where $$d_{ij} := \dim \Ext^1_{A_T}(S_{T_i},S_{T_j}).$$ This is known as the *Gabriel quiver* of $A_T$.

Let now $\i=(i_r,\ldots,i_1)$ be a reduced expression for $w$. Following [@BFZ], we define a quiver $\G_\i$ as follows. The vertex set of $\G_\i$ is equal to $\{1,\ldots,r\}$. For $1 \le k \le r$, let $$\begin{aligned}
\label{kminus}
k^- &:= \max\left(\{0\}\cup \{1 \le s \le k-1 \mid i_s = i_k \}\right),\\
k^+ &:= \min\left(\{ k+1 \le s \le r \mid i_s = i_k \} \cup \{r+1\}\right).\end{aligned}$$ For $1 \le s,\,t \le r$ such that $i_s \not = i_t$, there are $|c_{i_s,i_t}|$ arrows from $s$ to $t$ provided $t^+ \ge s^+ > t > s$. (Here, as in §\[sect3.1\], the $c_{ij}$’s are the entries of the Cartan matrix.) These are called the [*ordinary arrows*]{} of $\G_\i$. Furthermore, for each $1 \le s \le r$ there is an arrow $s \to s^-$ provided $s^- > 0$. These are the [*horizontal arrows*]{} of $\G_\i$.

\[main6\] The module $V_\i$ is a $\CC_w$-cluster-tilting module, and we have $\G_{V_\i} = \G_\i$.

\[Example4.8\] We continue Example \[Example4.4\]. For $\i=(2,1,2,1)$, the Gabriel quiver $\G_\i$ of ${\operatorname{End}}_\L(V_\i)^{\rm op}$ is: $$\xymatrix@-1.5pc{
4 \ar@{->}[rrrr] \ar@<.5ex>@{<-}[rrdd]\ar@<-.5ex>@{<-}[rrdd]& & & & 
\ar@<.5ex>@{->}[lldd]\ar@<-.5ex>@{->}[lldd]2 \ar@<.5ex>@{<-}[rrdd]\ar@<-.5ex>@{<-}[rrdd]\\
&&&\\
&& 3 \ar@{->}[rrrr] & & & & 1 
