Number of cluster variables

Question

In the paper Hernandez and Leclerc - Cluster algebras and quantum affine algebras, Section 13.5, it is said that when $\mathfrak{g}$ is of type $A_2$ and $\ell=2$, then the corresponding cluster algebra $\mathscr{A}_2$ for $U_q(\widehat{\mathfrak{g}})$ is of finite cluster type $D_4$. There are 16 cluster variables.

It is said that (1) when $\mathfrak{g}$ is of type $A_2$ and $\ell=3$, (2) when $\mathfrak{g}$ is of type $A_2$ and $\ell=4$, (3) when $\mathfrak{g}$ is of type $A_3$ and $\ell=2$, and (4) when $\mathfrak{g}$ is of type $A_4$ and $\ell=2$, the corresponding cluster algebras are also finite type cluster algebras. How many cluster variables do we have in the cases (1), (2), (3), and (4)? Thank you very much.

Answer 1

For $A_k$ of level $\ell$, the cluster types are given by square grids of size $k \times \ell$.

Therefore the types you ask for are $E_6$ and $E_8$ (see Scott - Grassmannians and cluster algebras), for which you can easily find the number of clusters using the usual formula expressing them in terms of the exponents (see the Y-system article Fomin and Zelevinsky - $Y$-systems and generalized associahedra in the Annals for example). The numbers of clusters are 833 and 25080.

The number of cluster variables are smaller: 42 and 128.