Quiver Mutations

A very helpful tool in understanding cluster algebra is the concept of “mutations.” Although in this section, we will not yet introduce any algebra, quiver mutations represent the structure of cluster algebras in some sense.

In the context of cluster algebra, “mutation” refers to a very specific action on clusters, which can be shown visually on quivers. As described in [2], there are three steps to mutation relative to a chosen vertex k:

1.) For all length-three subquivers i → k → j, a new directed edge of i → j must be added to the graph.

2.) All edges directly connected to k must be reversed.

3.) If there are any 1-cycles or 2-cycles that appear, remove them from the graph.

For example, as shown in the diagram below (which is Figure 1 in [2]), we can see that a mutation with respect to vertex “1” has occurred between the left and right graphs. Firstly, the subquivor 4 → 1 → 2 called for the addition of 4 → 2 to the new graph. Secondly, 1 → 2 and 4 → 1 were each reversed. Thirdly, there were no 1-cycles or 2-cycles, so none were removed.

Quiver Mutation Example

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