R_{i+1}\circ R_{i+2},             
              \end{aligned}$$ and $$\begin{aligned}
              R_{i+2}\circ R_{i+1}\circ({\mathrm{id}}-R_{i+2})
              &=R_{i+2}\circ({\mathrm{id}}-f_{\tau^{-1}(i+1)}\circ f^*_{\tau^{-1}(i+1)})\circ (f_{\tau^{-1}(i+2)}\circ      
              f^*_{\tau^{-1}(i+2)})\\
              &=R_{i+2}\circ(f_{\tau^{-1}(i+2)}\circ f^*_{\tau^{-1}(i+2)})\\
              &=R_{i+2}\circ({\mathrm{id}}-R_{i+2})=0.
              \end{aligned}$$ The relation ${\mathbb{R}}_{i+2}{\mathbb{R}}_{i+1}{\mathbb{R}}_{i+2}f^*={\mathbb{R}}_{i+1}{\mathbb{R}}_{i+2}{\mathbb{R}}_{i+1}f^*$ is shown similarly. The relation $(\ref{braid-3})$ is obvious by the definition.

Let $\mathrm{Br}_m$ denote the braid group of $m$-strands. For an element $\sigma\in\mathrm{Br}_m$, let $\bar{\sigma}$ denote the image of $\sigma$ via the quotient map $\mathrm{Br}_m\to S_m$.

\[braid action\] For $\sigma\in \mathrm{Br}_m$, define a functor $\mathbb{M}_\sigma:{\sf Std}_{\bm k}({C},\tau){\xrightarrow{\sim}}{\sf Std}_{\bm k}({C},\bar{\sigma}\circ\tau)$ as the composition of mutations defined above.

Stokes factors and Mutations
----------------------------

Fix a finite subset $C\subset {\mathbb{C}}$ and a bijection $\tau:C{\xrightarrow{\sim}}\{1,\dots, m\}$. Let $\big{(}(V,T),(V_{c},T_{c})_{{c}\in {C}}, f, f^*\big{)}$ be an object of ${{\sf Std}}(C,\tau)$.

Let $S_m$ be the symmetric group of degree $m$, generated by permutations $s_i:=(i\ i+1)$, $(i=1,2,\dots, m-1)$. We define a map $(\cdot)_R:S_m\to\mathrm{Br}_m$ by $(s)_R:=\sigma_{i_1}\cdots\sigma_{i_k}$ for a reduced expression $s=s_{i_1}\cdots s_{i_k}$. This map is well defined $($see, for example, [@boulie2 Chapter 4, §1.5, Proposition 5], [@matgen Theorem 2]$)$.

For $s\in S_m$, set $I_i(s):=\left\{j\in\{1,\dots, m\}\mid i<j, s(i)>s(j)\right\}$, and $I(s):=\{(i, j)\mid j\in I_i(s)\}$.

Let $A=\{a_1,a_2,\dots, a_k\}$ be a finite ordered set with $a_1<a_2<\cdots<a_k$. Let $(h_a)_{a\in A}$ be a sequence of endomorphisms of $V$ indexed by $A$. Then, we use the following notation$:$ $$\begin{aligned}
           \prod_{a\in A}^{\leftarrow}h_a:=h_{a_k}\circ h_{a_{k-1}}\circ\cdots\circ h_{a_1}, 
           & \
           \prod_{a\in A}^{\rightarrow}h_a:=h_{a_1}\circ h_{a_{2}}\circ\cdots\circ h_{a_k}.           
          