Easy way to understand theta basis for X-cluster algebras of finite type? For $\mathcal A$-cluster algebras of finite type, it is very easy to describe the theta-basis: it consists of the cluster monomials. Is there any similarly easy way to describe the theta-basis for $\mathcal X$-cluster algebras?
For $\mathcal A$-cluster algebras not of finite type, the cluster monomials always form a subset of the theta-basis, and if the description I asked for above extends to the non-finite type case in a similar way, I would like to know that too, but I would still be very happy with an answer which doesn't say anything about non finite type.
 A: There's another answer which doesn't work in full generality, but is often easier to work with, especially if you know the A-type theta functions really well.
When the exchange matrix has full rank, X-type theta functions are certain A-type theta functions after a change of variables
Specifically, let $B$ be an $d\times r$ extended exchange matrix whose rank is equal to its width $r$. Choose a skew-symmetrizable matrix $\widehat{B}$ whose first $r$-many columns are $B$, and let $B'$ denote the first $r$-many columns of $\widehat{B}^\top$. (Note that $B'$ does not depend on the choice of $\widehat{B}$, and $B'=B^\top$ if $B$ is square)
Let $\rho_{\widehat{B}^\top}:\mathbb{Z}[y^{\mathbb{Z}^d}]\rightarrow \mathbb{Z}[x^{\mathbb{Z}^d}]$ be the ring homomorphism which sends $y^n$ to $x^{\widehat{B}^\top n}$. This is an augmentation map in the sense of Fock-Goncharov (or at least, it is when restricted to the cluster algebras/varieties).
Then
$$\rho_{\widehat{B}^\top}(\vartheta_{\mathfrak{s}^\vee}[n])
= \vartheta_{\mathfrak{s}'}[\widehat{B}^\top n]
$$
where the latter theta function is an A-type theta function with exchange matrix $B'$.
In general, this doesn't determine the dual theta function. However, when $B$ has full rank (rank equal to its width), then this uniquely determines $\vartheta_{\mathfrak{s}^\vee}[n]$, which can be computed by replacing each $x^{\widehat{B}^\top n}$ by $y^n$.

As an example, let
$$B =\widehat{B} = \begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$$
Then $\rho_{\widehat{B}^\top}(y^{(n_1,n_2)}) = x^{(2n_2,-n_1)}$, and
$$ \rho_{\widehat{B}^\top}(\vartheta_{\mathfrak{s}^\vee}[1,-1])
= \vartheta_{\mathfrak{s}^\top}[-2,-1]
$$
where the latter theta function is in the A-type cluster algebra of $B'=B^\top= \begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$. One may compute that this is a cluster monomial; specifically,
\begin{align*}
\vartheta_{\mathfrak{s}^\top}[-2,-1] 
&= \vartheta_{\mathfrak{s}^\top}[-1,0]^2\vartheta_{\mathfrak{s}^\top}[0,-1]\\
&=(x^{(-1,0)}+x^{(-1,-1)}+x^{(1,-1)})^2( x^{(0,-1)} + x^{(2,-1)})\\
&= x^{(-2,-1)}(1+x^{(0,-1)}+x^{(2,-1)})^2( 1 + x^{(2,0)})
\end{align*}
The last expression has been factored so the exponent of each monomial is in the image of $\widehat{B}^\top$. Replacing each $x^{\widehat{B}^\top n}$ with $y^n$ yields the corresponding X-type theta function.
$$\vartheta_{\mathfrak{s}^\vee}[1,-1] 
= y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1+y^{(0,1)})
$$
