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Periodicity, linearizability, and integrability in seed mutations of type \(\boldsymbol{A}_N^{(1)}\). (English) Zbl 1475.37064

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This paper regards connections between the theories of dynamical systems and cluster algebras. The authors propose studying the dynamics of seed mutations in some suitable cluster algebras. The approach allows giving results concerning the Laurent phenomenon, the periodicity conjecture, and the discrete integrability in such algebras.
The analysis starts by defining the object of study, which are generalized Cartan matrices of type \(A^{(1)}_{N}\) associated with a unique path \(w=(t_{0}, t_{1},t_{2},\dots)\) in a \(n\)-regular tree \(\mathbb{T}_{N+1}\). The partial assignment \(w\subset\mathbb{T}_{N+1} \rightarrow \Sigma=(\Sigma_{0},\Sigma_{1},\Sigma_{2},\dots)\) is defined where for \(t\geq 0\), \(\Sigma_{t}=(\textbf{x}_{t}, \textbf{y}_{t}, B_{t}=(b^{t}_{ij}))\), is a seed with \(\textbf{x}_{t}=(x_{1;t},\dots, x_{n;t})\), \(\textbf{y}_{t}=(y_{1;t},\dots, y_{n;t})\), \(\textbf{x}_{t}\) is the cluster of the seed, \(\textbf{y}_{t}\) is the coefficient tuple and \(B_{t}\) is a suitable exchange matrix associated with a point \(t\in\mathbb{T}_{n}\). The map \(f: w\subset \mathbb{T}_{N+1}\rightarrow \Sigma\) with \(f(t_{l(N+1)+k})=\Sigma_{l(N+1)+k}\) is said to be the subcluster pattern. The periodicity of the quivers \(Q_{l(N+1)+k}\) associated with the exchange matrices \(B_{l(N+1)+k}\) is given by the following identity:
\[Q_{l(N+1)+k}=(\sigma_{N+1})^{k}Q_{} \] where \(Q_{0}\) has the shape
\[Q_0=\quad \begin{tikzcd}[row sep=0pt] \bigcirc\ar[rrrrrr, bend left=25]\rar&\bigcirc\rar&\bigcirc\rar&\dots\rar&\bigcirc\rar&\bigcirc\rar&\bigcirc\\ N+1 & N & N-1 & & 3 & 2 & 1 \end{tikzcd}\tag{F}\]
and \(\sigma_{N+1}\) is the \(N+1\)-permutation \((N+1,1,2,\dots, N-1,N)\).
In the third section, the authors study the dynamics of the coefficients \(\textbf{y}_{t}\) by proving the following identity via mutations and the periodicity of the exchange matrices (\(B_{l(N+1)+k}=B_{k}\)) :
\[v^{l+1}_{j}=\sigma_{N}v^{l}_{j}=v^{l}_{j+1} \]
which means that for \(l\geq1\) and \(j=1,2,\dots, N\), \(v^{l}_{j}\) has period \(N\) on \(l\). Where,
\[\begin{aligned} v^l_1 &=y_{N+1;l}y_{1;l(N+1)},\\ v^l_j &=y_{j;l(N+1)},\quad j=2,3,\dots, N. \end{aligned} \]
In the general case, the authors give explicit solutions \(y^{l}=(y^{l}_{1}, y^{l}_{2},\dots, y^{l}_{N+1})\) to the dynamical system governed by the map \(y^{l+1}=\psi(y^{l})\). Thus, an explicit formula for coefficients \(y_{k, l(N+1)+k-1}\) is given based on the coefficient \(y_{N+1}\) and some suitable constants.
The fourth section of the paper is devoted to the dynamics of the cluster variables, firstly it is proved that the cluster variables assigned to the path \(w\) of type \(A^{(1)}_{N}\) are given by using the solutions to the dynamical systems:
\[ \begin{aligned} z^{t+1}_{i}&=\frac{z^{t+1}_{i-1}z^{t}_{i+1}+1}{z^{t}_{i}},\quad i=1,2,\dots, N+1,\quad t\geq 1,\\ z^{t+1}_{0}&=z^{t}_{N+1},\\ z^{t}_{N+2}&=z^{t+1}_{1}. \end{aligned} \tag{1}\]
The proof of this result is based on the exchange rules and the explicit formulas given by the dynamic system governed by \(\psi\).
The authors also prove that for any \(t\geq 1\) and \(i=1,2,\dots, n\), it holds that \(\lambda^{t+N}_{i}=\lambda^{t}_{i}\), where
\[ \begin{aligned} \lambda_{i}&=\lambda_{i}(z^{1}_{1},\dots, z^{1}_{N+1})=\frac{z^{1}_{1}+z^{1}_{i+2}}{z^{1}_{i+1}},\\ \lambda_{N}&=\lambda_{N}(z^{1}_{1},\dots, z^{1}_{N+1})=\frac{z^{1}_{1}z^{1}_{N}+z^{1}_{2}z^{1}_{N+1}+1}{z^{1}_{1}z^{1}_{N+1}}. \end{aligned} \]
The system (1) can be linearized and the general solution to the dynamical system governed by the birational map \(\varphi\) such that \(z^{t+1}=\varphi(z^{t})\) is explicitly given. This results allows concluding that the cluster variables \(x^{t}_{1}, x^{t}_{2},\dots x^{t}_{N+1} \) assigned to the path \(w\) of type \(A^{(1)}_{N}\) exhibit the Laurent phenomenon.
Some properties of the birational map \(\varphi\) are given in the appendix of the paper.

MSC:

37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
37J70 Completely integrable discrete dynamical systems
13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
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References:

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