We mutate these quivers at the vertex V
There is nothing special with this point, only that it is randomly chosen. Feel free to try mutating at any other point.
Mutation of quivers
Mutation of quivers was jointly discovered by S. Fomin and A. Zelevinsky. The concept found its origin in the study of cluster algebras and their mutations.
Example of Quiver mutation
Mutation of quivers is like playing a simple game on the quiver diagrams and rules of this game are quite simple. Mutation can be performed at any point of the quiver to give another quiver,called, the mutated quiver. The rules of this game are given through examples highlighted at each step.
Step 1.
Points of the quiver remain unchanged
Step 2.
Arrows into V in the original quiver gives arrows out of V in the new quiver.
Step 3.
Arrows out of V in the original quiver gives arrows into V in the new quiver.
Step 4.
Arrows not to or from V in the original quiver remain unchanged in the new quiver.
Step 5.
Any path of length 2 through V in the original quiver gives an arrow fron the start to the end of the path in the new quiver
Step 6.
We simplify the mutated quiver by getting the difference between arrows joining any two points but in opposite direction.
The game of mutation is simple to follow. As it turns out, we can get back to our original quiver by repeating the process of mutation at the very vertex, In our case, V.
The concept of mutation can be extended to a family of quivers called del Pezzo quivers