Visual Cluster Algebras


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Show d-vector rays
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We follow the setup in Lee-Li-Zelevinsky. For any integer matrix \[ B = \begin{pmatrix} 0 &b \\ -c &0 \end{pmatrix}, \] let \( \Sigma = ((x_1,x_2),B)\) be a seed in \(\mathbb{Q}(x_1,x_2)\). Define \(x_n\) for \(n \in \mathbb{Z}\) recursively by the relations \[ x_{n-1} x_{n+1} = \begin{cases} x_n^b+1, \quad \text{ for n odd }; \\ x_n^c+1, \quad \text{ for n even.}\end{cases} \] The cluster algebra with trivial coefficients \( \mathcal{A} = \mathcal{A}(\Sigma)\) is the \(\mathbb{Z}\)-subalgebra of \(\mathbb{Q}(x_1,x_2)\) generated by the set \(\{x_n \mid n \in \mathbb{Z}\}\). The elements \(x_n\) are called cluster variables, and the pairs \((x_n, x_{n+1})\) are called clusters. Each cluster variable \(x = x_n\) is parameterised by an integer vector \(d = d_n\), called the d-vector of x with respect to the initial cluster \((x_1,x_2)\). Each d-vector \(d_n\) gives rise to a ray \( r_n = \mathbb{R}_{\geq 0} \cdot d_n\) in \(\mathbb{R}^2\).
When \(bc \geq 4\), the elements \(\dots ,d_{-1}, d_0, d_1, d_2, d_3, \dots\) are all distinct. Consider the infinite sequence \((r_n \mid n = 1, 2, ...)\) (respectively the infinite sequence \((r_n \mid n = 0, -1, ...)\)). This sequence converges to a ray denoted \(r_{\infty}\) (resp. converges to a ray denoted \(r_{-\infty}\)). We call these rays "extremal rays".

Parameter notations

1) The integers \(b\) and \(c\) denote the entries of the mutation matrix \(B\).
2) Given \(m \in \mathbb{Z}_{\geq 0}\), the software computes the rays \[ r_{-m}, \dots , r_0 , r_1 , r_2 , r_3 , \dots , r_{m+3}, \] which appears as black lines on the plane. Note that for any choice of \((b,c)\), we have that \(d_0, d_1, d_2\) and \(d_3\) are respectively \(\begin{pmatrix} 0\\1 \end{pmatrix}, \begin{pmatrix} -1\\0 \end{pmatrix}, \begin{pmatrix} 0\\-1 \end{pmatrix}\) and \(\begin{pmatrix} 1\\0 \end{pmatrix}\); thus the corresponding rays overlap with the standard coordinate axes of the plane. The remaining \(2m\) rays all appear in the first quadrant.
3) The extremal rays (when \(bc>3\)) are drawn in red.