WOLFRAM|DEMONSTRATIONS PROJECT

Cluster Algebras

​
Quiver
D6
E6
Pentagram Map
Number of Mutations
Mutation Sequence
0
0
0
0
0
-1
1
0
0
0
1
0
-1
-1
1
0
-1
1
0
0
-1
1
0
1
0
0
-1
0
0
-1
1
1
0
-1
0
0
-1
0
1
0
{
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
}
Cluster algebras are certain commutative rings introduced by Fomin and Zelevinsky. Their generators (called cluster variables) and algebraic relations are defined through an iterative process known as "seed mutations." The latter are defined in terms of a skew-symmetric matrix that can be represented as a quiver (a directed graph). To each node of the quiver, there is associated a transformation (called a mutation) that changes the original "seed," that is, the original skew matrix (or quiver) and the associated cluster variables
x
. This Demonstration shows several examples of how Mathematica can be used to generate a cluster algebra.
Use "Quiver" to select the seed from which you wish to start. Only three examples have been provided, but more can easily be added by changing the source code. "Number of Mutations" lets you control the number of mutations you wish to perform in one step. It changes the number of drop-down menus for the "Mutation Sequence." Choose the number of a vertex for performing a mutation and the value "0" for not performing one. The mutations are executed from left to right. The resulting quivers might become quite complicated and it can be helpful to rearrange the quiver to your liking. You can drag the vertices of the quiver to any position you like.
Displayed are the transformation of the quiver, the associated skew-symmetric matrix, and the cluster variables
x
under the specified sequence of mutations. Try the mutation sequence "5-3-1-6" with the
D
6
quiver in order to compare with the example in Keller's Java applet [1].