2.1 Tropical patterns via tropical points

The starting point of our paper is a simple observation that both equations (1.2) and (1.3) are tropicalizations of certain relations in a field of rational functions. To see this, we first fix some notation to be used throughout the paper.

Notation 2.1. By ${\mathbb Z}^{ {\mathrm {max}} }$ we mean the semi-field $({\mathbb Z}, \cdot , \oplus )$ with $a \cdot b = a+b$ and $a \oplus b = \mathrm {max}(a, b)$ for $a, b \in {\mathbb Z}$ . For any semi-field $\mathbb {P}$ , let $\mathrm {Hom}_{\mathrm {sf}}(\mathbb {P}, {\mathbb Z}^{\mathrm {max}})$ be the set of all semi-field homomorphisms from $\mathbb {P}$ to ${\mathbb Z}^{\mathrm {max}}$ . For r independent variables $\mathbf {x} = (x_1, \ldots , x_r)$ , let ${\mathbb Q}(\mathbf {x})_{>0}$ be the semi-field of rational functions in $\mathbf {x}$ over ${\mathbb Q}$ that have subtraction-free expressions in $\mathbf {x}$ . We also write

$$\begin{align*}\mathrm{val}_\delta(x) = \delta(x) \hspace{.2in} \;\;\; \text{for}\;\;\; \delta \in \mathrm{Hom}_{\mathrm{sf}}({\mathbb Q}(\mathbf{x})_{>0}, {\mathbb Z}^{ {\mathrm{max}} }) \;\; \text{and} \;\; x \in {\mathbb Q}(\mathbf{x})_{>0}. \end{align*}$$

Let $A = (a_{i, j})$ be any $r \times r$ symmetrizable generalized Cartan matrix, that is, an integral $r \times r$ matrix such that $a_{i, i} = 2$ for all $i \in [1, r]$ , $a_{i, j} \leq 0$ for all $i \neq j$ , and $DA$ is symmetric for some diagonal matrix D with positive integers on the diagonal. Consider two families $\{x(i,m)\}_{(i, m) \in [1,r] \times {\mathbb Z}}$ and $\{y(i,m)\}_{(i, m) \in [1,r] \times {\mathbb Z}}$ of commuting variables, respectively satisfying, for all $(i, m) \in [1, r] \times {\mathbb Z}$ ,

(2.1) $$ \begin{align} x(i,m) \, x(i,m+1) &= 1 + \prod_{j=i+1}^r x(j,m)^{-a_{j,i}} \; \prod_{j=1}^{i-1} x(j,m+1)^{-a_{j,i}},\qquad \end{align} $$

(2.2) $$ \begin{align} y(i,m) \, y(i,m+1) &= \prod_{j=i+1}^r (1+y(j,m))^{-a_{j,i}} \; \prod_{j=1}^{i-1} (1+y(j,m+1))^{-a_{j,i}}. \end{align} $$

As equations (2.1) and (2.2) are recursive with respect to the order in (1.4), setting

$$\begin{align*}\mathbf{x} = (x(1, 0), \ldots, x(r, 0)) \hspace{.2in} \text{and} \hspace{.2in} \mathbf{y} = (y(1, 0), \ldots, y(r, 0)) \end{align*}$$

as independent variables, one can respectively regard $\{x(i,m)\}_{(i, m) \in [1,r] \times {\mathbb Z}}$ and $\{y(i,m)\}_{(i, m) \in [1,r] \times {\mathbb Z}}$ as systems of elements in ${\mathbb Q}(\mathbf {x})_{>0}$ and ${\mathbb Q}(\mathbf {y})_{>0}$ .

For $\delta \in \mathrm {Hom}_{\mathrm {sf}}({\mathbb Q}(\mathbf {x})_{>0}, {\mathbb Z}^{ {\mathrm {max}} })$ and $\rho \in \mathrm {Hom}_{\mathrm {sf}}({\mathbb Q}(\mathbf {y})_{>0}, {\mathbb Z}^ {\mathrm {max}} )$ , define

$$ \begin{align*} &f_\delta:\; [1, r] \times {\mathbb Z} \longrightarrow {\mathbb Z}, \;\; f_\delta(i, m) = \mathrm{val}_\delta (x(i, m)),\\ &k_\rho:\; [1, r] \times {\mathbb Z} \longrightarrow {\mathbb Z}, \;\; k_\rho(i, m) = \mathrm{val}_\rho (y(i, m)). \end{align*} $$

Lemma 2.2. For any symmetrizable generalized Cartan matrix A, one has bijections

(2.3) $$ \begin{align} &\mathrm{Hom}_{\mathrm{sf}}({\mathbb Q}(\mathbf{x})_{>0},\; {\mathbb Z}^ {\mathrm{max}} ) \longrightarrow C_{\mathrm{frieze}}(A), \;\; \delta \longmapsto f_\delta, \end{align} $$

(2.4) $$ \begin{align} &\mathrm{Hom}_{\mathrm{sf}}({\mathbb Q}(\mathbf{y})_{>0},\; {\mathbb Z}^ {\mathrm{max}} ) \longrightarrow C_{\mathrm{add}}(A), \;\; \rho \longmapsto k_\rho. \end{align} $$

Tropicalization is best formulated via the notion of positive spaces, as introduced in [Reference Fock and Goncharov21, §1.1].

Definition 2.3. A positive space of dimension r is a pair

$$\begin{align*}\mathbb{X} = (\mathbb{F}, \,\{\mathbf{x}_t\}_{t \in {\mathbb T}}), \end{align*}$$

where $\mathbb {F}$ is a field isomorphic to the field of rational functions over ${\mathbb Q}$ in r independent variables, ${\mathbb T}$ is any nonempty index set, and $\{\mathbf {x}_t\}_{t \in {\mathbb T}}$ is an assignment of an ordered set of free generators of $\mathbb {F}$ over ${\mathbb Q}$ to each $t \in {\mathbb T}$ , such that ${\mathbb Q}(\mathbf {x}_t)_{>0} = {\mathbb Q}(\mathbf {x}_{t'})_{>0} \subset \mathbb {F}$ for all $t, t' \in {\mathbb T}$ . We also write $\mathbb {F} = \mathbb {F}(\mathbb {X})$ .

For a positive space $\mathbb {X} = (\mathbb {F}(\mathbb {X}), \{\mathbf {x}_t\}_{t \in {\mathbb T}})$ , we call each $\mathbf {x}_t$ , $t \in {\mathbb T}$ , a coordinate chart, or a chart, of $\mathbb {X}$ . Define the ambient semi-field of $\mathbb {X}$ as $\mathbb {F}_{>0}(\mathbb {X}) = {\mathbb Q}(\mathbf {x}_t)_{>0}$ using any $t \in {\mathbb T}$ , and set

$$\begin{align*}\mathbb{X}({\mathbb Z}^{\mathrm{max}}) = \mathrm{Hom}_{\mathrm{sf}}(\mathbb{F}_{>0}(\mathbb{X}), \, {\mathbb Z}^{\mathrm{max}}). \end{align*}$$

Elements in $\mathbb {X}({\mathbb Z}^{\mathrm {max}})$ are called ${\mathbb Z}^{\mathrm {max}}$ -tropical points, or simply tropical points, of $\mathbb {X}$ . Note that one can define a tropical point $\delta $ of $\mathbb {X}$ by specifying its coordinate vector $(\mathrm {val}_\delta (x_{t; 1}), \ldots , \mathrm {val}_\delta (x_{t; r}))$ using any chart $\mathbf {x}_t = (x_{t; 1}, \ldots , x_{t; r})$ of $\mathbb {X}$ .

The theory of cluster algebras famously gives rise to two classes of positive spaces. Indeed, recall that a skew-symmetrizable integral matrix is also called a mutation matrix. Associated to an $r \times r$ mutation matrix B, one has two r-dimensional positive spaces

$$\begin{align*}\mathbb{A}_B = (\mathbb{F}(\mathbb{A}_B), \; \{\mathbf{x}_t\}_{t \in {\mathbb T}_r}) \hspace{.2in} \text{and} \hspace{.2in} \mathbb{Y}_B = (\mathbb{F}(\mathbb{Y}_B), \; \{\mathbf{y}_t\}_{t \in {\mathbb T}_r}), \end{align*}$$

where ${\mathbb T}_r$ is a rooted r-regular tree, $\{\mathbf {x}_t\}_{t \in {\mathbb T}_r}$ is the collection of $\mathbf {A}$ -clusters, pairwise related by sequences of $\mathbf {A}$ -variable (i.e., cluster variable) mutations (see (3.2)), and $\{\mathbf {y}_t\}_{t \in {\mathbb T}_r}$ is the collection of $\mathbf {Y}$ -clusters, pairwise related by sequences of $\mathbf {Y}$ -variable mutations (see (3.5)). The subalgebras

$$\begin{align*}{\mathcal{U}}(\mathbb{A}_B) = \bigcap_{t \in {\mathbb T}_r} {\mathbb Z}[\mathbf{x}_t^{\pm 1}] \subset \mathbb{F}(\mathbb{A}_B) \hspace{.2in} \text{and} \hspace{.2in} {\mathcal{U}}(\mathbb{Y}_B) = \bigcap_{t \in {\mathbb T}_r} {\mathbb Z}[\mathbf{y}_t^{\pm 1}] \subset \mathbb{F}(\mathbb{Y}_B) \end{align*}$$

are respectively called the upper cluster algebras (with trivial coefficients) of $\mathbb {A}_B$ and of $\mathbb {Y}_B$ , and they are the central objects of study in the theory of cluster algebras [Reference Berenstein, Fomin and Zelevinsky3, Reference Fomin and Zelevinsky23]. By respectively tropicalizing the $\mathbf {A}$ -variable and $\mathbf {Y}$ -variable mutation rules, one obtains the respective mutation rules for the coordinate vectors of tropical points of $\mathbb {A}_B$ and tropical points of $\mathbb {Y}_B$ . See Lemma 3.6 and Lemma 3.13.

For an $r \times r$ symmetrizable generalized Cartan matrix $A = (a_{i, j})$ , we define the mutation matrix

(2.5) $$ \begin{align} B = B_A = \begin{pmatrix} 0 & a_{1,2} & a_{1,3} & \dots & a_{1,r} \\ -a_{2,1} & 0 & a_{2,3} & \dots & a_{2,r} \\ -a_{3,1} & -a_{3,2} & 0 & \dots & a_{3,r} \\ \dots& \dots & \dots & \dots & \dots \\ -a_{r,1} & -a_{r,2} & -a_{r,3}& \dots &0 \end{pmatrix}. \end{align} $$

Note that $B_A$ is acyclic in the sense of [Reference Berenstein, Fomin and Zelevinsky3, Definition 1.14], and every acyclic mutation matrix is, up to simultaneous relabeling of rows and columns, of the form $B_A$ for a symmetrizable generalized Cartan matrix A [Reference Berenstein, Fomin and Zelevinsky3, p.36].

The starting point of our paper is the fact that the relations in (2.1) and (2.2) are respectively satisfied by certain $\mathbf {A}$ -variables and certain $\mathbf {Y}$ -variables in the Fock-Goncharov dual pair

(2.6) $$ \begin{align} \mathbb{O}_A \, \stackrel{\mathrm{def}}{=}\, (\mathbb{A}_{B^T}, \; \mathbb{Y}_B). \end{align} $$

To give more detail, let $t_0$ be the root vertex of ${\mathbb T}_r$ and label by $t(i, m)$ , $(i, m) \in [1, r] \times {\mathbb Z}_{\geq 0}$ , the vertices on the infinite path in ${\mathbb T}_r$ that starts from $t_0$ and follows the sequence of edges $(1, 2, \ldots , r, 1, 2, \ldots )$ via

(2.7)

Similarly, label by $t(i, m)$ , $(i, m) \in [1, r] \times {\mathbb Z}_{\leq -1}$ , the vertices on the infinite path in ${\mathbb T}_r$ that starts from $t_0$ and follows the sequence of edges $(r, r-1, \ldots ,2, 1, r, r-1, \ldots )$ via

(2.8)

Notation 2.5. Denote the $\mathbf {A}$ -clusters of $\mathbb {A}_{B^T}$ by $\{\mathbf {x}_t^{\scriptscriptstyle \vee }\}_{t \in {\mathbb T}_r}$ and the $\mathbf {Y}$ -clusters of $\mathbb {Y}_B$ by $\{\mathbf {y}_t\}_{t \in {\mathbb T}_r}$ . Let $\mathfrak {X}(\mathbb {A}_{B^T})$ be the set of all cluster variables of $\mathbb {A}_{B^T}$ and $\mathfrak {X}(\mathbb {Y}_B)$ that of all $\mathbf {Y}$ -variables of $\mathbb {Y}_B$ . For $(i, m) \in [1, r] \times {\mathbb Z}$ , let $x^{\scriptscriptstyle \vee }(i, m) \in \mathfrak {X}(\mathbb {A}_{B^T})$ be the $i^{\mathrm {th}}$ variable in $\mathbf {x}^{\scriptscriptstyle \vee }_{t(i, m)}$ , and let $y(i, m) \in \mathfrak {X}(\mathbb {Y}_B)$ be the $i^{\mathrm {th}}$ variable in $\mathbf {y}_{t(i, m)}$ . Consider the maps

(2.9) $$ \begin{align} &[1, r] \times {\mathbb Z} \longrightarrow \mathfrak{X}(\mathbb{A}_{B^T}), \;\; (i, m) \longmapsto x^{\scriptscriptstyle \vee}(i, m), \end{align} $$

(2.10) $$ \begin{align} &[1, r] \times {\mathbb Z} \longrightarrow \mathfrak{X}(\mathbb{Y}_{B}), \;\; (i, m) \longmapsto y(i, m). \end{align} $$

For $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ and $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , let $(\delta ^{\scriptscriptstyle \vee }_{t; 1}, \ldots , \delta ^{\scriptscriptstyle \vee }_{t; r})$ and $(\rho _{t; 1}, \ldots , \rho _{t; r})$ be the respective coordinate vectors of $\delta ^{\scriptscriptstyle \vee }$ and $\rho $ at $t \in {\mathbb T}_r$ , and define $f_{{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}}, k_\rho : [1, r] \times {\mathbb Z} \to {\mathbb Z}$ by

(2.11) $$ \begin{align} &f_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}(i, m) = \mathrm{val}_{\delta^{\scriptscriptstyle \vee}}(x^{\scriptscriptstyle \vee}(i, m)) = \delta^{\scriptscriptstyle \vee}_{t(i, m); i}, \hspace{.2in} \hspace{.2in} (i, m) \in [1, r] \times {\mathbb Z}, \end{align} $$

(2.12) $$ \begin{align} &k_\rho(i,m) = \mathrm{val}_\rho(y(i, m)) = \rho_{t(i, m); i}, \hspace{.2in} \hspace{.2in} (i, m) \in [1, r] \times {\mathbb Z}. \end{align} $$

The following is a combination of Proposition 6.2, Proposition 6.5, and Proposition 6.11.

Theorem A. For any symmetrizable generalized Cartan matrix A and with $B = B_A$ as in (2.5), the map in (2.9) satisfies equation (2.1) with A replaced by $A^T$ , that is, for all $(i, m) \in [1, r] \times {\mathbb Z}$ ,

(2.13) $$ \begin{align} x^{{{\scriptscriptstyle \vee}}}(i,m) \, x^{{{\scriptscriptstyle \vee}}}(i,m+1) = 1 + \prod_{j=i+1}^r x^{{\scriptscriptstyle \vee}}(j,m)^{-a_{i, j}} \; \prod_{j=1}^{i-1} x^{{\scriptscriptstyle \vee}}(j,m+1)^{-a_{i, j}}, \end{align} $$

while the map in (2.10) satisfies equation (2.2). Consequently, one has the well-defined bijections

$$ \begin{align*} \mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}) &\longrightarrow C_{\mathrm{frieze}}(A^T), \;\; \delta^{\scriptscriptstyle \vee} \longmapsto f_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}, \\ \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}) &\longrightarrow C_{\mathrm{add}}(A), \;\; \rho \longmapsto k_\rho. \end{align*} $$

We refer to the bijections in Theorem A as realizations of tropical friezes and cluster-additive functions via tropical points of the Fock-Goncharov dual pair $(\mathbb {A}_{B^T}, \mathbb {Y}_B)$ .

The map in (2.9), called the generic $\mathbf {A}$ -frieze pattern Footnote ¹ , has been studied in [Reference Assem, Reutenauer and Smith2, Reference Germain, Huang and Lu16]. Correspondingly, we call the map in (2.10) the generic $\mathbf {Y}$ -frieze pattern. In view of Theorem A, tropical friezes are tropicalizations of generic $\mathbf {A}$ -frieze patterns, while cluster-additive functions are tropicalizations of generic $\mathbf {Y}$ -frieze patterns. We can thus also call cluster-additive functions tropical $\mathbf {Y}$ -friezes.

Theorem A sets the stage for the main investigation in our paper, namely that of the interplay between tropical friezes, cluster-additive functions, and Fock-Goncharov duality. For this purpose, we devote $\S $ 4 and $\S $ 5 to a general discussion, summarized in the next $\S $ 2.2, on certain aspects of Fock-Goncharov duality via the notion of admissible functions on Fock-Goncharov dual pairs.

2.2 Fock-Goncharov duality via admissible functions

Let B be any $r \times r$ mutation matrix, not necessarily of the form $B_A$ in (2.5), and consider the Fock-Goncharov dual pair $(\mathbb {A}_{B^T}, \mathbb {Y}_B)$ .

Following [Reference Gross, Hacking, Keel and Kontsevich27, Definition 0.1], for $\mathbb {X} = \mathbb {A}_{B^T}$ or $\mathbb {Y}_B$ , elements in ${\mathcal {U}}(\mathbb {X})$ that are Laurent monomials in some cluster of $\mathbb {X}$ are called global monomials on $\mathbb {X}$ . Global monomials on $\mathbb {A}_{B^T}$ are precisely the cluster monomials on $\mathbb {A}_{B^T}$ defined in [Reference Fomin and Zelevinsky22] (see Lemma 4.2). By [Reference Gross, Hacking, Keel and Kontsevich27] (see Lemma 4.8 and Lemma-Definition 4.17), global monomials on $\mathbb {A}_{B^T}$ are in bijection with their g-vectors, which lie in $\mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , and global monomials on $\mathbb {Y}_B$ are in bijection with their g-vectors, which lie in $\mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ .

V. V. Fock and A. B. Goncharov conjectured in [Reference Fock and Goncharov21, §Reference Bobiński4] that there is a ${\mathbb Z}$ -basis $\{x^{\scriptscriptstyle \vee }_\rho \}$ of ${\mathcal {U}}(\mathbb {A}_{B^T})$ , parametrized by $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ and extending the parametrization of cluster monomials on $\mathbb {A}_{B^T}$ by their g-vectors, and a ${\mathbb Z}$ -basis $\{y_{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}\}$ of ${\mathcal {U}}(\mathbb {Y}_B)$ , parametrized by $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ and extending the parametrization of global monomials on $\mathbb {Y}_B$ by their g-vectors, such that

(2.14) $$ \begin{align} \mathrm{val}_{\delta^{\scriptscriptstyle \vee}}(x^{\scriptscriptstyle \vee}_\rho) = \mathrm{val}_\rho(y_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}), \hspace{.2in} \;\;\forall \; \delta^{\scriptscriptstyle \vee} \in \mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}), \; \; \rho \in \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}). \end{align} $$

The common integer value in (2.14) is the conjectured Fock-Goncharov pairing between the tropical points $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ and $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ . The conjecture as stated does not hold in general [Reference Gross, Hacking and Keel26], as ${\mathcal {U}}(\mathbb {Y}_B)$ may have “too few” elements. In [Reference Gross, Hacking, Keel and Kontsevich27], the authors modified the conjecture by constructing the so-called theta functions $\theta _\rho \in {\mathcal {U}}(\mathbb {A}_{B^T})$ and $\theta _{\delta ^{\scriptscriptstyle \vee }} \in {\mathcal {U}}(\mathbb {Y}_B)$ for $\rho $ in a certain subset of $\mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ and $\delta ^{\scriptscriptstyle \vee }$ in a certain subset of $\mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ , and it is conjectured in [Reference Gross, Hacking, Keel and Kontsevich27, Remark 9.11] that equation (2.14) holds for $x^{\scriptscriptstyle \vee }_\rho = \theta _\rho $ and $y_{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}= \theta _{\delta ^{\scriptscriptstyle \vee }}$ . When B is skew-symmetric, the conjecture in [Reference Gross, Hacking, Keel and Kontsevich27, Remark 9.11] has been proved in [Reference Cheung, Magee, Mandel and Muller14] (see Remark 5.2).

While theta functions are important, their construction in [Reference Gross, Hacking, Keel and Kontsevich27] requires the rather involved techniques of scattering diagrams and broken lines in tropical spaces. In place of theta functions, in this paper we introduce and work with the so-called admissible functions. More precisely, for any $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , an element $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}(\mathbb {A}_{B^T})$ is $\rho $ -admissible, and we write

(2.15) $$ \begin{align} x^{\scriptscriptstyle \vee} \in {\mathcal{U}}^{\mathrm{admi}}_\rho(\mathbb{A}_{B^T}), \end{align} $$

if the Laurent expansion of $x^{\scriptscriptstyle \vee }$ in every cluster $\mathbf {x}_t^{\scriptscriptstyle \vee }$ of $\mathbb {A}_{B^T}$ has a certain prescribed form involving the coordinate vector of $\rho $ at t (see Definition 4.4 for details). An element $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}(\mathbb {A}_{B^T})$ is called an admissible function on $\mathbb {A}_{B^T}$ if it is $\rho $ -admissible for some $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ . Similarly, for $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ , we define $\delta ^{\scriptscriptstyle \vee }$ -admissible functions on $\mathbb {Y}_B$ (see Definition 4.14), and we write

(2.16) $$ \begin{align} y \in {\mathcal{U}}^{\mathrm{admi}}_{\delta^{\scriptscriptstyle \vee}}(\mathbb{Y}_B) \end{align} $$

if $y \in {\mathcal {U}}(\mathbb {Y}_B)$ is $\delta ^{\scriptscriptstyle \vee }$ -admissible. An element in ${\mathcal {U}}(\mathbb {Y}_B)$ is called an admissible function on $\mathbb {Y}_B$ if it is $\delta ^{\scriptscriptstyle \vee }$ -admissible for some $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ . Admissible functions, while still capturing the essential properties of theta functions, are more general (see $\S $ 4.6 and Example 4.5) and easier to work with due to the descriptive nature of their definition.

We regard the relations expressed in (2.15) and (2.16) as correspondences between tropical points in one positive space and global functions on the other in the dual pair $(\mathbb {A}_{B^T}, \mathbb {Y}_B)$ . The correspondences are in general not bijective (see Example 4.5). Inspired by [Reference Fock and Goncharov21, Conjectures 4.3] of Fock and Goncharov, we ask the following natural question (a similar question has been studied by J. Fei [Reference Fei19]).

Question. (Question 5.1) Let B be any $r \times r$ mutation matrix. When does one have

(2.17) $$ \begin{align} \mathrm{val}_{\delta^{\scriptscriptstyle \vee}}(x^{\scriptscriptstyle \vee}) = \mathrm{val}_\rho(y), \end{align} $$

where $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ , $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {admi}}_\rho (\mathbb {A}_{B^T})$ , and $y \in {\mathcal {U}}^{\mathrm {admi}}_{\delta ^{\scriptscriptstyle \vee }}(\mathbb {Y}_{B})$ ?

For the purpose of this paper, it is enough to consider admissible functions on positive $\mathbf {A}$ -spaces and positive $\mathbf {Y}$ -spaces with trivial coefficients (except in $\S $ 4.3, where we also consider principal coefficients). When the mutation matrices are of full rank, admissible functions are precisely the universally positive and compatibly pointed elements in the sense of F. Qin [Reference Qin38] (see Remark 4.6). Admissible functions on positive spaces with frozen variables (and full-rank extended mutation matrices) are used by the first author in [Reference Cao8] to study F-invariants.

In $\S $ 4, we review the bijections of global monomials on $\mathbb {A}_{B^T}$ and on $\mathbb {Y}_B$ with their g-vectors. For a cluster monomial $x^{\scriptscriptstyle \vee }$ on $\mathbb {A}_{B^T}$ , let $\rho (x^{\scriptscriptstyle \vee }) \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ be its g-vector, and for a global monomial y on $\mathbb {Y}_B$ , let $\delta ^{\scriptscriptstyle \vee }(y) \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ be its g-vector. The following is a summary of our results on admissible functions.

In Proposition 5.4, we show that the identity (2.17) holds if either $x^{\scriptscriptstyle \vee }$ or y is a global monomial.

2.3 Tropical friezes and cluster-additive functions via admissible functions

Let A be any $r \times r$ symmetrizable generalized Cartan matrix, let $B = B_A$ as in (2.5), and consider again the Fock-Goncharov dual pair $\mathbb {O}_A \, = \, (\mathbb {A}_{B^T}, \;\mathbb {Y}_{B})$ . Recall from (2.9) and (2.10) the generic frieze patterns

$$\begin{align*}(i, m) \longmapsto x^{\scriptscriptstyle \vee}(i, m) \in \mathfrak{X}(\mathbb{A}_{B^T}) \hspace{.2in} \text{and} \hspace{.2in} (i, m) \longmapsto y(i, m) \in \mathfrak{X}(\mathbb{Y}_B). \end{align*}$$

We show in Lemma 6.1 that each $y(i, m)$ is a global variable of $\mathbb {Y}_B$ , that is, a $\mathbf {Y}$ -variable of $\mathbb {Y}_B$ that is also a global monomial. For $(i, m) \in [1, r] \times {\mathbb Z}$ , denote the g-vectors of $x^{\scriptscriptstyle \vee }(i, m)$ and $y(i, m)$ respectively by

$$\begin{align*}\rho(i,m) \in \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}) \hspace{.2in} \text{and} \hspace{.2in} \delta^{\scriptscriptstyle \vee}(i, m) \in \mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}). \end{align*}$$

For $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {A}_{B^T})$ and $y \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {Y}_B)$ , define $f_y, k_{x^{\scriptscriptstyle \vee }}: [1, r]\times {\mathbb Z} \to {\mathbb Z}$ by

(2.18) $$ \begin{align} f_y(i, m) = \mathrm{val}_{\rho(i,m)}(y) \hspace{.2in} \text{and} \hspace{.2in} k_{x^{\scriptscriptstyle \vee}}(i,m) = \mathrm{val}_{\delta^{\scriptscriptstyle \vee}(i,m)}(x^{\scriptscriptstyle \vee}) \end{align} $$

for $(i, m) \in [1, r]\times {\mathbb Z}$ . Recall from Theorem A the bijections

$$ \begin{align*} \mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}) &\longrightarrow C_{\mathrm{frieze}}(A^T), \;\; \delta^{\scriptscriptstyle \vee} \longmapsto f_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}, \\ \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}) &\longrightarrow C_{\mathrm{add}}(A), \;\; \rho \longmapsto k_\rho. \end{align*} $$

The following theorem follows from Proposition 6.7 and Proposition 6.15.

Theorem C. Let A be any symmetrizable generalized Cartan matrix, and let $B = B_A$ as in (2.5).

1) One has $f_{y} \in C_{\mathrm {frieze}}(A^T)$ for every $y\in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {Y}_B)$ , and the map

$$\begin{align*}{\mathcal{U}}^{\mathrm{admi}}(\mathbb{Y}_B) \longrightarrow C_{\mathrm{frieze}}(A^T), \;\; y \longmapsto f_{y}, \end{align*}$$

is surjective. Moreover, for $y \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {Y}_B)$ and $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm { max}})$ , one has $f_{y} = f_{{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}}$ if and only if y is $\delta ^{\scriptscriptstyle \vee }$ -admissible.

2) One has $k_{x^{\scriptscriptstyle \vee }} \in C_{\mathrm {add}}(A)$ for every $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {A}_{B^T})$ , and the map

(2.19) $$ \begin{align} {\mathcal{U}}^{\mathrm{admi}}(\mathbb{A}_{B^T}) \longrightarrow C_{\mathrm{add}}(A), \;\; x^{\scriptscriptstyle \vee} \longmapsto k_{x^{\scriptscriptstyle \vee}}, \end{align} $$

is surjective. Moreover, for $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {A}_{B^T})$ and $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , one has $k_{x^{\scriptscriptstyle \vee }} = k_\rho $ if and only if $x^{\scriptscriptstyle \vee }$ is $\rho $ -admissible.

Combining Theorem A and Theorem C, we have thus two realizations of each $f \in C_{\mathrm {frieze}}(A^T)$ and $k \in C_{\mathrm {add}}(A)$ using the Fock-Goncharov dual pair $\mathbb {O}_A = (\mathbb {A}_{B^T}, \mathbb {Y}_B)$ , namely

(2.20) $$ \begin{align} &f = f_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}} = f_{y} \;\; \text{for a unique}\;\, \delta^{\scriptscriptstyle \vee} \in \mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}) \; \,\text{and any} \;\, y \in {\mathcal{U}}^{\mathrm{admi}}_{\delta^{\scriptscriptstyle \vee}}(\mathbb{Y}_B), \end{align} $$

(2.21) $$ \begin{align} &k = k_\rho = k_{x^{\scriptscriptstyle \vee}} \;\; \text{for a unique}\;\, \rho \in \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}) \; \,\text{and any} \;\, x^{\scriptscriptstyle \vee} \in {\mathcal{U}}^{\mathrm{admi}}_\rho(\mathbb{A}_{B^T}). \end{align} $$

Definition 2.6. For a map $g: [1, r]\times {\mathbb Z} \to {\mathbb Z}$ and for any $m \in {\mathbb Z}$ , define the $m^{\mathrm {th}}$ slice of g to be

$$\begin{align*}g|_{\{m\}} = (g(1, m), \, \ldots, \, g(r, m))^T \in {\mathbb Z}^r. \end{align*}$$

We have the following consequences of Theorem C.

When the Cartan matrix A is of finite and simply laced type, for each $(i, m) \in [1, r] \times {\mathbb Z}$ , Ringel defined in [Reference Ringel40] the cluster-hammock function $h_{(i, m)}$ to be the unique element in $C_{\mathrm {add}}(A)$ satisfying

(2.22) $$ \begin{align} h_{(i, m)} (j, m) = -\delta_{j, i}, \hspace{.2in} j \in [1, r]. \end{align} $$

Defining $h_{(i, m)} \in C_{\mathrm {add}}(A)$ as in (2.22) for an arbitrary symmetrizable generalized Cartan matrix A, we show in Proposition 6.23 and Corollary 6.25 that for every $(i, m) \in [1, r] \times {\mathbb Z}$ ,

(2.23) $$ \begin{align} h_{(i, m)} = k_{x^{\scriptscriptstyle \vee}(i, m)} \in C_{\mathrm{add}}(A) \cap C_{\mathrm{frieze}}(A). \end{align} $$

As a consequence of equation (2.23), we generalize in Proposition 6.27 a result of N. Reading and S. Stella [Reference Reading and Stella39, Theorem 2.2 and Theorem 2.3] on a duality of d-compatibility degrees in the finite type and rank $2$ cases.

2.4 The case of finite type: Ringel’s conjecture and applications

In $\S $ 7, we apply the results summarized in $\S $ 2.1– $\S $ 2.3 to the case when the Cartan matrices are of finite type.

Assume thus that A is an $r \times r$ Cartan matrix of finite type, and let $B = B_A$ as in (2.5). We show in Proposition 7.2 that both generic frieze patterns

$$\begin{align*}(i, m) \longmapsto x^{\scriptscriptstyle \vee}(i, m)\in \mathfrak{X}(\mathbb{A}_{B^T}) \hspace{.2in} \text{and} \hspace{.2in} (i, m) \longmapsto y(i, m) \in \mathfrak{X}(\mathbb{Y}_B) \end{align*}$$

are invariant under a bijection ${\mathfrak {F}}_A: [1, r] \times {\mathbb Z} \rightarrow [1, r] \times {\mathbb Z}$ , which is described using the root system of A and generalizes the gliding symmetry of the classical Coxeter frieze patterns (see $\S $ 7.2 and Remark 7.3). Combining with results in Theorem A, Theorem C, and 4) of Theorem C’, we prove the following generalization of Ringel’s conjecture, which is stated in [Reference Ringel40, $\S$ 6] for the simply laced case.

Decomposition of tropical friezes will depend on a better understanding of the fan structure of the tropical space $\mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ (but see Lemma 6.10) and will be investigated in a future paper.

Turning in the other direction, since A is of finite type, the full Fock-Goncharov conjectures hold for the Fock-Goncharov pair $\mathbb {O}_A = (\mathbb {A}_{B^T}, \mathbb {Y}_B)$ . Namely, one has the bijective Fock-Goncharov correspondences

(2.24) $$ \begin{align} &\mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}) \longrightarrow {\mathcal{U}}^{\mathrm{mono}}(\mathbb{Y}_B): \;\;\; \delta^{\scriptscriptstyle \vee}\; \longmapsto \;y_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}, \end{align} $$

(2.25) $$ \begin{align} &\mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}) \longrightarrow {\mathcal{U}}^{\mathrm{mono}}(\mathbb{A}_{B^T}): \;\;\;\rho\; \longmapsto \;x_{\rho}^{\scriptscriptstyle \vee}, \end{align} $$

where $y_{{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}}$ has g-vector $\delta ^{\scriptscriptstyle \vee }$ and $x_{\rho }^{\scriptscriptstyle \vee }$ has g-vector $\rho $ , as well as the Fock-Goncharov pairing

(2.26) $$ \begin{align} \langle \delta^{\scriptscriptstyle \vee}, \; \rho\rangle = \mathrm{val}_{\delta^{\scriptscriptstyle \vee}} (x^{\scriptscriptstyle \vee}_\rho) = \mathrm{ val}_\rho(y_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}), \hspace{.2in} \delta^{\scriptscriptstyle \vee} \in \mathbb{A}_{B^T}({\mathbb Z}^{\mathrm{max}}), \; \, \rho \in \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}). \end{align} $$

Let ${\mathcal {D}}_A \subset [1, r]\times {\mathbb Z}$ be the fundamental domain of ${\mathfrak {F}}_A$ given in (7.4), so one has the bijection

$$\begin{align*}{\mathcal{D}}_A \longrightarrow \mathfrak{X}(\mathbb{A}_{B^T}), \;\; (i, m) \longmapsto x^{\scriptscriptstyle \vee}(i,m). \end{align*}$$

In particular, $|{\mathcal {D}}_A| = r + \frac {1}{2} |\Phi _A|$ , where $\Phi _A$ is the root system of A. For $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ and ${\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})}$ , let $f_{{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}} \in C_{\mathrm {frieze}}(A^T)$ and $k_\rho \in C_{\mathrm {add}}(A)$ be as in Theorem A. As applications of tropical friezes and cluster-additive functions to Fock-Goncharov duality, we prove in $\S $ 7.5 the following theorem.

Theorem E. Let $B = B_A$ as in (2.5) for a Cartan matrix A of finite type.

1) For every $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , one has

$$\begin{align*}x^{\scriptscriptstyle \vee}_\rho= \prod_{(i,m) \in {\mathcal{D}}_A} x^{\scriptscriptstyle \vee} (i,m)^{[-k_\rho(i,m)]_+}; \end{align*}$$

2) For every $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ and every $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , one has

$$\begin{align*}\langle \delta^{\scriptscriptstyle \vee} , \rho \rangle = \sum_{(i,m) \in {\mathcal{D}}_A} f_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}}(i,m) [-k_{\rho}(i,m)]_+; \end{align*}$$

3) For every $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ with (row) coordinate vector $\delta ^{\scriptscriptstyle \vee }_{t_0}$ at $t_0$ , letting $-p(\delta ^{\scriptscriptstyle \vee })$ be the unique point in $\mathbb {Y}_{-B^T}({\mathbb Z}^{\mathrm { max}})$ with coordinate vector $-\delta ^{\scriptscriptstyle \vee }_{t_0} B^T$ at $t_0$ , one has $k_{-p(\delta ^{\scriptscriptstyle \vee })} \in C_{\mathrm {add}}(A^T)$ and

(2.27) $$ \begin{align} y_{{\scriptstyle{\delta^{\scriptscriptstyle \vee}}}} = \mathbf{y}_{t_0}^{-(\delta_{t_0}^{\scriptscriptstyle \vee})^T} \prod_{(i, m) \in {\mathcal{D}}_A} \left(F(i, m)(\mathbf{ y}_{t_0})\right)^{[-k_{-p(\delta^{\scriptscriptstyle \vee})}(i, m-1)]_+}, \end{align} $$

where $\mathbf {y}_{t_0}$ is the $\mathbf {Y}$ -cluster of $\mathbb {Y}_B$ at $t_0$ , and $F(i, m)$ for $(i, m) \in {\mathcal {D}}_A$ is the $i^{\mathrm {th}}\, F$ -polynomial at $t(i, m)$ associated to the mutation matrix B with principal coefficients at $t_0$ .

We remark that a geometric model of $\mathbb {A}_{B^T}$ is given in [Reference Yang and Zelevinsky44] using certain Coxeter reduced double Bruhat cells in the complex semi-simple Lie group associated to $A^T$ , and the cluster variables $x^{\scriptscriptstyle \vee }(i, m)$ and the F-polynomials $F(i, m)$ , for $(i, m) \in {\mathcal {D}}_A$ , are explicitly expressed using generalized minors.

We also remark that C. Melo and A. Nájera Chávez obtain in [Reference Melo and Chávez35, Lemma 4.1] a formula expressing each global monomial $y_{{\scriptstyle {\delta ^{\scriptscriptstyle \vee }}}}$ on $\mathbb {Y}_B$ in terms of $\delta _{t_0}^{\scriptscriptstyle \vee }$ and F-polynomials from a vertex $w \in {\mathbb T}_r$ at which $\delta ^{\scriptscriptstyle \vee }$ is optimized (see §5.1 for the definition).

The formulas in Theorem E can be easily implemented using a computer program. Some of this can be done using Keller’s Java applets [Reference Keller31]. A more direct program is available at the Visual Cluster Algebra website [43] maintained by the second author.

2.5 Notation

For an integer $r \geq 1$ , we denote by ${\mathbb Z}^r$ the set of all column integral vectors of size r and by ${\mathbb Z}^r_{\mathrm {row}}$ that of row vectors. The standard basis of ${\mathbb Z}^r_{\mathrm {row}}$ is denoted as $(e^1, \ldots , e^r)$ . For an integral matrix M, we write $M\geq 0$ if all of its entries are nonnegative.

For r independent variables $\mathbf {x} = (x_1, \ldots , x_r)$ , ${\mathbb Z}_{\geq 0}[\mathbf {x}^{\pm 1}]$ denotes the set of all Laurent polynomials in $\mathbf {x}$ with nonnegative integer coefficients. If R is a commutative ring and $R^\times $ its group of units, for $\mathbf {a} = (a_1, \ldots , a_n) \in (R^\times )^n$ , we write $\mathbf {a}^L = a_1^{l_1} \cdots a_n^{l_n} \in R^\times $ for $L = (l_1, \ldots , l_n)^T \in {\mathbb Z}^n$ , and if L is an $n \times m$ integral matrix with columns $L_1, \ldots , L_m \in {\mathbb Z}^n$ , we write $\mathbf {a}^L = (\mathbf {a}^{L_1}, \ldots , \mathbf {a}^{L_m}) \in (R^\times )^m$ .

For two sets T and S, we will sometimes use $\{s_t\}_{t \in T}$ to denote (not a subset of S but) a map $T \to S, t \mapsto s_t$ , and we call it an assignment. For an r-tuple of elements $\mathbf {s} = (s_1, \ldots , s_r)$ in S, we also write $s \in \mathbf {s}$ to mean that $s = s_i$ for some $i \in [1, r]$ .

3.1 Tropical points and global functions on positive spaces

Let $\mathbb {X} = (\mathbb {F}(\mathbb {X}), \,\{\mathbf {x}_t\}_{t \in {\mathbb T}})$ be an r-dimensional positive space as in Definition 2.3, and recall that

$$\begin{align*}\mathbb{X}({\mathbb Z}^{\mathrm{max}}) = \mathrm{Hom}_{\mathrm{sf}}(\mathbb{F}_{>0}(\mathbb{X}), \, {\mathbb Z}^{\mathrm{max}}). \end{align*}$$

For $\rho \in \mathbb {X}({\mathbb Z}^{\mathrm {max}})$ and $t \in {\mathbb T}$ , the vector $\mathrm {val}_\rho (\mathbf {x}_t) \in {\mathbb Z}^r_{\mathrm {row}}$ is called the coordinate vector of $\rho $ at $t \in {\mathbb T}$ . We also say that an assignment $\{\rho _t \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}}$ is a tropical point of $\mathbb {X}$ if there exists $\rho \in \mathbb {X}({\mathbb Z}^{\mathrm {max}})$ such that $\rho _t = \mathrm {val}_\rho (\mathbf {x}_t)$ for every $t \in {\mathbb T}$ , and in this case we write $\rho = \{\rho _t \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}}\in \mathbb {X}({\mathbb Z}^{\mathrm {max}})$ . Define

$$\begin{align*}{\mathcal{U}}(\mathbb{X}) = \bigcap_{t \in {\mathbb T}} {\mathbb Z}[\mathbf{x}_t^{\pm 1}] \subset \mathbb{F}(\mathbb{X}) \hspace{.2in} \text{and} \hspace{.2in} {\mathcal{U}}^+(\mathbb{X}) = \bigcap_{t \in {\mathbb T}} {\mathbb Z}_{\geq 0}[\mathbf{x}_t^{\pm 1}] \subset {\mathcal{U}}(\mathbb{X}). \end{align*}$$

An element of $\mathbb {F}(\mathbb {X})$ that lies in ${\mathcal {U}}(\mathbb {X})$ (resp. in ${\mathcal {U}}^+(\mathbb {X})$ ) is also said to be universally Laurent (resp. universally positive Laurent). Elements in ${\mathcal {U}}(\mathbb {X})$ are also called global functions on $\mathbb {X}$ .

3.2 Matrix patterns

Recall again that an $r \times r$ skew-symmetrizable matrix with integer entries is called a mutation matrix of size r. Given a mutation matrix $B = (b_{i, j})$ of size r and $k \in [1,r]$ , the mutation of B in direction k is the $r \times r$ matrix $B' = (b_{i,j}')$ , where

(3.1) $$ \begin{align} b_{i,j}' = \begin{cases} -b_{i,j}, & \text{ if } i = k \text{ or } j = k, \\ b_{i,j} + [b_{i,k}]_+[b_{k,j}]_+ - [-b_{i,k}]_+[-b_{k,j}]_+, &\text{ otherwise}, \end{cases} \end{align} $$

and we write $B' =\mu _k(B)$ . Then $\mu _k(B)$ is again a mutation matrix and $\mu _k^2(B)= B$ .

Throughout the paper, ${\mathbb T}_r$ denotes an r-regular tree with the r edges from each vertex labeled bijectively by the set $[1, r]$ , and we write $t \in {\mathbb T}_r$ for a vertex t of ${\mathbb T}_r$ . If $t, t' \in {\mathbb T}_r$ are joined by an edge labeled by $k \in [1, r]$ , we write . We also fix $t_0 \in {\mathbb T}_r$ and call it the root of ${\mathbb T}_r$ .

3.3 Positive $\mathbf {A}$ -spaces, tropical mutation rule, and d-compatibility degrees with cluster variables

Let $\mathbb {F}$ be a field isomorphic to the field of rational functions over ${\mathbb Q}$ in r independent variables.

Given a seed $(\mathbf {x}, B)$ in $\mathbb {F}$ and $k \in [1,r]$ , the $\mathbf {A}$ -mutation of $(\mathbf {x}, B)$ in direction k is the seed $(\mathbf {x}', \;\mu _k(B))$ , where $\mu _k(B)$ is given in equation (3.1) and $\mathbf {x}' = (x_1', \dots , x_r')$ is given by

(3.2) $$ \begin{align} x_i' = \begin{cases} x_k^{-1}(\prod_{j=1}^r x_j^{[b_{j,k}]_+} +\prod_{j=1}^r x_j^{[-b_{j,k}]_+}) ,& i = k; \\ x_i,& \text{otherwise}. \end{cases} \end{align} $$

For the positive space $\mathbb {A}_B$ with matrix pattern $\{B_t = (b^t_{i, j}) \}_{t \in {\mathbb T}_r}$ , the following is obtained by tropicalizing equation (3.2).

Lemma 3.6. An assignment $\{\delta _t \!= (\delta _{t;1}, \dots , \delta _{t;r}) \!\in \!{\mathbb Z}^r_{\mathrm {row}}\}_{t\in \mathbb T_r}$ is a tropical point of $\mathbb {A}_B$ if and only if

(3.3) $$ \begin{align} \delta_{t';i} = \begin{cases} -\delta_{t;k} + {\mathrm{max}} (\sum_{j=1}^r [b_{j,k}^t]_+\delta_{t;j}\,, \; \sum_{j=1}^r [-b_{j,k}^t]_+\delta_{t;j}),& i = k; \\ \delta_{t;i},& \text{otherwise}, \end{cases} \end{align} $$

for all $i \in [1, r]$ whenever in ${\mathbb T}_r$ .

Let $\mathfrak {X}(\mathbb {A}_B)$ denote the set of cluster variables of $\mathbb {A}_B$ . Two cluster variables x and $x'$ are said to be compatible if there exists a cluster of $\mathbb {A}_B$ containing both x and $x'$ .

Proof. Recall that $(e^1, \ldots , e^r)$ is the standard basis of ${\mathbb Z}^r_{\mathrm {row}}$ . Choose any $t \in {\mathbb T}_r$ and $i \in [1,r]$ such that $x = x_{t;i}$ , and define $d_x \in \mathbb {A}_B({\mathbb Z}^{\mathrm {max}})$ by setting $\mathrm {val}_{d_x}(\mathbf {x}_t) = -e^i$ . Suppose that $x= x_{t'; i'}$ for some $t' \in {\mathbb T}_r$ and $i' \in [1,r]$ . By [Reference Cao and Li11, Theorem 10], there exists a sequence

in ${\mathbb T}_r$ such that $k_j \neq i$ for every $j \in [1, l-1]$ and $\mathbf {x}_{t_l} = \mathbf {x}_{t^\prime }$ up to a reordering of their variables. By (3.3), one has $\mathrm {val}_{d_x}(\mathbf {x}_{t_2}) = -e^i$ , and by induction, $\mathrm {val}_{d_x}(\mathbf {x}_{t_l}) = -e^i$ . Thus $\mathrm {val}_{d_x}(\mathbf {x}_{t'}) = -e^{i'}$ . This shows that $d_x$ is independent of the choice of t and i and has the two properties as described.

Recall that elements in the upper cluster algebra ${\mathcal {U}}(\mathbb {A}_B) =\bigcap _{t \in {\mathbb T}_r} {\mathbb Z}[\mathbf {x}^{\pm 1}_t]$ are also called global functions on $\mathbb {A}_B$ , and ${\mathcal {U}}^+(\mathbb {A}_B) \stackrel {\text {def}}{=} \bigcap _{t \in {\mathbb T}_r} {\mathbb Z}_{\geq 0}[\mathbf {x}^{\pm 1}_t]$ .

Definition 3.8. For $x\in \mathfrak {X}(\mathbb {A}_B)$ and $x' \in \mathbb {F}_{>0}(\mathbb {A}_B)$ , define the d-compatibility degree of $x'$ with x to be

(3.4) $$ \begin{align} ( x \, \|\, x' )_d \stackrel{\text{def}}{=} \mathrm{val}_{d_x}(x'). \end{align} $$

Recall that for any $x' \in {\mathcal {U}}^+(\mathbb {A}_B)\backslash \{0\}$ and any $t \in {\mathbb T}_r$ , one can write

$$\begin{align*}x^\prime = \frac{P(\mathbf{x}_t)}{\mathbf{x}_t^{\mathbf{d}}}, \end{align*}$$

where $\mathbf {d} \in {\mathbb Z}^r$ and $P(\mathbf {x}_t) \in {\mathbb Z}_{\geq 0}[\mathbf {x}_t]$ is not divisible by $x_{t; i}$ for any $i \in [1, r]$ . The vector $\mathbf {d}$ is called the denominator vector of $x^\prime $ at t. For any $x \in \mathfrak {X}(\mathbb {A}_B)$ and $x^\prime \in {\mathcal {U}}^+(\mathbb {A}_B)\backslash \{0\}$ , the d-compatibility degree $(x \, \|\, x')_d$ is thus the $i^{\mathrm {th}}$ component of the denominator vector of $x'$ at t for any $t \in {\mathbb T}_r$ such that $x_{t; i} = x$ (see [Reference Cao and Li12] and [Reference Cao, Keller and Qin10, Remark 2.31]). The following facts on $(\cdot \, \|\, \cdot )_d$ will be used in $\S $ 6.6.

Proof. Choose any $t \in {\mathbb T}_r$ such that $u = x_{t; j_1}^{m_1} \cdots x_{t; j_l}^{m_l}$ , where $\{j_1, \ldots , j_l\} \subset [1, r]$ and $m_{k}> 0$ for each $k \in [1, l]$ . Suppose that v is a cluster monomial in the cluster $\mathbf {x}_{t'}$ for some $t' \in {\mathbb T}_r$ . By Lemma 3.9, for each $i \in [1, r]$ , one has $(x_{t'; i} \, \| \, v)_d \leq 0$ . Thus

$$\begin{align*}(x_{t'; i} \, \| \, u)_d \leq (x_{t'; i} \, \| \, v)_d \leq 0, \hspace{.2in} \forall \; i \in [1, r]. \end{align*}$$

Suppose that $i \in [1, r]$ is such that $x_{t'; i} \notin \{x_{t; j_1}, \ldots , x_{t; j_l}\}$ . It then follows from Lemma 3.9 and

$$\begin{align*}(x_{t'; i} \, \| \, u)_d = m_1 (x_{t';i} \, \| \, x_{t; j_1})_d + \cdots + m_l (x_{t';i} \, \| \, x_{t; j_l})_d \leq 0 \end{align*}$$

that $(x_{t';i} \, \| \, x_{t; j_k})_d = 0$ for every $k \in [1, l]$ . Thus the elements of the set $S=\{x_{t'; 1}, \ldots , x_{t'; r}, x_{t; j_1}, \ldots , x_{t; j_l}\}$ are pairwise compatible, that is, S is a compatible subset of $\mathfrak {X}(\mathbb {A}_B)$ . By [Reference Cao and Li11, Theorem 13], S is a subset of some cluster of $\mathbb {A}_B$ . We can thus assume without loss of generality that $t = t'$ . For each $i \in [1, r]$ , one then has $(x_{t; i} \, \| \, u)_d \leq (x_{t; i} \, \| \, v)_d$ , and, if $x_{t; i}^\prime $ is the unique new variable in $\mu _i(\mathbf {x}_t)$ , then one also has

$$\begin{align*}-(x_{t; i} \, \| \, u)_d = (x_{t; i}^\prime \, \| \, u)_d \leq (x_{t; i}^\prime \, \| \, v)_d = -(x_{t; i} \, \| \, v)_d. \end{align*}$$

So $(x_{t; i} \, \| \, u)_d = (x_{t; i} \, \| \, v)_d$ for each $i\in [1,r]$ . It follows that $u = v$ .

3.4 Positive $\mathbf {Y}$ -spaces, tropical mutation rule, and d-compatibility degrees with $\mathbf {Y}$ -variables

Let again $\mathbb {F}$ be a field isomorphic to the field of rational functions over ${\mathbb Q}$ in r independent variables.

Given a seed $(\mathbf {y}, B)$ in $\mathbb {F}$ and $k \in [1,r]$ , the $\mathbf {Y}$ -mutation of $(\mathbf {y}, B)$ in direction k is the seed $(\mathbf {y}^\prime , \mu _k(B))$ , where $\mu _k(B)$ is given in equation (3.1) and $\mathbf {y}' = (y_1', \dots , y_r')$ is given byFootnote ²

(3.5) $$ \begin{align} y_i' = \begin{cases} y_k^{-1}, & i = k; \\ y_i y_k^{[b_{k,i}]_+} (1+y_k)^{-b_{k,i}}, &\text{otherwise}. \end{cases} \end{align} $$

For the positive space $\mathbb {Y}_B$ with $\mathbf {Y}$ -clusters $\{\mathbf {y}_t = (y_{t; 1}, \ldots , y_{t; r})\}_{t \in {\mathbb T}_r}$ and matrix pattern ${\{B_t = (b^t_{i, j}) \}_{t \in {\mathbb T}_r}}$ , the following is obtained by tropicalizing equation (3.5).

Lemma 3.13. An assignment $\{\rho _t = (\rho _{t;1}, \dots , \rho _{t;r}) \in {\mathbb Z}_{\mathrm {row}}^r\}_{t\in \mathbb T_r}$ is a tropical point of $\mathbb {Y}_B$ if and only if

(3.6) $$ \begin{align} \rho_{t';i} = \begin{cases} -\rho_{t;k}, & i = k; \\ \rho_{t;i} + [b_{k,i}^t]_+\, \rho_{t;k} - b^t_{k,i} \, [\rho_{t;k}]_+, & \text{otherwise}, \end{cases} \end{align} $$

for all $i \in [1, r]$ whenever .

Denote by $\mathfrak {X}(\mathbb {Y}_B)$ the set of $\mathbf {Y}$ -variables of $\mathbb {Y}_B$ . Two elements $y, y' \in \mathfrak {X}(\mathbb {Y}_B)$ are said to be compatible if there exists a $\mathbf {Y}$ -cluster of $\mathbb {Y}_B$ containing both y and $y'$ .

Proof. Choose any $t \in {\mathbb T}_r$ and $i \in [1,r]$ such that $y = y_{t;i}$ , and define $d_y \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ by setting $\mathrm {val}_{d_y}(\mathbf {y}_t) = -e^i$ , where again $(e^1, \ldots , e^r)$ is the standard basis of ${\mathbb Z}^r_{\mathrm {row}}$ . Suppose that $y= y_{t'; i'}$ for some $t' \in {\mathbb T}_r$ and $i' \in [1,r]$ . By [Reference Cao, Keller and Qin10, Theorem 7.10], there exists a sequence

in ${\mathbb T}_r$ such that $y \in \mathbf {y}_{t_j}$ for every $j \in [1, l]$ and $\mathbf {y}_{t_l} = \mathbf {y}_{t^\prime }$ up to a reordering of their variables. By the mutation formula in (3.5) and by the algebraic independence of the variables in any $\mathbf {Y}$ -cluster, one sees that $k_j \neq i$ and $b^{t_j}_{k_j, i}= 0$ for every $j \in [1, l-1]$ , and $y = y_{t_j; i}$ for every $j \in [1, l]$ . By (3.6), one has $\mathrm {val}_{d_y}(\mathbf {y}_{t_2}) = -e^i$ , and by induction, $\mathrm {val}_{d_y}(\mathbf {y}_{t_l}) = -e^i$ . Thus $\mathrm {val}_{d_y}(\mathbf {y}_{t'}) = -e^{i'}$ . This shows that $d_y$ is well-defined and has the two required properties.

Recall that elements in ${\mathcal {U}}(\mathbb {Y}_B) = \bigcap _{t \in {\mathbb T}_r} {\mathbb Z}[\mathbf {y}^{\pm 1}_t]$ are called global functions on $\mathbb {Y}_B$ , and we have set ${\mathcal {U}}^+(\mathbb {Y}_B) = \bigcap _{t \in {\mathbb T}_r} {\mathbb Z}_{\geq 0}[\mathbf {y}^{\pm 1}_t]$ .

Definition 3.15. For $y\in \mathfrak {X}(\mathbb {Y}_B)$ and $y' \in \mathbb {F}_{>0}(\mathbb {Y}_B)$ , define the d-compatibility degree of $y'$ with y to be

$$\begin{align*}( y \, \|\, y' )_d \stackrel{\text{def}}{=} \mathrm{val}_{d_y}(y'). \end{align*}$$

For $y' \in {\mathcal {U}}^+(\mathbb {Y}_B)\backslash \{0\}$ , the integer $( y \, \|\, y' )_d$ is the y-exponent of the denominator of $y'$ as a Laurent polynomial in $\mathbf {y}_t$ for any $t \in {\mathbb T}_r$ such that $y \in \mathbf {y}_t$ .

Remark 3.16. Note that $\mathfrak {X}(\mathbb {Y}_B) \subset \mathbb {F}_{>0}(\mathbb {Y}_B)$ but, unlike the case of positive $\mathbf {A}$ -spaces, $\mathfrak {X}(\mathbb {Y}_B)$ is not necessarily a subset of ${\mathcal {U}}^+(\mathbb {Y}_B)$ . Moreover, for $y, y' \in \mathfrak {X}(\mathbb {Y}_B)$ and $y \neq y'$ , if y and $y'$ are compatible, then $(y\, \|\, y')_d = (y'\, \|\, y)_d = 0$ by Lemma-Definition 3.14, but the converse is not necessarily true. Indeed, for $B = \left (\begin {array}{cc} 0 & -1\\ 1 & 0\end {array}\right )$ and $\mathbf {y}_{t_0} = (y_1, y_2)$ , we have

$$ \begin{align*} (y_1, \, y_2) &\stackrel{\mu_1}{\longrightarrow} \left(\frac{1}{y_1}, \, y_2(1+y_1)\right) \stackrel{\mu_2}{\longrightarrow} \left(\frac{1+y_2+y_1y_2}{y_1}, \; \frac{1}{y_2(1+y_1)}\right) \\ &\stackrel{\mu_1}{\longrightarrow} \left(\frac{y_1}{1+y_2+y_1y_2}, \; \frac{1+y_2}{y_1y_2}\right) \stackrel{\mu_2}{\longrightarrow} \left(\frac{1}{y_2}, \; \frac{y_1y_2}{1+y_2}\right) \stackrel{\mu_1}{\longrightarrow} (y_2, \, y_1). \end{align*} $$

Thus $\mathfrak {X}(\mathbb {Y}_B)$ has $10$ elements, while we have

$$\begin{align*}\mathfrak{X}(\mathbb{Y}_B) \cap {\mathcal{U}}^+(\mathbb{Y}_B) = \left\{ y_1, \; y_2(1+y_1), \; \frac{1+y_2+y_1y_2}{y_1}, \; \frac{1+y_2}{y_1y_2}, \; \frac{1}{y_2}\right\}. \end{align*}$$

Let $z = y_2(1+y_1)$ . One also has $(y_1\, \|\, z)_d = (z\, \|\, y_1)_d = 0$ , but $y_1$ and z are not compatible.

3.5 The ensemble map

Let again B be any mutation matrix of size r. Recall from Definition 2.4 that the pair $(\mathbb {A}_B, \mathbb {Y}_B)$ of positive spaces is called a Fock-Goncharov ensemble. Following [Reference Fock and Goncharov21], the map

$$\begin{align*}p_{\!{}_{B}}: \;\mathbb{A}_B({\mathbb Z}^{ {\mathrm{max}} }) \longrightarrow \mathbb{Y}_B({\mathbb Z}^{ {\mathrm{max}} }) \end{align*}$$

induced by the semi-field homomorphism

$$\begin{align*}p_{\!{}_{B}}^*: \;\;\mathbb{F}_{>0}(\mathbb{Y}_B) \longrightarrow \mathbb{F}_{>0}(\mathbb{A}_B), \;\; p_{\!{}_{B}}^*(\mathbf{y}_{t_0}) = \mathbf{x}_{t_0}^B, \end{align*}$$

is called the p-map, or the ensemble map, of $(\mathbb {A}_B, \mathbb {Y}_B)$ . By the well-known fact [Reference Fomin and Zelevinsky23, Proposition 3.9] that $p_{\!{}_B}^*(\mathbf {y}_t) = \mathbf {x}_t^{B_t}$ for every $t \in {\mathbb T}_r$ , where $\{B_t\}_{t \in {\mathbb T}_r}$ is the matrix pattern with $B_{t_0} = B$ . Thus the map $p_{\!{}_{B}}$ is linear in each coordinate chart of $\mathbb {A}_B({\mathbb Z}^{\mathrm {max}})$ , that is,

(3.7) $$ \begin{align} p_{\!{}_{B}}:\; \mathbb{A}_B({\mathbb Z}^{ {\mathrm{max}} }) \longrightarrow \mathbb{Y}_B({\mathbb Z}^{ {\mathrm{max}} }),\;\; \{\delta_t\in\mathbb Z_{\mathrm{row}}^r\}_{t\in{\mathbb T}_r}\longmapsto \{\delta_tB_t\in\mathbb Z_{\mathrm{row}}^r\}_{t \in {\mathbb T}_r}. \end{align} $$

3.6 Positive spaces with principal coefficients

Let l be any nonnegative integer, and consider any $(r+l) \times (r+l)$ skew-symmetrizable matrix

(3.8) $$ \begin{align} \breve{B} = \left(\begin{array}{cc} B & Q \\ P & R\end{array}\right), \end{align} $$

where B again has size $r \times r$ . We then have the Fock-Goncharov ensemble $(\mathbb {A}_{\breve {B}}, \mathbb {Y}_{\breve {B}})$ , written as

$$\begin{align*}\mathbb{A}_{\breve{B}} = (\mathbb{F}(\mathbb{A}_{\breve{B}}), \, \{{\mathbf{x}}_t\}_{t \in {\mathbb T}_{r+l}}) \quad \text{ and } \quad \mathbb{Y}_{\breve{B}} = (\mathbb{F}(\mathbb{Y}_{\breve{B}}), \,\{{\mathbf{y}}_t\}_{t \in {\mathbb T}_{r+l}}). \end{align*}$$

Identify the root $t_0$ of ${\mathbb T}_r$ with that of ${\mathbb T}_{r+l}$ and ${\mathbb T}_r$ with the subtree of ${\mathbb T}_{r+l}$ with edges labeled by indices from $[1, r] \subset [1, r+l]$ . Let $\{\breve {B}_t\}_{t \in {\mathbb T}_{r+l}}$ be the matrix pattern with $\breve {B}_{t_0} = \breve {B}$ , and let

$$\begin{align*}\breve{B}_t = \left(\begin{array}{cc} B_t & Q_t \\ P_t & R_t\end{array}\right) \hspace{.2in} \text{and} \hspace{.2in} (B_t \backslash\!\!\backslash P_t) = \left(\begin{array}{c} B_t \\ P_t\end{array}\right), \hspace{.2in} t \in {\mathbb T}_{r+l}. \end{align*}$$

One checks directly that $\{(\mathbf {x}_t, \, (B_t \backslash \!\!\backslash P_t))\}_{t \in {\mathbb T}_r}$ depends only on the $(r+l) \times r$ matrix $(B \backslash \!\!\backslash P)$ , and that $\{(\mathbf {y}_t, \, (B_t \; Q_t))\}_{t \in {\mathbb T}_r}$ depends only on the $r \times (r+l)$ matrix $(B \; Q)$ . Correspondingly, one has two positive spaces

$$\begin{align*}\mathbb{A}_{(B \backslash\!\!\backslash P)} = (\mathbb{F}(\mathbb{A}_{(B \backslash\!\!\backslash P)}), \; \{\mathbf{x}_t\}_{t \in {\mathbb T}_r}) \hspace{.2in} \text{and} \hspace{.2in} \mathbb{Y}_{(B \; Q)} = (\mathbb{F}(\mathbb{Y}_{(B \; Q)}), \; \{\mathbf{y}_t\}_{t \in {\mathbb T}_r}). \end{align*}$$

The last l elements in $\mathbf {x}_t$ , being the same for all $t \in {\mathbb T}_r$ , are called the frozen variables of $\mathbb {A}_{(B \backslash \!\!\backslash P)}$ . Tropical points of $\mathbb {A}_{(B \backslash \!\!\backslash P)}$ are thus assignments $\{{\delta }_t \in {\mathbb Z}^{r+l}_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ that obey the mutation rule (3.3) with B replaced by $\breve {B}$ , which depends only on $(B \backslash \!\!\backslash P)$ when t is restricted to ${\mathbb T}_r$ . Similarly, tropical points of $\mathbb {Y}_{(B \; Q)}$ are assignments $\{{\rho }_t \in {\mathbb Z}^{r+l}_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ obeying the mutation rule (3.6) with B replaced by $\breve {B}$ , which depends only on $(B \; Q)$ when restricted to $t \in {\mathbb T}_r$ .

Lemma 3.17. For any $\breve {B}$ as in (3.8), the p-map for the ensemble $(\mathbb {A}_{\breve {B}}, \mathbb {Y}_{\breve {B}})$ gives the map

$$\begin{align*}p_{\!{}_{\breve{B}}}: \mathbb{A}_{(B \backslash\!\!\backslash P)}({\mathbb Z}^{\mathrm{max}}) \longrightarrow \mathbb{Y}_{(B \; Q)}({\mathbb Z}^{\mathrm{max}}), \; \{\widetilde{\delta}_t \in {\mathbb Z}^{r+l}_{\mathrm{row}}\}_{t \in {\mathbb T}_r} \longmapsto \{\widetilde{\delta}_t \breve{B}_t\in {\mathbb Z}^{r+l}_{\mathrm{row}}\}_{t \in {\mathbb T}_r}, \end{align*}$$

which is bijective if $\breve {B} \in GL(r+l, {\mathbb Z})$ . Furthermore, one has the well-defined maps

$$ \begin{align*} &\varepsilon_{\!{}_{(B\backslash\!\!\backslash P)}}: \mathbb{A}_B({\mathbb Z}^{\mathrm{max}}) \longrightarrow \mathbb{A}_{(B \backslash\!\!\backslash P)}({\mathbb Z}^{\mathrm{max}}), \, \{\delta_t \in {\mathbb Z}^r_{\mathrm{row}}\}_{t \in {\mathbb T}_r} \longmapsto \!\{(\delta_t, 0) \in {\mathbb Z}^{r+l}_{\mathrm{row}}\}_{t \in {\mathbb T}_r},\\ &\omega_{\!{}_{(B \; Q)}}: \mathbb{Y}_{(B \; Q)}({\mathbb Z}^{\mathrm{max}}) \longrightarrow \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}), \, \{(\rho_t, \, \rho^\prime_t) \in {\mathbb Z}^{r+l}_{\mathrm{row}}\}_{t \in {\mathbb T}_r} \!\longmapsto \!\{\rho_t \in {\mathbb Z}^{r}_{\mathrm{row}}\}_{t \in {\mathbb T}_r}, \end{align*} $$

with $\varepsilon _{\!{}_{(B\backslash \!\!\backslash P)}}$ injective, $\omega _{\!{}_{(B \; Q)}}$ surjective, and $\omega _{\!{}_{(B \; Q)}} \circ p_{\!{}_{\breve {B}}} \circ \varepsilon _{\!{}_{(B\backslash \!\!\backslash P)}} = p_{\!{}_{B}}$ .

For any $r \times r$ mutation matrix B, define the positive $\mathbf {A}$ - and $\mathbf {Y}$ -spaces with principal coefficients associated to B as

$$\begin{align*}\mathbb{A}_B^{\mathrm{prin}} = \mathbb{A}_{(B \backslash\!\!\backslash I_r)} \hspace{.2in} \text{and} \hspace{.2in} \mathbb{Y}_B^{\mathrm{prin}} = \mathbb{Y}_{(B \; I_r)}, \end{align*}$$

where $I_r$ is the identity matrix of size r. Let

(3.9) $$ \begin{align} \overline{B} = \begin{pmatrix} B & -I_r \\ I_r & 0 \end{pmatrix}, \end{align} $$

and let $\{\overline B_t\}_{t\in \mathbb T_{2r}}$ be the matrix pattern with $\overline B_{t_0}=\overline B$ . For $t \in {\mathbb T}_{r}$ , write

(3.10) $$ \begin{align} \widetilde{B}_t = \begin{pmatrix} B_t \\ C_t \end{pmatrix} = (B_t \backslash\!\!\backslash C_t) \hspace{.2in} \text{if} \hspace{.2in} \overline{B}_t = \begin{pmatrix} B_t & Q_t \\ C_t & R_t \end{pmatrix}. \end{align} $$

The following fact will be used in $\S $ 4.4.

Lemma 3.18. With the notation as above, one has the well-defined injective map $\beta _{B^T}: \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}}) \longrightarrow \mathbb {Y}^{\mathrm {prin}}_{B^T}({\mathbb Z}^{\mathrm {max}})$ defined by

(3.11) $$ \begin{align} \{\delta_t^{\scriptscriptstyle \vee} \in {\mathbb Z}^r_{\mathrm{row}}\}_{t \in {\mathbb T}_r} \longmapsto \{(\delta^{\scriptscriptstyle \vee}_t B_t^T, \, \delta_t^{\scriptscriptstyle \vee} C_t^T) \in {\mathbb Z}^{2r}_{\mathrm{row}}\}_{t \in {\mathbb T}_r}. \end{align} $$

Remark 3.19. The embedding $\beta _{B^T}$ in (3.11) appears in [Reference Gross, Hacking, Keel and Kontsevich27, p. 554]. In [Reference Gross, Hacking, Keel and Kontsevich27, Definition 7.12], $\mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ is identified with its image under $\beta _{B^T}$ as a linear subspace of $\mathbb {Y}_{B^T}^{\mathrm {prin}}({\mathbb Z}^{\mathrm {max}})$ . Indeed, if $G_t = G_t^{B, t_0}$ for $t \in {\mathbb T}_r$ in the notation of [Reference Fomin and Zelevinsky23, §Reference Buan, Marsh, Reineke, Reiten and Todorov6], then $G_tB_t = BC_t$ for every $t \in {\mathbb T}_r$ by [Reference Fomin and Zelevinsky23, (6.14)]. If we regard $\beta _{B^T}$ in (3.11) as a map from the set $R^{\scriptscriptstyle \vee }$ of all assignments $\delta ^{\scriptscriptstyle \vee } =\{\delta _t^{\scriptscriptstyle \vee } \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ to the set $\widetilde {R}$ of all assignments $\widetilde {\rho } =\{(\xi _t, \eta _t) \in {\mathbb Z}^r_{\mathrm {row}} \times {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ , then $\widetilde {\rho } \in \widetilde {R}$ belongs to the image of $\beta _{B^T}$ if and only if

$$\begin{align*}G_t \xi_t^T - B \eta_t^T = 0, \hspace{.2in} \;\; \forall \; t \in {\mathbb T}_r, \end{align*}$$

and in that case $\widetilde {\rho } = \beta _{B^T}(\delta ^{\scriptscriptstyle \vee })$ with $\delta ^{\scriptscriptstyle \vee } = \{\eta _t(C_t^{-1})^T\}_{t \in {\mathbb T}_r} \in R^{\scriptscriptstyle \vee }$ . Moreover, for any $\delta ^{\scriptscriptstyle \vee } \in R^{\scriptscriptstyle \vee }$ , one has $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ if and only if $\beta _{B^T}(\delta ^{\scriptscriptstyle \vee }) \in \mathbb {Y}_{B^T}^{\mathrm {prin}}({\mathbb Z}^{\mathrm {max}})$ .

4.1 Global monomials

Let B be any $r \times r$ mutation matrix, and write

(4.1) $$ \begin{align} \mathbb{A}_{B^T} &= (\mathbb{F}(\mathbb{A}_{B^T}), \{\mathbf{x}^{\scriptscriptstyle \vee}_t = (x^{\scriptscriptstyle \vee}_{t; 1}, \ldots, x^{\scriptscriptstyle \vee}_{t; r})\}_{t \in {\mathbb T}_r}), \nonumber\\ \mathbb{Y}_B &= (\mathbb{F}(\mathbb{Y}_B), \{\mathbf{y}_t=(y_{t; 1}, \ldots, y_{t; r})\}_{t \in {\mathbb T}_r}). \end{align} $$

Elements in $\mathbb {F}(\mathbb {A}_{B^T})$ of the form $(\mathbf {x}_t^{\scriptscriptstyle \vee })^{\mathbf {m}}$ (resp. in $\mathbb {F}(\mathbb {Y}_B)$ of the form $\mathbf {y}_t^{\mathbf {m}}$ ) for some $t \in {\mathbb T}_r$ and $\mathbf {m} \in {\mathbb Z}^r$ are called local Laurent monomials on $\mathbb {A}_{B^T}$ (resp. on $\mathbb {Y}_B$ ).

Recall that a cluster monomial on $\mathbb {A}_{B^T}$ is an element of the form $(\mathbf {x}_t^{\scriptscriptstyle \vee })^{\mathbf {m}}$ for some $t \in {\mathbb T}_r$ and $\mathbf {m} \in {\mathbb Z}_{\geq 0}^r$ . We prove the following well-known fact (see [Reference Gross, Hacking, Keel and Kontsevich27, Definition 0.1]) for the convenience of the reader.

Proof. By the Laurent phenomenon, 1) implies 2). By definition, 3) implies 1). It remains to show that 2) implies 3). Let $x^{\scriptscriptstyle \vee } = (\mathbf { x}_t^{\scriptscriptstyle \vee })^{\mathbf {m}}$ , where $t \in {\mathbb T}_r$ and $\mathbf {m} = (m_1, \ldots , m_k)^T \in {\mathbb Z}^r$ , and assume that $x^{\scriptscriptstyle \vee }$ is a global monomial. If $m_k < 0$ for some $k \in [1, r]$ , then with $t' \in {\mathbb T}_r$ such that , the mutation relation (3.2) gives

$$\begin{align*}x^{\scriptscriptstyle \vee} = M \, \left(\frac{P_k^+ + P_k^-}{x_{t';k}^{\scriptscriptstyle \vee}}\right)^{m_k} \end{align*}$$

for some M, $P_k^+$ , and $P_k^-$ that are monomials in the variables in $\mathbf {x}_t^{\scriptscriptstyle \vee } \cap \mathbf {x}_{t'}^{\scriptscriptstyle \vee }$ . So $x^{\scriptscriptstyle \vee }$ cannot be in ${\mathbb Z}[(\mathbf {x}_{t'}^{\scriptscriptstyle \vee })^{\pm 1}]$ , contradicting the assumption that $x^{\scriptscriptstyle \vee }$ is a global monomial. Thus $\mathbf {m} \geq 0$ .

Let $\{B_t\}_{t \in {\mathbb T}_r}$ be the matrix pattern with $B_{t_0} = B$ .

4.2 Admissible functions on $\mathbb {A}_{B^T}$

Let again B be any $r \times r$ mutation matrix and let $(\mathbb {A}_{B^T}, \mathbb {Y}_B)$ be the Fock-Goncharov dual pair as in (4.1).

Definition 4.4. 1) Given $\rho = \{\rho _t \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r} \in \mathbb {Y}_{B}({\mathbb Z}^{\mathrm {max}})$ , an element $x^{\scriptscriptstyle \vee } \in \mathcal {U}(\mathbb {A}_{B^T})$ is said to be $\rho $ -admissible if for each $t \in {\mathbb T}_r$ , one has

(4.2) $$ \begin{align} x^{\scriptscriptstyle \vee} = (\mathbf{x}_t^{\scriptscriptstyle \vee})^{-\rho_t^T} \left(1 + \sum_{0 \neq \mathbf{u} \in U_t} \lambda_{\mathbf{u}} (\mathbf{ x}_t^{\scriptscriptstyle \vee})^{B_t^T\mathbf{u}}\right) \end{align} $$

for some finite subset $U_t$ of ${\mathbb Z}_{\geq 0}^r$ and $\lambda _{\mathbf {u}} \in {\mathbb Z}_{\geq 0}$ for every $\mathbf {u} \in U_t$ . Denote by ${\mathcal {U}}^{\mathrm {admi}}_{\rho }(\mathbb {A}_{B^T})$ the set of all $\rho $ -admissible elements in ${\mathcal {U}}(\mathbb {A}_{B^T})$ .

2) An element in ${\mathcal {U}}(\mathbb {A}_{B^T})$ is said to be admissible, and called an admissible function on $\mathbb {A}_{B^T}$ , if it is $\rho $ -admissible for some $\rho \in \mathbb {Y}_{B}({\mathbb Z}^{\mathrm {max}})$ . The set of all admissible functions on $\mathbb {A}_{B^T}$ is denoted by ${\mathcal {U}}^{\mathrm {admi}}(\mathbb {A}_{B^T})$ .

For a given mutation matrix B, a natural question is whether or not ${\mathcal {U}}^{\mathrm {admi}}_{\rho }(\mathbb {A}_{B^T}) \neq \emptyset $ for every $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ . We will show in Corollary 4.21 that the answer is positive when B is mutation equivalent to an acyclic matrix. See also Remark 4.22. On the other hand, the set ${\mathcal {U}}^{\mathrm {admi}}_{\rho }(\mathbb {A}_{B^T})$ may contain more than one element as in the following example.

Remark 4.6. Suppose that B has full rank. Then for every $t \in {\mathbb T}_r$ , $B_t^T$ has full rank and one has the partial order $\preceq _t$ , called the dominance order in [Reference Qin24], on ${\mathbb Z}^{r}$ , defined as

$$\begin{align*}\mathbf{m} \preceq \mathbf{m}' \hspace{.2in} \text{if and only if} \hspace{.2in} \mathbf{m}' = \mathbf{m} + {{B}_t^T \mathbf{u}} \;\;\; \text{for some}\;\;\; \mathbf{u} \in {\mathbb Z}_{\geq 0}^{r}. \end{align*}$$

For any $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {A}_{B^T})$ and $t \in {\mathbb T}_r$ , $-{\rho }_t^T$ is the unique minimal element among all the exponent vectors in the Laurent expansion of $x^{\scriptscriptstyle \vee }$ in $\mathbf {x}^{\scriptscriptstyle \vee }_t$ . In particular, when B has full rank, every $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {A}_{B^T})$ is $\rho $ -admissible for a unique $\rho \in \mathbb {Y}_B({\mathbb Z}^{\mathrm { max}})$ , and admissible functions on $\mathbb {A}_{B^T}$ are precisely all the universally positive and compatibly pointed elements in ${\mathcal {U}}(\mathbb {A}_{B^T})$ in the sense of [Reference Qin38, Definition 3.4.2]. If B does not have full rank, an arbitrary element $x^{\scriptscriptstyle \vee } \in {\mathcal {U}}(\mathbb {A}_{B^T})$ may have more than one expression of the form in (4.2) at some $t \in {\mathbb T}_r$ , so there is no well-defined vector $\rho _t \in {\mathbb Z}^r_{\mathrm {row}}$ associated to $x^{\scriptscriptstyle \vee }$ and $t \in {\mathbb T}_r$ .

Proof. We prove 2) first. Let t, $\mathbf {m}$ , and $\rho (x^{\scriptscriptstyle \vee }) \in \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ be as stated. For $w \in {\mathbb T}_r$ , let $G_t^w:=G_t^{B_w^T; w}$ be the G-matrix and $\mathbf {F}_t^w := (F_{t; 1}^{B_w^T; w}, \ldots , F_{t; r}^{B_w^T; w})$ be the F-polynomials in r independent variables defined in [Reference Fomin and Zelevinsky23, §Reference Buan, Marsh, Reineke, Reiten and Todorov6]. Then $G_t^t = I_r$ and $\mathbf {x}_t^{\scriptscriptstyle \vee } = (\mathbf {x}_w^{\scriptscriptstyle \vee })^{G_t^w} \mathbf {F}_t^w((\mathbf {x}_w^{\scriptscriptstyle \vee })^{B_w^T})$ . It follows that

$$\begin{align*}x^{\scriptscriptstyle \vee} = (\mathbf{x}_t^{\scriptscriptstyle \vee})^{\mathbf{m}} = (\mathbf{x}_w^{\scriptscriptstyle \vee})^{G_t^w\mathbf{m}} \left(\mathbf{ F}_t^w((\mathbf{x}_w^{\scriptscriptstyle \vee})^{B_w^T})\right)^{\mathbf{m}}, \hspace{.2in} w \in {\mathbb T}_r. \end{align*}$$

Let $i \in [1, r]$ and $\rho _{(t; i)} =\{-(G_t^we_i)^T \in {\mathbb Z}_{\mathrm {row}}^r \}_{w\in {\mathbb T}_r}$ , where $e_i$ is the $i^{\mathrm {th}}$ standard basis vector of $\mathbb Z^r$ . By [Reference Nakanishi and Zelevinsky36, Proposition 4.2], the sign-coherence of c-vectors [Reference Gross, Hacking, Keel and Kontsevich27, Corollary 5.5] implies that $\rho _{(t; i)} \in \mathbb {Y}_{B}({\mathbb Z}^{\mathrm {max}})$ and the F-polynomial $F_{t; i}^{B_w^T, w}$ has constant term $1$ for $i\in [1,r]$ . By the G-matrix sign-coherence [Reference Gross, Hacking, Keel and Kontsevich27, Theorem 5.11] and the fact that $\mathbf {m} \in {\mathbb Z}_{\geq 0}^r$ , the assignment $\left \{-(G_t^w \mathbf {m})^T \in {\mathbb Z}_{\mathrm {row}}^r\right \}_{w\in {\mathbb T}_r}$ is a tropical point of $\mathbb {Y}_B$ , which must equal $\rho (x^{\scriptscriptstyle \vee })$ as the two have the same coordinate vector $-\mathbf {m}^T$ at t. Moreover, for every $w \in {\mathbb T}_r$ , one knows by [Reference Gross, Hacking, Keel and Kontsevich27, Corollary 0.4] that the F-polynomial $F_{t; i}^{B_w^T, w}$ has nonnegative integer coefficients for every $i \in [1, r]$ , so $\left (\mathbf {F}_t^w((\mathbf {x}_w^{\scriptscriptstyle \vee })^{B_w^T})\right )^{\mathbf {m}}$ is a polynomial in $(\mathbf {x}_w^{\scriptscriptstyle \vee })^{B_w^T}$ with nonnegative integer coefficients and constant term $1$ . It follows that $x^{\scriptscriptstyle \vee }$ is $\rho (x^{\scriptscriptstyle \vee })$ -admissible.

Suppose now that $\rho = \{\rho _w \in {\mathbb Z}^r_{\mathrm {row}}\}_{w \in {\mathbb T}_r}$ is any point in $\mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ such that $x^{\scriptscriptstyle \vee }$ is $\rho $ -admissible. Then

$$\begin{align*}(\mathbf{x}_t^{\scriptscriptstyle \vee})^{\mathbf{m}} = x^{\scriptscriptstyle \vee} =(\mathbf{x}_t^{\scriptscriptstyle \vee})^{-\rho_t^T} F_t((\mathbf{ x}_t^{\scriptscriptstyle \vee})^{B_t^T}) \end{align*}$$

for some $F_t \in {\mathbb Z}_{\geq 0}[y_1, \ldots , y_r]$ with constant term $1$ . By the linear independence of Laurent monomials in $\mathbf {x}_{t}^{\scriptscriptstyle \vee }$ , one must have $F_t = 1$ and $\rho _t = -\mathbf {m}^T$ . Thus $\rho = \rho (x^{\scriptscriptstyle \vee })$ , which concludes the proof of 2).

1) is [Reference Gross, Hacking, Keel and Kontsevich27, (2) of Lemma 7.10]. However, 1) also follows easily from 2). Indeed, assume that $x^{\scriptscriptstyle \vee }=(\mathbf {x}_{t'}^{\scriptscriptstyle \vee })^{\mathbf {m}'}$ for some $t'\in \mathbb T_r$ and $\mathbf {m}'\in \mathbb Z_{\geq 0}^r$ . Let $\rho '$ be the unique tropical point of $\mathbb {Y}_B$ with coordinate vector $-(\mathbf {m}^\prime )^T$ at $t'$ . Then by 2), $x^{\scriptscriptstyle \vee }$ is $\rho '$ -admissible, and $\rho ' = \rho (x^{\scriptscriptstyle \vee })$ .

Let $\Delta ^+(\mathbb {Y}_B, {\mathbb Z}) \subset \mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ be the setFootnote ⁴ of g-vectors of all cluster monomials on $\mathbb {A}_{B^T}$ , that is,

(4.3) $$ \begin{align} \Delta^+(\mathbb{Y}_B, {\mathbb Z})=\{\rho = \{\rho_t\in {\mathbb Z}^r_{\mathrm{row}}\}_{t \in {\mathbb T}_r}\in \mathbb{Y}_B({\mathbb Z}^{\mathrm{max}}): -\rho_t \geq 0\, \text{for some}\, t \in {\mathbb T}_r\}. \end{align} $$

Denote by ${\mathcal {U}}^{\mathrm {mono}}(\mathbb {A}_{B^T})$ the set of all cluster monomials on $\mathbb {A}_{B^T}$ . By Lemma-Definition 4.7, one has

$$\begin{align*}{\mathcal{U}}^{\mathrm{mono}}(\mathbb{A}_{B^T}) \subset {\mathcal{U}}^{\mathrm{admi}}(\mathbb{A}_{B^T}) \subset {\mathcal{U}}(\mathbb{A}_{B^T}). \end{align*}$$

Lemma 4.8. For any mutation matrix B, one has the well-defined bijection

(4.4) $$ \begin{align} {\mathcal{U}}^{\mathrm{mono}}(\mathbb{A}_{B^T}) \longrightarrow \Delta^+(\mathbb{Y}_B, {\mathbb Z}),\;\; x^{\scriptscriptstyle \vee} \longmapsto \rho(x^{\scriptscriptstyle \vee}). \end{align} $$

Proof. By Lemma-Definition 4.7, the map in (4.4) is well-defined and it is surjective. The injectivity of the map in (4.4) seems well known and is implied in [Reference Gross, Hacking, Keel and Kontsevich27, (5) of Theorem 0.3] via the construction of theta functions, but we provide an elementary proof for the convenience of the reader.

Suppose that $x^{\scriptscriptstyle \vee }, z^{\scriptscriptstyle \vee } \in {\mathcal {U}}^{\mathrm {mono}}(\mathbb {A}_{B^T})$ are such that $\rho (x^{\scriptscriptstyle \vee }) = \rho (z^{\scriptscriptstyle \vee }) = \{\rho _t \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ , and let $t \in {\mathbb T}_r$ be such that $-\rho _t \geq 0$ and $x^{\scriptscriptstyle \vee }= (\mathbf {x}_t^{\scriptscriptstyle \vee })^{-\rho _t^T}$ . As $z^{\scriptscriptstyle \vee }$ is also $\rho $ -admissible, one has

$$\begin{align*}z^{\scriptscriptstyle \vee} = (\mathbf{x}_t^{\scriptscriptstyle \vee})^{-\rho_t^T} F((\mathbf{x}_t^{\scriptscriptstyle \vee})^{B_t^T}) = x^{\scriptscriptstyle \vee} F((\mathbf{x}_t^{\scriptscriptstyle \vee})^{B_t^T}) \end{align*}$$

for some $F \in {\mathbb Z}_{\geq 0}[y_1, \ldots , y_r]$ with constant term $1$ . Let $f =F((\mathbf {x}_t^{\scriptscriptstyle \vee })^{B_t^T})-1 \in \mathbb {F}(\mathbb {A}_{B^T})$ , so either $f = 0$ or f is in $\mathbb {F}_{>0}(\mathbb {A}_{B^T})$ . Similarly, there exist $t^\prime \in {\mathbb T}_r$ and $F^\prime \in {\mathbb Z}_{\geq 0}[y_1, \ldots , y_r]$ with constant term $1$ such that $x^{\scriptscriptstyle \vee } = z^{\scriptscriptstyle \vee } F^\prime ((\mathbf {x}_{t'}^{\scriptscriptstyle \vee })^{B_{t'}^T})$ . Let $f' = F^\prime ((\mathbf { x}_{t'}^{\scriptscriptstyle \vee })^{B_{t'}^T}) - 1$ , which is again either $0$ or in $\mathbb {F}_{>0}(\mathbb {A}_{B^T})$ . It follows that $1 = F((\mathbf {x}_t^{\scriptscriptstyle \vee })^{B_t^T})F^\prime ((\mathbf {x}_{t'}^{\scriptscriptstyle \vee })^{B_{t'}^T}) = (1+f) (1+f^\prime )$ , so $f + f^\prime + ff^\prime = 0$ , which can happen only if $f = f' = 0$ . Thus $x^{\scriptscriptstyle \vee } = z^{\scriptscriptstyle \vee }$ . Hence, the map in (4.4) is injective.

Remark 4.9. Consider the semi-field ${\mathbb R}^{\mathrm {max}} = ({\mathbb R}, \cdot , \oplus )$ with $a \cdot b = a+b$ and $a \oplus b = \mathrm {max}(a, b)$ for $a, b \in {\mathbb R}$ , and the set $\mathbb {Y}_B({\mathbb R}^{\mathrm {max}}) = \mathrm {Hom}_{\mathrm {sf}}(\mathbb {F}_{>0}(\mathbb {Y}_B), {\mathbb R}^{\mathrm {max}})$ of ${\mathbb R}^{\mathrm {max}}$ -tropical points of $\mathbb {Y}_B$ . A point in $\mathbb {Y}_B({\mathbb R}^{\mathrm {max}})$ is then an assignment $\rho = \{\rho _t \in {\mathbb R}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ to each $t \in {\mathbb T}_r$ of a (row) vector of size r with entries in ${\mathbb R}$ which obeys the same mutation rule (3.6) as points in $\mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})$ , so $\mathbb {Y}_B({\mathbb Z}^{\mathrm {max}})\subset \mathbb {Y}_B({\mathbb R}^{\mathrm {max}})$ as sets. For $t \in {\mathbb T}_r$ , let $C_t^+({\mathbb R}) \subset \mathbb {Y}_B({\mathbb R}^{\mathrm {max}})$ be the set of all $\rho \in \mathbb {Y}_B({\mathbb R}^{\mathrm {max}})$ such that $-\rho _t \geq 0$ , and call $C_t^+({\mathbb R})$ the Fock-Goncharov cluster chamber associated to t [Reference Gross, Hacking, Keel and Kontsevich27, Definition 2.9]. Define

$$\begin{align*}\Delta^+(\mathbb{Y}_B, {\mathbb R}) = \bigcup_{t \in {\mathbb T}_r} C_t^+({\mathbb R}) \subset \mathbb{Y}_B({\mathbb R}^{\mathrm{max}}). \end{align*}$$

By [Reference Gross, Hacking, Keel and Kontsevich27, Theorem 2.13], under the identification

(4.5) $$ \begin{align} \mathbb{Y}_B({\mathbb R}^{\mathrm{max}}) \cong {\mathbb R}^r, \;\; \rho = \{\rho_t \in {\mathbb R}^r_{\mathrm{row}}\}_{t \in {\mathbb T}_r} \longmapsto -\rho_{t_0}^T, \end{align} $$

the collection $\Delta ^+(\mathbb {Y}_B)$ of the subsets $C_t^+({\mathbb R})$ over $t \in {\mathbb T}_r$ forms the maximal cones of a simplicial fan in ${\mathbb R}^r$ , called the g-vector fan of $\mathbb {A}_{B^T}$ . Note that under the identification in (4.5), $\Delta ^+(\mathbb {Y}_B, {\mathbb Z})$ is exactly the set of all integral points in $\Delta ^+(\mathbb {Y}_B, {\mathbb R})$ .

4.3 Admissible functions on $\mathbb {A}_B^{\mathrm {prin}}$

For an arbitrary full-rank extended mutation matrix $(B \backslash \!\!\backslash P)$ as in $\S $ 3.6, one can define admissible functions on $\mathbb {A}_{(B\backslash \!\!\backslash P)}$ using tropical points of $\mathbb {Y}_{(B^T \; P^T)}$ , and we refer to [Reference Cao8] for this general setting. For our paper, we only need to consider admissible functions on positive $\mathbf {A}$ -spaces with principal coefficients.

Let B be any $r \times r$ mutation matrix, and consider the Fock-Goncharov dual pair

$$\begin{align*}(\mathbb{A}_B^{\mathrm{prin}} = \mathbb{A}_{(B \backslash\!\!\backslash I_r)}, \;\; \mathbb{Y}_{B^T}^{\mathrm{prin}} = \mathbb{Y}_{(B^T\; I_r)}). \end{align*}$$

For $t \in {\mathbb T}_r$ , let $\widetilde {B}_t = (B_t \backslash \!\!\backslash C_t)$ be given in equation (3.10), and denote the clusters of $\mathbb {A}_B^{\mathrm {prin}}$ as ${\{\widetilde {\mathbf {u}}_t = (\mathbf {u}_t, \mathbf {p})\}_{t \in {\mathbb T}_r}}$ , where $\mathbf {p} = (p_1, \ldots , p_r)$ are the frozen variables. Set $\widehat {\mathbf {y}_t}= \widetilde {\mathbf {u}}_t^{\widetilde B_t} =\mathbf {u}_t^{B_t}\mathbf {p}^{C_t}$ for $t \in {\mathbb T}_r$ .

An extended cluster monomial of $\mathbb {A}_B^{\mathrm {prin}}$ is any $\widetilde {u} \in \mathbb {F}_{>0}(\mathbb {A}_B^{\mathrm {prin}})$ such that $\widetilde {u} = \mathbf {u}_t^{\mathbf {m}} \mathbf {p}^{\mathbf {n}}$ for some $t \in {\mathbb T}_r$ , $\mathbf {m} \in {\mathbb Z}_{\geq 0}^r$ , and $\mathbf {n} \in {\mathbb Z}^r$ . Denote by ${\mathcal {U}}^{\mathrm {mono}}(\mathbb {A}_B^{\mathrm {prin}})$ the set of all extended cluster monomials on $\mathbb {A}_B^{\mathrm {prin}}$ . Let

(4.6) $$ \begin{align} \Delta^+(\mathbb{Y}_{B^T}^{\mathrm{prin}}, {\mathbb Z}) = \{\widetilde{\rho} \in \mathbb{Y}_{B^T}^{\mathrm{prin}}({\mathbb Z}^{\mathrm{max}}): \omega_{\!{}_{(B^T \; I_r)}}(\widetilde{\rho})_t \leq 0 \; \text{at some}\; t \in {\mathbb T}_r\}, \end{align} $$

where $\omega $ is defined in Lemma 3.17. The following statement and definition can again be found in [Reference Gross, Hacking, Keel and Kontsevich27, Definition 5.10 and Lemma 7.10]. We give a direct proof using Lemma-Definition 4.7.

Lemma-Definition 4.12. Let $\widetilde {u} \in {\mathcal {U}}^{\mathrm {mono}}(\mathbb {A}_B^{\mathrm {prin}})$ . Let $t \in {\mathbb T}_r$ , $\mathbf {m} \in {\mathbb Z}_{\geq 0}^r$ , and $\mathbf { n} \in {\mathbb Z}^r$ be such that $\widetilde {u} = \mathbf {u}_t^{\mathbf {m}} \mathbf {p}^{\mathbf {n}}$ , and let $\widetilde {\rho }(\widetilde {u})$ be the unique element in $\Delta ^+(\mathbb {Y}_{B^T}^{\mathrm {prin}}, {\mathbb Z})\subset \mathbb {Y}_{B^T}^{\mathrm {prin}}({\mathbb Z}^{\mathrm {max}})$ with coordinate vector $-(\mathbf {m}^T, \mathbf {n}^T)$ at t. Then the map

(4.7) $$ \begin{align} {\mathcal{U}}^{\mathrm{mono}}(\mathbb{A}_B^{\mathrm{prin}}) \longrightarrow \Delta^+(\mathbb{Y}_{B^T}^{\mathrm{prin}}, {\mathbb Z}), \;\; \widetilde{u} \longmapsto \widetilde{\rho}(\widetilde{u}), \end{align} $$

is a well-defined bijection and $\widetilde {u}$ is $\widetilde {\rho }(\widetilde {u})$ -admissible. We call $\widetilde {\rho }(\widetilde {u})$ the extended g-vector of $\widetilde {u} \in {\mathcal {U}}^{\mathrm {mono}}(\mathbb {A}_B^{\mathrm {prin}})$ .

Proof. Suppose that $\widetilde {u} = \mathbf {u}_t^{\mathbf {m}} \mathbf {p}^{\mathbf {n}}$ for some $t \in {\mathbb T}_r$ , $\mathbf {m} \in {\mathbb Z}_{\geq 0}^r$ , and $\mathbf {n} \in {\mathbb Z}^r$ . Then $\widetilde {u}^\prime := \mathbf {u}_t^{\mathbf {m}}$ is a cluster monomial on $\mathbb {A}_{\overline {B}} = (\mathbb {F}(\mathbb {A}_{\overline {B}}), \{(\mathbf {u}_w, \mathbf {p})\}_{w \in {\mathbb T}_{2r}})$ , where $\overline {B}$ is given in equation (3.9). By Lemma-Definition 4.7, $\widetilde {u}^\prime $ is $\overline {\rho }$ -admissible, where $\overline {\rho }$ is the tropical point in $\mathbb {Y}_{\overline {B}^T}({\mathbb Z}^{\mathrm {max}})$ with coordinate vector $(-\mathbf {m}^T,0)\in \mathbb Z_{\mathrm {row}}^{2r}$ at the vertex t. For any $w \in {\mathbb T}_r \subset {\mathbb T}_{2r}$ , one has

$$\begin{align*}\widetilde{u}^\prime =\mathbf{u}_t^{\mathbf{m}}= (\mathbf{u}_w, \mathbf{p})^{-\overline{\rho}_w^T} \, (\mathbf{F}_t^{w} ((\mathbf{u}_w, \mathbf{ p})^{\overline{B}_w}))^{\mathbf{m}'}, \quad \mathbf{m}' = \begin{pmatrix} \mathbf{m}\\ 0 \end{pmatrix}\in \mathbb Z^{2r}, \end{align*}$$

where $\{\overline {B}_w\}_{w\in {\mathbb T}_{2r}}$ is the matrix pattern of $\mathbb {A}_{\overline {B}}$ , and $\mathbf {F}^w_t = (F^{\overline {B}_w;w}_{t;1}, \ldots , F^{\overline {B}_w;w}_{t;2r})$ are F-polynomials (see [Reference Fomin and Zelevinsky23, §Reference Buan, Marsh and Reiten5]). Since $w \in {\mathbb T}_r \subset {\mathbb T}_{2r}$ and by a recursive argument using [Reference Fomin and Zelevinsky23, (5.2), (5.3) and (5.9)], one can show that $F^{\overline {B}_w;w}_{t;i}$ depends only on $\widehat {\mathbf {y}}_w=\mathbf {u}_w^{B_w}\mathbf {p}^{C_w}$ and that $F^{\overline {B}_w;w}_{t;r+i} = 1$ for each $i \in [1,r]$ . Moreover, it follows from the mutation rule of the coordinate vectors of tropical points of $\mathbb {Y}_{B^T}^{\mathrm {prin}}$ that $\overline {\rho }_w -\widetilde {\rho }(\widetilde {u})_w = (0, \mathbf {n}^T)$ for every $w \in {\mathbb T}_r$ . Applying Lemma-Definition 4.7 to $\widetilde {u}^\prime $ and using $\widetilde {u} = \mathbf {u}_t^{\mathbf {m}} \mathbf {p}^{\mathbf { n}}=\widetilde {u}^\prime \mathbf {p}^{\mathbf {n}}$ , one has that $\widetilde {u}$ is $\widetilde {\rho }(\widetilde {u})$ -admissible. By the same arguments as in the proof of Lemma-Definition 4.7, one can show that $\widetilde {\rho }(\widetilde {u})$ is independent of the choice of t and thus it is well-defined. By the same arguments as in the proof of Lemma 4.8, one can show that the map in (4.7) is bijective.

The following lemma will be used in §7.5.

Proof. Let $\mathrm {Trop}(\mathbf {p})$ be the tropical semi-field generated by $\mathbf {p} = (p_1, \ldots , p_r)$ , and let

(4.8) $$ \begin{align} T: \;\; \mathbb{F}_{>0}(\mathbb{A}_B^{\mathrm{prin}}) \longrightarrow \mathrm{Trop}(\mathbf{p}) \end{align} $$

be the unique semi-field homomorphism which sends $p_i \in \mathbb {F}_{>0}(\mathbb {A}_{B}^{\mathrm {prin}})$ to $p_i \in \mathrm {Trop}(\mathbf {p})$ for every $i \in [1, r]$ and sends every cluster variable of $\mathbb {A}_B^{\mathrm {prin}}$ to $1 \in \mathrm {Trop}(\mathbf {p})$ (see [Reference Germain, Huang and Lu16, Lemma 2.2]). Suppose that $t \in {\mathbb T}_r$ is such that $\widetilde {u} = \mathbf {u}_t^{-\rho _t^T} \mathbf {p}^{-\mu _t^T}$ . Then $T(\widetilde {u}) = \mathbf {p}^{-\mu _t^T}$ . On the other hand, since $\widetilde {u}$ is $\widetilde {\rho }$ -admissible,

(4.9) $$ \begin{align} \widetilde{u} = \mathbf{u}_{t_0}^{-\rho_{t_0}^T} \mathbf{p}^{-\mu_{t_0}^T} F_{t_0}(\widehat{\mathbf{y}}_{t_0}) \end{align} $$

for some polynomial $F_{t_0}$ with nonnegative integer coefficients and constant term $1$ . Since $T(\widehat {\mathbf {y}}_{t_0})=\mathbf {p}$ ,

$$\begin{align*}T(\widetilde{u}) = T\left(\mathbf{u}_{t_0}^{-\rho_{t_0}^T} \mathbf{p}^{-\mu_{t_0}^T} F_{t_0}(\widehat{\mathbf{y}}_{t_0})\right) =\mathbf{p}^{-\mu_{t_0}^T} T(F_{t_0}(\widehat{\mathbf{y}}_{t_0})) = \mathbf{p}^{-\mu_{t_0}^T}. \end{align*}$$

It follows that $\mu _t = \mu _{t_0}$ .

4.4 Admissible functions on $\mathbb {Y}_B$

We continue with the notation in (4.1).

Definition 4.14. 1) Given $\delta ^{\scriptscriptstyle \vee } = \{\delta _t^{\scriptscriptstyle \vee } \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r} \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ , an element $y \in \mathcal {U}(\mathbb {Y}_{B})$ is said to be $\delta ^{\scriptscriptstyle \vee }$ -admissible if for each $t \in {\mathbb T}_r$ , one has

(4.10) $$ \begin{align} y = (\mathbf{y}_t)^{-(\delta_t^{\scriptscriptstyle \vee})^T} \left(1 + \sum_{0 \neq \mathbf{v} \in V_t} \mu_{\mathbf{v}} \mathbf{y}_t^{\mathbf{v}}\right) \quad \end{align} $$

for some finite subset $V_t$ of ${\mathbb Z}_{\geq 0}^r$ and $\mu _{\mathbf {v}} \in {\mathbb Z}_{\geq 0}$ for every $\mathbf {v} \in V_t$ . The set of all $\delta ^{\scriptscriptstyle \vee }$ -admissible elements of ${\mathcal {U}}(\mathbb {Y}_B)$ is denoted by ${\mathcal {U}}^{\mathrm {admi}}_{\delta ^{\scriptscriptstyle \vee }}(\mathbb {Y}_B)$ .

2) An element in ${\mathcal {U}}(\mathbb {Y}_B)$ that is $\delta ^{\scriptscriptstyle \vee }$ -admissible for some $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ is called an admissible function on $\mathbb {Y}_B$ . The set of all admissible functions on $\mathbb {Y}_B$ is denoted by ${\mathcal {U}}^{\mathrm {admi}}(\mathbb {Y}_B)$ .

Remark 4.15. Simpler than the case of positive $\mathbf {A}$ -spaces, every nonzero $y \in {\mathcal {U}}^+(\mathbb {Y}_B)$ gives rise to a well-defined assignment $\delta ^{\scriptscriptstyle \vee } = \{\delta _t^{\scriptscriptstyle \vee } \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ via its Laurent expansion

$$\begin{align*}y = \mathbf{y}_t^{-(\delta_t^{\scriptscriptstyle \vee})^T} F_t(\mathbf{y}_t) \end{align*}$$

at every $t \in {\mathbb T}_r$ , where $F_t(\mathbf {y}_t) \in {\mathbb Z}_{\geq 0}[\mathbf {y}_t]$ is not divisible by $y_{t; i}$ for any $i \in [1, r]$ , and $(\delta _t^{\scriptscriptstyle \vee })^T \in {\mathbb Z}^r$ is the denominator vector of y at t. Then $y \in {\mathcal {U}}^+(\mathbb {Y}_B)$ is admissible if and only if $\delta ^{\scriptscriptstyle \vee } = \{\delta _t^{\scriptscriptstyle \vee } \in {\mathbb Z}^r_{\mathrm {row}}\}_{t \in {\mathbb T}_r}$ is a tropical point of $\mathbb {A}_{B^T}$ and $F_t$ has constant term $1$ for every $t \in {\mathbb T}_r$ . In particular, every $y \in {\mathcal {U}}^{\mathrm {admi}}(\mathbb {Y}_B)$ is $\delta ^{\scriptscriptstyle \vee }$ -admissible for a unique $\delta ^{\scriptscriptstyle \vee } \in \mathbb {A}_{B^T}({\mathbb Z}^{\mathrm {max}})$ .