How to draw a quiver for a pseudoline arragement?

Question

In the lecture notes, on page 24, there is an example of drawing a quiver for a pseudoline arragement. What is the rule to draw a quiver for a pseudoline arragement? I don't know how to put the directions of arrows. Thank you very much.

Answer 1

Recall how the quiver and its mutations encode the clusters the their mutations. The vertices of the quiver correspond to cluster variables. When we mutate at a vertex $x$ the arrows change as prescribed while all vertices except $x$ stay the same. The vertex $x$ is replaced by $x'$ and we have the relation $$xx' = M_{in} + M_{out}$$ where $M_{in}$ and $M_{out}$ are monomials consisting of vertices adjacent to $x$ by incoming and out going arrows respectively. See pages 23 and 26 of the linked notes for details.

Now look at the exchange relations for the chamber minors on page 19. This tells us what the arrows must look like for the quiver mutations to correspond to the braid moves. For this example direct horizontal arrows left and direct vertical arrows right. In the picture on the lower left of page 24 move the vertex corresponding the $\Delta_{13}$ up so it is to the left of $\Delta_{23}$, then it follows this rule.