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Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.

We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the minor a of a matrix consisting of the $i_1,\ldots, i_n$-th rows and $j_1,\ldots, j_n$-th columns.

A basis of $\mathbb{C}[Mat_3]$ is $x_{ij}$, $i,j=1,2,3$, where $x_{ij}(a) = a_{ij}$, $a \in \mathbb{C}[Mat_3]$.

Is $$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$$ a basis of $\mathbb{C}[Mat_3]?$ How to obtain $x_{11}$ from $$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}?$$

The 9-tuple $$(\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3} \mid \Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123})$$ is an extended cluster ($\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3}$ are cluster variables and the rest are frozen variables). Using the mutation: $$\Delta_{12}^{12} \Delta_{23}^{23} = \Delta_{12}^{23}\Delta_{23}^{12} + \Delta_{2}^{2} \Delta_{123}^{123},$$ we obtain $\Delta_{12}^{12}$ and another extended cluster $$(\Delta_2^2, \Delta_{12}^{12}, \Delta_{3}^{2},\Delta_{2}^{3} \mid \Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}).$$ How many clusters do this cluster algebra have? Thank you very much.

Edit: learn this example from a talk by M. Shapiro.

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.

We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the minor a of a matrix consisting of the $i_1,\ldots, i_n$-th columns and $j_1,\ldots, j_n$-th rows.

A basis of $\mathbb{C}[Mat_3]$ is $x_{ij}$, $i,j=1,2,3$, where $x_{ij}(a) = a_{ij}$, $a \in \mathbb{C}[Mat_3]$.

Is $$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$$ a basis of $\mathbb{C}[Mat_3]?$ How to obtain $x_{11}$ from $$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}?$$

The 9-tuple $$(\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3} \mid \Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123})$$ is an extended cluster ($\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3}$ are cluster variables and the rest are frozen variables). Using the mutation: $$\Delta_{12}^{12} \Delta_{23}^{23} = \Delta_{12}^{23}\Delta_{23}^{12} + \Delta_{2}^{2} \Delta_{123}^{123},$$ we obtain $\Delta_{12}^{12}$ and another extended cluster $$(\Delta_2^2, \Delta_{12}^{12}, \Delta_{3}^{2},\Delta_{2}^{3} \mid \Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}).$$ How many clusters do this cluster algebra have? Thank you very much.

Edit: I learn this example from the beginning of a talk by M. Shapiro.

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.

We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the minor a of a matrix consisting of the $i_1,\ldots, i_n$-th rows and $j_1,\ldots, j_n$-th columns.

A basis of $\mathbb{C}[Mat_3]$ is $x_{ij}$, $i,j=1,2,3$, where $x_{ij}(a) = a_{ij}$, $a \in \mathbb{C}[Mat_3]$.

Is $\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ a basis of $\mathbb{C}[Mat_3]$? How to obtain $x_{11}$ from $\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$?

$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ is a cluster. Using the mutation: $\Delta_{12}^{12} \Delta_{23}^{23} = \Delta_{12}^{23}\Delta_{23}^{12} + \Delta_{2}^{2} \Delta_{123}^{123}$, we obtain $\Delta_{12}^{12}$ and another cluster $\Delta_2^2, \Delta_{12}^{12}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ is a cluster. How many clusters do this cluster algebra have? Thank you very much.

Edit: learn this example from a talk by M. Shapiro.

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.

We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the minor a of a matrix consisting of the $i_1,\ldots, i_n$-th rows and $j_1,\ldots, j_n$-th columns.

A basis of $\mathbb{C}[Mat_3]$ is $x_{ij}$, $i,j=1,2,3$, where $x_{ij}(a) = a_{ij}$, $a \in \mathbb{C}[Mat_3]$.

Is $$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$$ a basis of $\mathbb{C}[Mat_3]?$ How to obtain $x_{11}$ from $$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}?$$

The 9-tuple $$(\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3} \mid \Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123})$$ is an extended cluster ($\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3}$ are cluster variables and the rest are frozen variables). Using the mutation: $$\Delta_{12}^{12} \Delta_{23}^{23} = \Delta_{12}^{23}\Delta_{23}^{12} + \Delta_{2}^{2} \Delta_{123}^{123},$$ we obtain $\Delta_{12}^{12}$ and another extended cluster $$(\Delta_2^2, \Delta_{12}^{12}, \Delta_{3}^{2},\Delta_{2}^{3} \mid \Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}).$$ How many clusters do this cluster algebra have? Thank you very much.

Edit: learn this example from a talk by M. Shapiro.

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.

We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the minor a of a matrix consisting of the $i_1,\ldots, i_n$-th rows and $j_1,\ldots, j_n$-th columns.

A basis of $\mathbb{C}[Mat_3]$ is $x_{ij}$, $i,j=1,2,3$, where $x_{ij}(a) = a_{ij}$, $a \in \mathbb{C}[Mat_3]$.

Is $\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ a basis of $\mathbb{C}[Mat_3]$? How to obtain $x_{11}$ from $\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$?

$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ is a cluster. Using the mutation: $\Delta_{12}^{12} \Delta_{23}^{23} = \Delta_{12}^{23}\Delta_{23}^{12} + \Delta_{2}^{2} \Delta_{123}^{123}$, we obtain $\Delta_{12}^{12}$ and another cluster $\Delta_2^2, \Delta_{12}^{12}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ is a cluster. How many clusters do this cluster algebra have? Thank you very much.

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$.

We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the minor a of a matrix consisting of the $i_1,\ldots, i_n$-th rows and $j_1,\ldots, j_n$-th columns.

A basis of $\mathbb{C}[Mat_3]$ is $x_{ij}$, $i,j=1,2,3$, where $x_{ij}(a) = a_{ij}$, $a \in \mathbb{C}[Mat_3]$.

Is $\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ a basis of $\mathbb{C}[Mat_3]$? How to obtain $x_{11}$ from $\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$?

$\Delta_2^2, \Delta_{23}^{23}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ is a cluster. Using the mutation: $\Delta_{12}^{12} \Delta_{23}^{23} = \Delta_{12}^{23}\Delta_{23}^{12} + \Delta_{2}^{2} \Delta_{123}^{123}$, we obtain $\Delta_{12}^{12}$ and another cluster $\Delta_2^2, \Delta_{12}^{12}, \Delta_{3}^{2},\Delta_{2}^{3},\Delta_{1}^{3},\Delta_{3}^{1},\Delta_{12}^{23},\Delta_{23}^{12},\Delta_{123}^{123}$ is a cluster. How many clusters do this cluster algebra have? Thank you very much.

Edit: learn this example from a talk by M. Shapiro.