This is a simple Demonstration of how Mathematica
can be used to facilitate computations with cluster algebras. Cluster algebras were introduced by Fomin and Zelevinsky [2–4] and are a rapidly expanding subject in mathematics . They have found applications in many different areas, including combinatorics, representation theory, number theory, and physics.
A cluster algebra can be generated from certain data (seeds), which consist of a skew-symmetric exchange matrix
representing a quiver
(without loops or two-cycles) and cluster variables
. For simplicity, we treat here only the case without coefficients, but the source code can be manipulated to include the latter as well. This data,
, is subject to certain transformations called "mutations"; see  and  for an introduction.
A mutation around a vertex
of the quiver changes the exchange matrix
according to the formula (matrix mutation)
and the variables
according to the exchange relation
(Cluster algebras). Let
be a quiver without loops or two-cycles and with vertex set
. Given the initial seed
, define the clusters
with respect to
to be the sets
appearing in seeds
via iterated mutations, the cluster variables
to be the elements of all clusters, and the cluster algebra associated with
to be the
-subalgebra of the field
generated by all the cluster variables.
This Demonstration shows how the mutations change the initial seed for several examples. The important feature we would like to emphasize is that Mathematica
allows one to combine graphical interactive displays of the quivers arising in cluster algebras with the needed symbolic computational power to manipulate the cluster variables. Fomin and Zelevinsky have shown that the cluster variables are always Laurent polynomials, that is, their denominators are always monomials. This Demonstration lets you verify this remarkable fact on simple examples.
The set of cluster variables in cluster algebras may be finite or infinite. According to a classification theorem by Fomin and Zelevinsky , the set of cluster-finite cluster algebras is parametrized by the finite root systems of semisimple Lie algebras; we have included here the examples
. The example related to the pentagram map  yields a cluster algebra of infinite type, since the mutation sequence 5-6-4-5-3 produces a matrix
; compare with .
 S. Fomin and A. Zelevinsky, "Cluster Algebras I: Foundations," Journal of the American Mathematical Society
(2), 2002 pp. 497–529. doi:10.1090/S0894-0347-01-00385-X
 S. Fomin and A. Zelevinsky, "Cluster Algebras IV: Coefficients," Compositio Mathematica
(1), 2007 pp. 112–164. doi:10.1112/S0010437X06002521
 A. Berenstein, S. Fomin, and A. Zelevinsky, "Cluster Algebras III: Upper Bounds and Double Bruhat Cells," Duke Mathematical Journal
(1), 2005 pp. 1–52. doi:10.1215/S0012-7094-04-12611-9
 S. Fomin and A. Zelevinsky, "Cluster Algebras II: Finite Type Classification," Inventiones Mathematicae
(1), 2003 pp. 63–121. doi:10.1007/s00222-003-0302-y