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](http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=5866).



Cluster algebra structure on the coordinate ring of $Mat_3$
Let 
$$F = \big\{
\Delta\_2^2, \Delta\_{23}^{23}, \Delta\_{3}^{2}, \Delta\_{2}^{3}, \Delta\_{1}^{3}, \Delta\_{3}^{1}, \Delta\_{12}^{23}, \Delta\_{23}^{12}, \Delta\_{123}^{123}
\big\}.$$ 


The claim in the linked talk is not that $F$ is a linear basis for $\mathbb{C}[Mat\_3] = \mathbb{C}[x\_{ij}]$, but that each $x\_{ij}$ can be expressed as a rational function in the $\Delta^J\_I \in F$. The claim is actually a bit more, it is that each minor of the matrix $M = (x\_{ij})$ is a subtraction-free rational function in the elements of $F$. This means if each minor in $F$ is positive, then the matrix is totally positive.


Here is how to recover $x\_{11}$. Notice we are given $x\_{22}, x\_{23}, x\_{32}, x\_{13}$, and $x\_{31}$. We can then recover
$$x\_{33} = \frac{\Delta^{23}\_{23} + x\_{23}x\_{32}}{x\_{22}}$$
since we when also given $\Delta^{23}\_{23}$. We can then similarly recover $x\_{12}$ and $x\_{21}$ since we have the minors $\Delta^{23}\_{12}$ and $\Delta^{12}\_{23}$ along with all but one relevant coordinate in each case. We can then recover
$$\Delta\_{12}^{12} = \frac{x\_{22}\Delta\_{123}^{123} + \Delta^{12}\_{23}\Delta^{23}\_{12}}{\Delta^{23}\_{23}}$$
using the [Lewis Carroll identity](https://en.wikipedia.org/wiki/Dodgson_condensation#Desnanot.E2.80.93Jacobi_identity_and_proof_of_correctness_of_the_condensation_algorithm). Finally we get $x\_{11}$ using $\Delta\_{12}^{12}$ along with $x\_{12}, x\_{21}$, and $x\_{22}$.


