Visual Cluster Algebras

Mutation patterns

Following Fomin and Zelevinsky's definitions in Cluster algebras IV. Coefficients (2007), this tool computes:

  1. (Tall) matrix mutations.
  2. \(\mathbb{Z}^{\rm min} / \mathbb{Z}^{\rm max}\)-tropical point \(\mathcal{A}\)-mutations.
  3. \(\mathbb{Z}^{\rm min} / \mathbb{Z}^{\rm max}\)-tropical point \(\mathcal{X}\)-mutations.

Greedy elements

Greedy elements are distinguished elements in rank 2 cluster algebras, defined recursively in an article by Lee, Li and Zelevinsky in 2012. This tool computes greedy elements in any rank 2 cluster algebra.

Denominator vector fan

The Laurent phenomenon insures each cluster variable has a denominator vector with respect to a fixed initial seed. The set of all denominator vectors defines a simplicial fan, called the d-vector fan (with respect to a fixed seed). This tool provides a plot of the d-vector fan for any rank 2 cluster algebra.

Acyclic cluster algebras

Acyclic cluster algebras are parametrised by symmetrisable Cartan matrices, and the underlying combinatorics appear in many different contexts. This tool computes:

  1. \(\mathbb{Q}\)-valued frieze patterns and Y-frieze patterns.
  2. Cluster-additive functions and tropical friezes.
  3. Cluster monomials and global monomials (in finite type).