I was looking at P. Plamondon's paper titled "GENERIC BASES FOR CLUSTER ALGEBRAS FROM THE CLUSTER CATEGORY". I'm confused about the calculation in Example 4.3. The example starts with a quiver $Q$ and potential $W$, and then it considers a certain module $M$, and applies the functor $F$ to it. But, I don't understand why the calculation of the cluster character $X'_{M}$ is done by considering $FM$ as a representation over the opposite quiver of $Q$?
The general setting I am interested in is the following: Let $Q$ be a finite acyclic quiver, and let $ A = P_1 \oplus \dotsb \oplus P_n$ be its path algebra, expressed as the direct sum of the indecomposable projective modules $P_i$ at the vertices $i\in Q_0$. Consider the quiver with potential $(Q,0)$. Applying the Derksen–Weyman–Zelevinsky mutation procedure for quivers with potential, I mutate $(Q,0)$ along some fixed sequence to obtain a new quiver with potential $(\Gamma,W)$.
Independently, by using the same mutation sequence, starting with mutation of a summand of $A$, I obtain a cluster-tilting object $T$ in the cluster category $C(Q)$. Let $B = \operatorname{End}_{C(Q)}(T)$ be the associated cluster-tilted algebra. I also have the standard functor $F : C(Q) \longrightarrow \mathrm{mod}\text{-}B$, given by $X \longmapsto FX := \operatorname{Hom}_{C(Q)}(T,X)$.
I would like to understand the precise relationship between the DWZ-mutated quiver with potential $(\Gamma,W)$ and the algebra $B$.
Questions
- Is the quiver $\Gamma$, together with the relations determined by the Jacobian ideal, the ordinary quiver with relations for the algebra $B$?
- For an object $X \in C(Q)$, the functor produces a right $B$-module $FX = \operatorname{Hom}_{C(Q)}(T,X)$. Do these $B$-modules correspond to representations of the quiver $\Gamma$ (with its Jacobian relations), or do they correspond instead to representations of the opposite quiver $\Gamma^{\,op}$?
Thank you very much for your help!