Probably the shortest answer, especially in finite-type:

X-type theta functions are monomials times F-polynomials

Making this precise is a bit fiddly, because there are multiple conventions and (more importantly) two different notions of `the' dual cluster algebra to an A-type cluster algebra:

• The "Fock-Goncharov dual" (sometimes called the cluster dual or the Langlands dual), which can be connected to the original algebra via an augmentation map
• The "GHKK dual" (sometimes called the mirror dual), which corresponds to the mirror dual directly constructed by the Gross-Siebert program.

They are closely related; if $B$ is the exchange matrix used to define one, the other is defined by $B^\top$. As a consequence, whenever $B$ is skew-symmetric and the full Fock-Goncharov conjecture holds, there is a twist isomorphism between them which sends theta functions to theta functions. However, even in this case, the set of theta functions have different parametrizations by the dual lattice $N$, and theta reciprocity only holds for the GHKK dual (without a twist of the parametrization).

With that caveat out of the way, I can tell you how to describe the X-type theta functions in the GHKK dual. If $\mathfrak{s}$ is the coefficient-free $A$-type seed defined by a skew-symmetrizable matrix $B$, then the GHKK dual theta function associated to $n\in N$ is

$$\vartheta_{\mathfrak{s}^\vee}[n] = y^n F_{B^\top}[B^\top n] $$

where $F_{B^\top}[B^\top n]$ is the F-polynomial obtained by taking the A-type seed of $B$, adjoining principal coefficients (denoted by $y$s), computing the A-type theta function $\vartheta_{B^\top}[B^\top n]$, and specializing the initial $x$-variables to $1$.

In finite-type, the theta functions correspond to cluster monomials and the F-polynomials here correspond to the F-polynomials of Fomin-Zelevinsky. The latter may be computed by any number of enumerative problems. For example, given a simply-laced Dynkin quiver, the F-polynomial will be a generating function of Euler characteristics of quiver Grassmannians.

An example of this construction

Let $B=\begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$. With the convention I'm using, the six A-type cluster variables are \begin{align*} \vartheta_\mathfrak{s}[1,0] &= x^{(1,0)} \\ \vartheta_\mathfrak{s}[0,1] &= x^{(0,1)} \\ \vartheta_\mathfrak{s}[-1,0] &= x^{(-1,0)}+x^{(-1,2)} \\ \vartheta_\mathfrak{s}[0,-1] &= x^{(0,-1)}+x^{(-1,-1)}+x^{(-1,1)} \\ \vartheta_{\mathfrak{s}}[1,-2] &= x^{(1,-2)} + 2x^{(0,-2)} + x^{(-1,-2)}+x^{(-1,0)} \\ \vartheta_{\mathfrak{s}}[1,-1] &= x^{(1,-1)} + x^{(0,-1)} \end{align*} All theta functions are cluster monomials (products of adjacent pairs above). The scattering diagram and the Newton polytope of several theta functions (with a dot at $m$) are pictured below.

To cook up the F polynomials, we compute the six A-type cluster variables of $\mathsf{B}^\top=\begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$ with principal coefficients. \begin{align*} \vartheta_{\mathfrak{s}^\top}[1,0] &= x^{(1,0)} \\ \vartheta_{\mathfrak{s}^\top}[0,1] &= x^{(0,1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,1] &= x^{(-1,1)}+y^{(1,0)}x^{(-1,0)} \\ \vartheta_{\mathfrak{s}^\top}[-2,1] &= x^{(-2,1)} + 2y^{(1,0)}x^{(-2,0)} + y^{(2,0)} x^{(-2,-1)}+y^{(2,1)}x^{(0,-1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,0] &= x^{(-1,0)}+y^{(1,0)}x^{(-1,-1)}+y^{(1,1)}x^{(1,-1)} \\ \vartheta_{\mathfrak{s}^\top}[0,-1] &= x^{(0,-1)} + y^{(0,1)}x^{(2,-1)} \end{align*} and specialize the $x$-variables to $1$: \begin{align*} F_{B^\top}[1,0] &= 1 \\ F_{B^\top}[0,1] &= 1 \\ F_{B^\top}[-1,1] &= 1+y^{(1,0)} \\ F_{B^\top}[-2,1] &= 1 + 2y^{(1,0)} + y^{(2,0)} + y^{(2,1)} \\ F_{B^\top}[-1,0] &= 1+y^{(1,0)}+y^{(1,1)} \\ F_{B^\top}[0,-1] &= 1 + y^{(0,1)} \end{align*} All other F-polynomials in this case can be computed as products of adjacent F-polynomials above; e.g. $$ F_{B^\top}[-2,-1] = F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] =(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) $$

The X-type theta function in the GHKK dual with $n=(1,-1)$ is then \begin{align*} \vartheta_{\mathfrak{s}^\vee}[1,-1] &= y^{(1,-1)}F_{B^\top}[-2,-1] \\ % &= y^{(1,-1)}F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] \\ &= y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) \end{align*} We can check directly that theta reciprocity holds between $\vartheta_{\mathfrak{s}^\vee}[1,-1]$ and each of the cluster variables on the $A$-type side. $$\begin{array}{|c|ccccccc|} \hline m & (1,0) & (0,1) & (-1,0) & (0,-1) & (1,-2) & (1,-1) \\ \hline \mathrm{val}_{(1,-1)}(\vartheta_{\mathfrak{s}}[m]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \end{array}$$ Note that the latter row is given by the tropicalization of $\vartheta_{\mathfrak{s}^\vee}[1,-1]$: $$\mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) = (m_1-m_2) + 2\min(0,m_1,m_1+m_2)+ \min(0,m_2) $$

Probably the shortest answer, especially in finite-type:

X-type theta functions are monomials times F-polynomials

Making this precise is a bit fiddly, because there are multiple conventions and (more importantly) two different notions of `the' dual cluster algebra to an A-type cluster algebra:

• The "Fock-Goncharov dual" (sometimes called the cluster dual or the Langlands dual), which can be connected to the original algebra via an augmentation map
• The "GHKK dual" (sometimes called the mirror dual), which corresponds to the mirror dual directly constructed by the Gross-Siebert program.

They are closely related; if $B$ is the exchange matrix used to define one, the other is defined by $B^\top$. As a consequence, whenever $B$ is skew-symmetric and the full Fock-Goncharov conjecture holds, there is a twist isomorphism between them which sends theta functions to theta functions. However, even in this case, the set of theta functions have different parametrizations by the dual lattice $N$, and theta reciprocity only holds for the GHKK dual (without a twist of the parametrization).

With that caveat out of the way, I can tell you how to describe the X-type theta functions in the GHKK dual. If $\mathfrak{s}$ is the coefficient-free $A$-type seed defined by a skew-symmetrizable matrix $B$, then the GHKK dual theta function associated to $n\in N$ is

$$\vartheta_{\mathfrak{s}^\vee}[n] = y^n F_{B^\top}[B^\top n] $$

where $F_{B^\top}[B^\top n]$ is the F-polynomial obtained by taking the A-type seed of $B^\top$, adjoining principal coefficients (denoted by $y$s), computing the A-type theta function $\vartheta_{B^\top}[B^\top n]$, and specializing the initial $x$-variables to $1$.

In finite-type, the theta functions correspond to cluster monomials and the F-polynomials here correspond to the F-polynomials of Fomin-Zelevinsky. The latter may be computed by any number of enumerative problems. For example, given a simply-laced Dynkin quiver, the F-polynomial will be a generating function of Euler characteristics of quiver Grassmannians.

An example of this construction

Let $B=\begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$. With the convention I'm using, the six A-type cluster variables are \begin{align*} \vartheta_\mathfrak{s}[1,0] &= x^{(1,0)} \\ \vartheta_\mathfrak{s}[0,1] &= x^{(0,1)} \\ \vartheta_\mathfrak{s}[-1,0] &= x^{(-1,0)}+x^{(-1,2)} \\ \vartheta_\mathfrak{s}[0,-1] &= x^{(0,-1)}+x^{(-1,-1)}+x^{(-1,1)} \\ \vartheta_{\mathfrak{s}}[1,-2] &= x^{(1,-2)} + 2x^{(0,-2)} + x^{(-1,-2)}+x^{(-1,0)} \\ \vartheta_{\mathfrak{s}}[1,-1] &= x^{(1,-1)} + x^{(0,-1)} \end{align*} All theta functions are cluster monomials (products of adjacent pairs above). The scattering diagram and the Newton polytope of several theta functions (with a dot at $m$) are pictured below.

To cook up the F polynomials, we compute the six A-type cluster variables of $\mathsf{B}^\top=\begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$ with principal coefficients. \begin{align*} \vartheta_{\mathfrak{s}^\top}[1,0] &= x^{(1,0)} \\ \vartheta_{\mathfrak{s}^\top}[0,1] &= x^{(0,1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,1] &= x^{(-1,1)}+y^{(1,0)}x^{(-1,0)} \\ \vartheta_{\mathfrak{s}^\top}[-2,1] &= x^{(-2,1)} + 2y^{(1,0)}x^{(-2,0)} + y^{(2,0)} x^{(-2,-1)}+y^{(2,1)}x^{(0,-1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,0] &= x^{(-1,0)}+y^{(1,0)}x^{(-1,-1)}+y^{(1,1)}x^{(1,-1)} \\ \vartheta_{\mathfrak{s}^\top}[0,-1] &= x^{(0,-1)} + y^{(0,1)}x^{(2,-1)} \end{align*} and specialize the $x$-variables to $1$: \begin{align*} F_{B^\top}[1,0] &= 1 \\ F_{B^\top}[0,1] &= 1 \\ F_{B^\top}[-1,1] &= 1+y^{(1,0)} \\ F_{B^\top}[-2,1] &= 1 + 2y^{(1,0)} + y^{(2,0)} + y^{(2,1)} \\ F_{B^\top}[-1,0] &= 1+y^{(1,0)}+y^{(1,1)} \\ F_{B^\top}[0,-1] &= 1 + y^{(0,1)} \end{align*} All other F-polynomials in this case can be computed as products of adjacent F-polynomials above; e.g. $$ F_{B^\top}[-2,-1] = F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] =(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) $$

The X-type theta function in the GHKK dual with $n=(1,-1)$ is then \begin{align*} \vartheta_{\mathfrak{s}^\vee}[1,-1] &= y^{(1,-1)}F_{B^\top}[-2,-1] \\ % &= y^{(1,-1)}F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] \\ &= y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) \end{align*} We can check directly that theta reciprocity holds between $\vartheta_{\mathfrak{s}^\vee}[1,-1]$ and each of the cluster variables on the $A$-type side. $$\begin{array}{|c|ccccccc|} \hline m & (1,0) & (0,1) & (-1,0) & (0,-1) & (1,-2) & (1,-1) \\ \hline \mathrm{val}_{(1,-1)}(\vartheta_{\mathfrak{s}}[m]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \end{array}$$ Note that the latter row is given by the tropicalization of $\vartheta_{\mathfrak{s}^\vee}[1,-1]$: $$\mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) = (m_1-m_2) + 2\min(0,m_1,m_1+m_2)+ \min(0,m_2) $$

Probably the shortest answer, especially in finite-type:

X-type theta functions are F-polynomials, times a monomial

Making this precise is a bit fiddly, because there are multiple conventions and (more importantly) two different notions of `the' dual cluster algebra to an A-type cluster algebra:

• The "Fock-Goncharov dual" (sometimes called the cluster dual or the Langlands dual), which can be connected to the original algebra via an augmentation map
• The "GHKK dual" (sometimes called the mirror dual), which corresponds to the mirror dual directly constructed by the Gross-Siebert program.

They are closely related; if $B$ is the exchange matrix used to define one, the other is defined by $B^\top$. As a consequence, whenever $B$ is skew-symmetric and the full Fock-Goncharov conjecture holds, they are isomorphic and the theta functions coincide. However, even in this case, the set of theta functions have different parametrizations by the dual lattice $N$, and theta reciprocity only holds for the GHKK dual (without a twist or reparametrization).

With that caveat out of the way, I can tell you how to describe the X-type theta functions in the GHKK dual. If $\mathfrak{s}$ is the coefficient-free $A$-type seed defined by a skew-symmetrizable matrix $B$, then the GHKK dual theta function associated to $n\in N$ is

$$\vartheta_{\mathfrak{s}^\vee}[n] = y^n F_{B^\top}[B^\top n] $$

where $F_{B^\top}[B^\top n]$ is the F-polynomial obtained by taking the A-type seed of $B$, adjoining principal coefficients (denoted by $y$s), computing the A-type theta function $\vartheta_{B^\top}[B^\top n]$, and specializing the initial $x$-variables to $1$.

In finite-type, the theta functions correspond to cluster monomials, and the F-polynomials here correspond to the F-polynomials of Fomin-Zelevinsky, and may be computed by any number of enumerative problems. For example, given a simply-laced Dynkin quiver, the F-polynomial will be a generating function of Euler characteristics of quiver Grassmannians.

An example of this construction

Let $B=\begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$. With the convention I'm using, the six A-type cluster variables are \begin{align*} \vartheta_\mathfrak{s}[1,0] &= x^{(1,0)} \\ \vartheta_\mathfrak{s}[0,1] &= x^{(0,1)} \\ \vartheta_\mathfrak{s}[-1,0] &= x^{(-1,0)}+x^{(-1,2)} \\ \vartheta_\mathfrak{s}[0,-1] &= x^{(0,-1)}+x^{(-1,-1)}+x^{(-1,1)} \\ \vartheta_{\mathfrak{s}}[1,-2] &= x^{(1,-2)} + 2x^{(0,-2)} + x^{(-1,-2)}+x^{(-1,0)} \\ \vartheta_{\mathfrak{s}}[1,-1] &= x^{(1,-1)} + x^{(0,-1)} \end{align*} All theta functions are cluster monomials (products of adjacent pairs above). The scattering diagram and the Newton polytope of several theta functions (with a dot at $m$) are pictured below.

To cook up the F polynomials, we compute the six A-type cluster variables of $\mathsf{B}^\top=\begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$ with principal coefficients. \begin{align*} \vartheta_{\mathfrak{s}^\top}[1,0] &= x^{(1,0)} \\ \vartheta_{\mathfrak{s}^\top}[0,1] &= x^{(0,1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,1] &= x^{(-1,1)}+y^{(1,0)}x^{(-1,0)} \\ \vartheta_{\mathfrak{s}^\top}[-2,1] &= x^{(-2,1)} + 2y^{(1,0)}x^{(-2,0)} + y^{(2,0)} x^{(-2,-1)}+y^{(2,1)}x^{(0,-1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,0] &= x^{(-1,0)}+y^{(1,0)}x^{(-1,-1)}+y^{(1,1)}x^{(1,-1)} \\ \vartheta_{\mathfrak{s}^\top}[0,-1] &= x^{(0,-1)} + y^{(0,1)}x^{(2,-1)} \end{align*} and specialize the $x$-variables to $1$: \begin{align*} F_{B^\top}[1,0] &= 1 \\ F_{B^\top}[0,1] &= 1 \\ F_{B^\top}[-1,1] &= 1+y^{(1,0)} \\ F_{B^\top}[-2,1] &= 1 + 2y^{(1,0)} + y^{(2,0)} + y^{(2,1)} \\ F_{B^\top}[-1,0] &= 1+y^{(1,0)}+y^{(1,1)} \\ F_{B^\top}[0,-1] &= 1 + y^{(0,1)} \end{align*} All other F-polynomials in this case can be computed as products of adjacent F-polynomials above.

The X-type theta function in the GHKK dual with $n=(1,-1)$ is then \begin{align*} \vartheta_{\mathfrak{s}^\vee}[1,-1] &= y^{(1,-1)}F_{B^\top}[-2,-1] \\ &= y^{(1,-1)}F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] \\ &= y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) \end{align*} We can check directly that theta reciprocity holds between $\vartheta_{\mathfrak{s}^\vee}[1,-1]$ and each of the cluster variables on the $A$-type side. $$\begin{array}{|c|ccccccc|} \hline m & (1,0) & (0,1) & (-1,0) & (0,-1) & (1,-2) & (1,-1) \\ \hline \mathrm{val}_{(1,-1)}(\vartheta_{\mathfrak{s}}[m]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \end{array}$$ Note that the latter row is given by the tropicalization of $\vartheta_{\mathfrak{s}^\vee}[1,-1]$: $$\mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) = (m_1-m_2) + 2\min(0,m_1,m_1+m_2)+ \min(0,m_2) $$

Probably the shortest answer, especially in finite-type:

X-type theta functions are monomials times F-polynomials

Making this precise is a bit fiddly, because there are multiple conventions and (more importantly) two different notions of `the' dual cluster algebra to an A-type cluster algebra:

• The "Fock-Goncharov dual" (sometimes called the cluster dual or the Langlands dual), which can be connected to the original algebra via an augmentation map
• The "GHKK dual" (sometimes called the mirror dual), which corresponds to the mirror dual directly constructed by the Gross-Siebert program.

They are closely related; if $B$ is the exchange matrix used to define one, the other is defined by $B^\top$. As a consequence, whenever $B$ is skew-symmetric and the full Fock-Goncharov conjecture holds, there is a twist isomorphism between them which sends theta functions to theta functions. However, even in this case, the set of theta functions have different parametrizations by the dual lattice $N$, and theta reciprocity only holds for the GHKK dual (without a twist of the parametrization).

With that caveat out of the way, I can tell you how to describe the X-type theta functions in the GHKK dual. If $\mathfrak{s}$ is the coefficient-free $A$-type seed defined by a skew-symmetrizable matrix $B$, then the GHKK dual theta function associated to $n\in N$ is

$$\vartheta_{\mathfrak{s}^\vee}[n] = y^n F_{B^\top}[B^\top n] $$

where $F_{B^\top}[B^\top n]$ is the F-polynomial obtained by taking the A-type seed of $B$, adjoining principal coefficients (denoted by $y$s), computing the A-type theta function $\vartheta_{B^\top}[B^\top n]$, and specializing the initial $x$-variables to $1$.

In finite-type, the theta functions correspond to cluster monomials and the F-polynomials here correspond to the F-polynomials of Fomin-Zelevinsky. The latter may be computed by any number of enumerative problems. For example, given a simply-laced Dynkin quiver, the F-polynomial will be a generating function of Euler characteristics of quiver Grassmannians.

An example of this construction

Let $B=\begin{bmatrix} 0 & -1 \\ 2 & 0 \end{bmatrix}$. With the convention I'm using, the six A-type cluster variables are \begin{align*} \vartheta_\mathfrak{s}[1,0] &= x^{(1,0)} \\ \vartheta_\mathfrak{s}[0,1] &= x^{(0,1)} \\ \vartheta_\mathfrak{s}[-1,0] &= x^{(-1,0)}+x^{(-1,2)} \\ \vartheta_\mathfrak{s}[0,-1] &= x^{(0,-1)}+x^{(-1,-1)}+x^{(-1,1)} \\ \vartheta_{\mathfrak{s}}[1,-2] &= x^{(1,-2)} + 2x^{(0,-2)} + x^{(-1,-2)}+x^{(-1,0)} \\ \vartheta_{\mathfrak{s}}[1,-1] &= x^{(1,-1)} + x^{(0,-1)} \end{align*} All theta functions are cluster monomials (products of adjacent pairs above). The scattering diagram and the Newton polytope of several theta functions (with a dot at $m$) are pictured below.

To cook up the F polynomials, we compute the six A-type cluster variables of $\mathsf{B}^\top=\begin{bmatrix} 0 & 2 \\ -1 & 0 \end{bmatrix}$ with principal coefficients. \begin{align*} \vartheta_{\mathfrak{s}^\top}[1,0] &= x^{(1,0)} \\ \vartheta_{\mathfrak{s}^\top}[0,1] &= x^{(0,1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,1] &= x^{(-1,1)}+y^{(1,0)}x^{(-1,0)} \\ \vartheta_{\mathfrak{s}^\top}[-2,1] &= x^{(-2,1)} + 2y^{(1,0)}x^{(-2,0)} + y^{(2,0)} x^{(-2,-1)}+y^{(2,1)}x^{(0,-1)} \\ \vartheta_{\mathfrak{s}^\top}[-1,0] &= x^{(-1,0)}+y^{(1,0)}x^{(-1,-1)}+y^{(1,1)}x^{(1,-1)} \\ \vartheta_{\mathfrak{s}^\top}[0,-1] &= x^{(0,-1)} + y^{(0,1)}x^{(2,-1)} \end{align*} and specialize the $x$-variables to $1$: \begin{align*} F_{B^\top}[1,0] &= 1 \\ F_{B^\top}[0,1] &= 1 \\ F_{B^\top}[-1,1] &= 1+y^{(1,0)} \\ F_{B^\top}[-2,1] &= 1 + 2y^{(1,0)} + y^{(2,0)} + y^{(2,1)} \\ F_{B^\top}[-1,0] &= 1+y^{(1,0)}+y^{(1,1)} \\ F_{B^\top}[0,-1] &= 1 + y^{(0,1)} \end{align*} All other F-polynomials in this case can be computed as products of adjacent F-polynomials above; e.g. $$ F_{B^\top}[-2,-1] = F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] =(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) $$

The X-type theta function in the GHKK dual with $n=(1,-1)$ is then \begin{align*} \vartheta_{\mathfrak{s}^\vee}[1,-1] &= y^{(1,-1)}F_{B^\top}[-2,-1] \\ % &= y^{(1,-1)}F_{B^\top}[-1,0]^2F_{B^\top}[0,-1] \\ &= y^{(1,-1)}(1+y^{(1,0)}+y^{(1,1)})^2(1 + y^{(0,1)}) \end{align*} We can check directly that theta reciprocity holds between $\vartheta_{\mathfrak{s}^\vee}[1,-1]$ and each of the cluster variables on the $A$-type side. $$\begin{array}{|c|ccccccc|} \hline m & (1,0) & (0,1) & (-1,0) & (0,-1) & (1,-2) & (1,-1) \\ \hline \mathrm{val}_{(1,-1)}(\vartheta_{\mathfrak{s}}[m]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) & 1 & -1 & -3 & -2 & -1 & 1 \\ \hline \end{array}$$ Note that the latter row is given by the tropicalization of $\vartheta_{\mathfrak{s}^\vee}[1,-1]$: $$\mathrm{val}_{m}(\vartheta_{\mathfrak{s}^\vee}[1,-1]) = (m_1-m_2) + 2\min(0,m_1,m_1+m_2)+ \min(0,m_2) $$