             \label{endo 1}
                             \\
           &\mathrm{Sf}_{\tau}(s):=\left\{g\in {\mathrm{End}}^\diamond\left(\bigoplus_{c\in C}V_c\right)
                             \middle| g_{cc'} =0 
                             \text{ for }(\tau(c),\tau(c'))\notin I(s)\text{ with }\ c\neq c'\right\}, 
           \end{aligned}$$ where $s$ is a element of $S_m$, and $\tau $ is an isomorphism $\tau: C{\xrightarrow{\sim}}\{1,\dots, m\}$.

If $(i, j)\in I(s)$, then $i<j$. If $(i, j), (j, k)\in I(s)$, then $(i, k)\in I(s)$. These properties imply $\mathrm{Sf}_\tau(s)$ is a group.

Take $s\in S_m$ and put $\sigma:=(s)_R$, then $(\sigma f)^{-1}\circ f\in {\mathrm{Sf}}_\tau(s)$.

Since ${\mathrm{Sf}}_\tau(s)$ is a group, it is enough to show $f^{-1}\circ \sigma f\in {\mathrm{Sf}}_\tau (s)$. We have $$\begin{aligned}
\sigma f\mid_{V_c}
            &=\left(\prod_{i\in I_{\tau(c)}(s)}^{\leftarrow}R_i\right)\circ f_c \\
            &=\left(\prod^{\leftarrow}_{(\tau(c),\tau(c')\in I(s))}({\mathrm{id}}_{V} -f_{c'}\circ f_{c'}^*)\right)\circ f_c.           
            \end{aligned}$$ This implies $f^{-1}\circ \sigma f\in {\mathrm{Sf}}_\tau (s)$.

Stokes factors and mutations for Stokes filtered local systems {#factor and local system}
--------------------------------------------------------------

Let $C$ be a set of $m$-points in ${\mathbb{C}}$. Let $\theta_\circ$ be a $C$-generic real number. Recall that we have the bijection $\tau_{\theta_\circ}:C{\xrightarrow{\sim}}\{1,2,\dots, m\}$ such that $\tau_{\theta_\circ}(c)<\tau_{\theta_\circ}(c')\Leftrightarrow c<_{{\theta_\circ}+\pi/2}c'$. We put $c_i:=\tau_{\theta_\circ}^{-1}(i)$. We also have an equivalence of categories $$\mathfrak{A}_{\theta_\circ}:{\sf St}^C_{\bm k}\to {\sf Std}_{\bm k}(C,\tau_{\theta_\circ}).$$ We also recall that we have a functor $$\mathbb{M}_\sigma:{\sf Std}_{\bm k}(C,\tau_{\theta_\circ})
          {\xrightarrow{\sim}}{\sf Std}_{\bm k}(C,\bar{\sigma}\circ\tau_{\theta_\circ})$$ for $\sigma\in\mathrm{Br}_m$. The purpose of §\[factor and local system\] is to show the following:

\[mutation gives stokesfactor\] Let $\theta_\circ,\theta_\circ'$ be 