Both cluster algebras $A_n$ and $D_n$ admit an interpretation in terms of (tagged) triangulations of Riemann surfaces - respectively a Poincarè disk with n+3 punctures on the boundary and a Poincarè disk with n punctures on the boundary and one in the interior. The cluster variables are the lambda-lengths, i.e. geodesic distances between horocycles centered at the punctures. Flips of the triangulations correspond to mutations of the cluster variables.

Now, we can express the cluster variables explicitly in many ways. For example for $A_n$, if we use the hyperboloid model of the Poincarè disk, we can uniquely associate horocycles to light-like vectors $k_a$, then a lambda length associated to an edge joining puncture $i$ and $j$ is given by the (root of) Lorentz scalar product $k_i \cdot k_j$. Clearly it scales if we scale the vectors $k_i$. We can define scale invariant quantities of lambda lengths, i.e. cross ratios, which themselves satisfied mutation relations, and depend only on the light-rays rather than the vectors. If I am not mistaken, $\lambda$-lengths are the $\mathcal{A}$ variable of $A_n$ and cross ratios are the $\mathcal{X}$ variables.

How this picture generalize to $D_n$? What is the expression of the $\mathcal{X}$ variables in terms of the $\mathcal{A}$ variables for $D_n$ and is there some simple underling geometrical picture as before?

Both cluster algebras $A_n$ and $D_n$ admit an interpretation in terms of (tagged) triangulations of Riemann surfaces - respectively a Poincaré disk with n+3 punctures on the boundary and a Poincaré disk with n punctures on the boundary and one in the interior. The cluster variables are the lambda-lengths, i.e. geodesic distances between horocycles centered at the punctures. Flips of the triangulations correspond to mutations of the cluster variables.

Now, we can express the cluster variables explicitly in many ways. For example for $A_n$, if we use the hyperboloid model of the Poincaré disk, we can uniquely associate horocycles to light-like vectors $k_a$, then a lambda length associated to an edge joining puncture $i$ and $j$ is given by the (root of) Lorentz scalar product $k_i \cdot k_j$. Clearly it scales if we scale the vectors $k_i$. We can define scale invariant quantities of lambda lengths, i.e. cross ratios, which themselves satisfied mutation relations, and depend only on the light-rays rather than the vectors. If I am not mistaken, $\lambda$-lengths are the $\mathcal{A}$ variable of $A_n$ and cross ratios are the $\mathcal{X}$ variables.

How this picture generalize to $D_n$? What is the expression of the $\mathcal{X}$ variables in terms of the $\mathcal{A}$ variables for $D_n$ and is there some simple underling geometrical picture as before?