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Suppose we have a seed $(x,y,B)$ where $B$ is a skew-symmetrizable matrix, $x = \{x_1,\ldots,x_n\}$ and y is an n-tuple of elements in $Trop\{x_{n+1},\ldots,x_m\}$.

If there are finitely many cluster variables in the cluster algebra generated by this seed, then why are there only finitely many y-variables? (This is equivalent to asking why the extended part of $\tilde{B}$ stays bounded if there are only finitely many cluster variables.) -- it is mentioned here (https://arxiv.org/pdf/1707.07190.pdf) in the proof of Corollary 5.1.6 which makes me think it is trivial.

If there are finitely many cluster variables, it is clear why the mutation equivalence class of $B$ is finite, because otherwise there would be a mutation equivalent matrix $B'$ with an entry $b'_{ij} \geq 4$ and then performing a sequence of mutations at $i$ and $j$ would generate infinitely many cluster variables.

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Let $\Sigma:=(\mathbf{x},\mathbf{y},B)$ be an initial seed where $\mathbf{x}:=\{x_1,\ldots,x_n\},\mathbf{y}:=\{y_1,\ldots,y_n\}$ and let $\mathbf{c} := \{x_{n+1},\ldots,x_m\}$ denote the set of frozen variables.

By the Laurent phenomenon, we know that every cluster variable in the seed pattern generated by $\Sigma$ is a Laurent polynomial in the cluster variables $x_1,\ldots,x_n$ over the ring of frozen variables $\mathbb{Z}[\mathbf{c}]$.

If there are only finitely many cluster variables, then there exists $r>0$ such that: for any cluster variable in the seed pattern, no exponent of a frozen variable appearing in the cluster variable exceeds $r$ (when the cluster variable is expressed as a Laurent polynomial in $x_1,\ldots,x_n$ over $\mathbb{Z}[\mathbf{c}]$).

If there are infinitely many $y$-variables in the seed pattern then there exists a seed $(\overline{\mathbf{x}},\overline{\mathbf{y}},\overline{B})$ in the seed pattern where the exponent $\overline{b}_{kj}$ of a frozen variable $x_k$ appearing in $\overline{y}_{j} \in \overline{\mathbf{y}}$ is such that $|\overline{b}_{kj}|> 2r$ for some $n+1\leq k \leq m$ and $1 \leq j \leq n$.

To contradict the assumption there are infinitely many $y$-variables consider the following exchange relation:

$$\overline{x}_j\overline{x}'_j = \prod_{\overline{b}_{ij}>0}\overline{x}_i^{\overline{b}_{ij}}+ \prod_{\overline{b}_{ij}<0}\overline{x}_i^{-\overline{b}_{ij}}$$

On the LHS, the largest exponent of a frozen variable in $\overline{x}_j\overline{x}'_j$ is at most 2r. However, on the RHS, the frozen variable $\overline{x}_k = x_k$ appears with exponent greater than $2r$.

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