# Cluster patterns and scattering diagrams

Tomoki Nakanishi

###### Abstract.

In this survey we review some important results by Gross, Hacking, Keel, and
Kontsevich on cluster algebra theory, namely, the column sign-coherence of
$C$-matrices and the Laurent positivity, both of which were conjectured by
Fomin and Zelevinsky. We digest and reconstruct the proofs of these
conjectures by Gross et al. still based on their scattering diagram method,
however, without relying on toric geometry. At the same time, we also give a
detailed account of the correspondence between the notions of cluster patterns
and scattering diagrams. Most of the results in this article are found in or
translated from the known results in the literature. However, the approach,
the construction of logic and proofs, and the overall presentation are new.
Also, as an application of the results and the techniques in the article, we
show that there is a one-to-one correspondence between $g$-vectors and cluster
variables in cluster patterns with arbitrary coefficients.

###### Key words and phrases:

cluster algebra, scattering diagram

###### 2010 Mathematics Subject Classification:

Primary 13F60

This work was partially supported by JSPS KAKENHI Grant Number 16H03922.

###### Contents

  1. 1 Introduction
  2. 2 Cluster patterns and separation formulas
    1. 2.1 Seeds and mutations
    2. 2.2 Cluster patterns
    3. 2.3 $C$-matrices, $G$-matrices, and $F$-polynomials
      1. 2.3.1 Matrix notations
      2. 2.3.2 $C$-matrices
      3. 2.3.3 $G$-matrices
      4. 2.3.4 $F$-polynomials
    4. 2.4 Separation formulas and tropicalization
    5. 2.5 Sign-coherence of $C$-matrices and Laurent positivity
  3. 3 More about $C$\- and $G$-matrices
    1. 3.1 Mutations and second duality
    2. 3.2 Dual mutations and third duality
    3. 3.3 Principal extension of $B$-patterns
    4. 3.4 $G$-cones and $G$-fans
    5. 3.5 Rank 2 examples of $G$-fans
    6. 3.6 Piecewise linear isomorphisms between $G$-fans
    7. 3.7 Proof of Theorem 3.17
    8. 3.8 Properties of $\hat{c}$-vectors
  4. 4 Scattering diagrams and $C$\- and $G$-matrices
    1. 4.1 Supports of scattering diagrams
    2. 4.2 Proof of sign-coherence conjecture
  5. 5 More about $F$-polynomials
    1. 5.1 Fock-Goncharov decomposition
    2. 5.2 Nontropical parts and $F$-polynomials
    3. 5.3 Detropicalization
  6. 6 Scattering diagrams
    1. 6.1 Fixed data and seed
    2. 6.2 Walls
    3. 6.3 Scattering diagrams
    4. 6.4 Wall-crossing automorphisms
    5. 6.5 Rank 2 examples of consistent scattering diagrams $\mathfrak{D}_{\mathfrak{s}}$
  7. 7 Scattering diagrams and $F$-polynomials
    1. 7.1 Mutations of scattering diagrams
    2. 7.2 Relation between $\mathfrak{D}(\mathbf{\Sigma}_{\mathfrak{s},t_{0}})$ and $\mathfrak{D}_{\mathfrak{s}}$
    3. 7.3 $F$-polynomials and wall-crossing automorphisms
    4. 7.4 Theta functions and Laurent positivity
  8. 8 Some applications
    1. 8.1 Detropicalization revisited
    2. 8.2 Bijection between $g$-vectors and $x$-variables

##  1\. Introduction

In this survey we review some important results by Gross, Hacking, Keel, and
Kontsevich [GHKK18] on cluster algebra theory, namely, the column sign-
coherence of $C$-matrices and the Laurent positivity, both of which were
conjectured by Fomin and Zelevinsky [FZ07, FZ02].

Let us start by recalling the background. In the paper [FZ07] Fomin and
Zelevinsky developed a new method to study cluster patterns, which are the
underlying algebraic and combinatorial structure to define cluster algebras.
The basic ingredients of the method are $C$-matrices, $G$-matrices, and
$F$-polynomials associated with a given cluster pattern. Through the
separation formulas, the cluster pattern is recovered from them. Moreover, one
can regard $C$\- and $G$-matrices as the tropical part of the cluster pattern,
while $F$-polynomials as the nontropical part, so that the separation formulas
naturally separate these two parts. In many applications of cluster algebras
or cluster patterns, this separation plays a central role.

In [FZ07] several conjectures on the properties of $C$\- and $G$-matrices and
$F$-polynomials were also proposed. Among them, there are two prominent
conjectures, namely,

  * •

The sign-coherence conjecture: Every $C$-matrix is column sign-coherent.

  * •

The Laurent positivity conjecture: Every $F$-polynomial has no negative
coefficients. (It is equivalent to the earlier conjecture in [FZ02] on the
positivity of the coefficients in the Laurent polynomial expressions for
cluster variables.)

Due to their importance, these conjectures, together with other conjectures in
[FZ07], has been intensively studied, and proved in some special cases by
various methods. Most notably, in the skew-symmetric case, the sign-coherence
conjecture was proved by [DWZ10, Pla11, Nag13], and the Laurent positivity
conjecture was proved by [LS15, Dav18]. However, the extending these proofs to
the most general case, i.e., the skew-symmetrizable case, seemed difficult and
demanding some new ideas.

In [GHKK18] Gross, Hacking, Keel, and Kontsevich finally proved both
conjectures together in full generality by the scattering diagram method. The
scattering diagrams were originally introduced in [KS06] to study the toric
degenerations of Calabi-Yau manifolds in homological mirror symmetry in two
dimensions, and they were generalized to higher dimensions by [GS11].

cluster patterns |  | scattering diagrams
---|---|---
$G$-matrices |  | supports of walls
$C$-matrices |  | normal vectors of walls
$F$-polynomials |  | wall-crossing automorphisms
cluster variables |  | theta functions
Table 1. Correspondence between the notions of cluster patterns and scattering
diagrams

Roughly speaking, we have the correspondence between the notions in cluster
patterns and scattering diagrams as in Table 1. Let us admit this
correspondence. Then, by the definition/construction of scattering diagrams,
any normal vector of a wall is either positive or negative. This proves the
sign-coherence conjecture. On the other hand, any theta function has a
manifestly positive combinatorial description by broken lines as shown in
[GHKK18]. This proves the Laurent positivity conjecture. Therefore,
establishing the above correspondence is the crucial step to prove the
conjectures.

Due to the importance of these theorems in cluster algebra theory, it is
highly desirable to fully understand the above correspondence. However, it
seems that the content of [GHKK18] is accessible only to some experts in
cluster algebras, due to the following obstacles:

  * •

Since the scattering diagram method is still a new idea to researchers in
cluster algebras, the apparent complexity of the formalism does not allow the
readers to penetrate the essence of the method easily.

  * •

Moreover, the arguments and the proofs in [GHKK18] also depend on the
accompanying toric geometrical setting, because it is the motivation of the
paper by the authors of [GHKK18].

To remedy the situation, we digest and reconstruct the results and the proofs
of the above theorems in [GHKK18] still based on their scattering diagram
method, however, without relying on toric geometry. To be more specific, we
set the following guidelines while preparing the manuscript:

  * •

We assume that readers are familiar with basic concepts in cluster algebras to
some extent, for example, seeds and mutations, cluster variables,
coefficients, the Laurent phenomenon, the finite type classification, etc.

  * •

The goal of the article is to give reasonably self-contained proofs of the
sign-coherence and the Laurent positivity theorems based on the formulation of
cluster patterns by [FZ07] with the help of some key properties of scattering
diagrams from [GHKK18].

  * •

At the same time, we also give a detailed account of the correspondence
between the notions of cluster patterns and scattering diagrams in Table 1.

  * •

We employ the Fock-Goncharov decomposition of mutations with tropical sign to
work on $F$-polynomials. This substitutes and avoids the toric geometrical
setting in [GHKK18], including cluster varieties, which plays a key role at
several points in their original proofs of the conjectures.

  * •

We rely crucially on some basic properties of scattering diagrams in [GHKK18]
which are independent of toric geometrical setting. However, we omit the
proofs by two reasons. Firstly, they are rather long to present. Secondly, we
are not able to improve or shorten the original presentations in [GHKK18].
Therefore, we ask the readers to refer to [GHKK18] if necessary.

  * •

Our approach is entirely based on the formulation of cluster patterns in
[FZ07]. However, after the paper [FZ07], considerable progress has been made
in the subject. So, we also take it into account and reorganize and/or
streamline some part of the formulation. Most notably, we do not rely on the
notion of principal coefficients except for the polynomial property of
$F$-polynomials.

Most of the results in this article are found in or translated from the known
results in [GHKK18], and/or in other related literatures, especially, a series
of papers by Reading [RS16, Rea14, Rea20b, Rea20a]. However, the approach, the
construction of logic and proofs, and the overall presentation are new. For
example, we prove the sign-coherence conjecture inductively along the
$n$-regular tree graph together with establishing the correspondence in Table
1 for $C$\- and $G$-matrices. This approach is completely different from the
toric geometrical argument in [GHKK18]. On the other hand, it is close to the
approach in [Rea14, Rea20b] using mutation fans therein. In fact, many of our
results can be also obtained from the results on the mutations fans therein. A
subtle but important difference is that we work on $C$\- and $G$-matrices
without referring to mutation fans at all. Therefore, our approach is more
direct. To summarize, our presentation is complementary to the above
references, and we believe it serves as an alternative guide for the
scattering diagram method in the study of cluster algebras.

Finally let us mention that there are already several excellent applications
of the scattering diagram method to cluster algebra theory (e.g., [Mul16,
CGM+17, CL20, Zho20, Qin19]).

We thank Nathan Reading for helpful comments on the draft and Remark 7.15. We
also thank Kodai Matsushita for discussions on scattering diagrams.

##  2\. Cluster patterns and separation formulas

In this section we recall several basic notions and facts on cluster patterns
by following the formulation of Fomin and Zelevinsky especially in [FZ07].
Most of the materials are taken or adapted from [FZ07], and we assume that the
readers are familiar with them to some extent. Thus, we skip the proofs of the
results unless we think necessary.

###  2.1. Seeds and mutations

Below we fix a positive integer $n$, which is called the rank of the
forthcoming seeds, cluster patterns, etc.

######  Definition 2.1 (Skew-symmetrizable matrix).

An $n\times n$ integer square matrix $B=(b_{ij})_{i,j=1}^{n}$ is said to be
skew-symmetrizable if there is a positive rational diagonal matrix
$D=(d_{i}\delta_{ij})_{i,j=1}^{n}$ such that $DB$ is skew-symmetric, i.e,

(2.1) $\displaystyle d_{i}b_{ij}=-d_{j}b_{ji}$

holds. The matrix $D$ is called a (left) skew-symmetrizer of $B$.

######  Remark 2.2.

Usually a skew-symmetrizer $D$ is assumed to be an integer matrix. However, it
is natural to use a rational one in order to match with the convention in
[GHKK18]. See (6.9). So, we employ the condition here.

For a skew-symmetrizable matrix $B$, its diagonal entries $b_{ii}$ vanish due
to (2.1). Also, (2.1) is rephrased as

(2.2) $\displaystyle DBD^{-1}=-B^{T},$

where for any matrix $M$, $M^{T}$ stands for the transpose of $M$.

For our purpose, we especially employ the following (nonstandard) definition
of seeds.

######  Definition 2.3 (Seed).

Let $\mathcal{F}_{X}$ and $\mathcal{F}_{Y}$ be a pair of fields both of which
are isomorphic to the rational function field of $n$-variables with
coefficients in $\mathbb{Q}$. A (labeled) seed in
$(\mathcal{F}_{X},\mathcal{F}_{Y})$ is a triplet
$\Sigma=(\mathbf{x},\mathbf{y},B)$, where $\mathbf{x}=(x_{1},\dots,x_{n})$ and
$\mathbf{y}=(y_{1},\dots,y_{n})$ are $n$-tuples of free generating elements in
$\mathcal{F}_{X}$ and $\mathcal{F}_{Y}$, respectively, and
$B=(b_{ij})_{i,j=1}^{n}$ is an $n\times n$ skew-symmetrizable (integer)
matrix. We call $\mathbf{x}$, $\mathbf{y}$, and $B$, respectively, the
$x$-variables, the $y$-variables, and the exchange matrix of $\Sigma$. We also
call the pair $(\mathcal{F}_{X},\mathcal{F}_{Y})$ the ambient fields of a seed
$\Sigma$ and the forthcoming cluster patterns, etc.

######  Remark 2.4.

In the standard terminology of [FZ07], the $x$-variables above are cluster
variables without coefficients, while the $y$-variables above are coefficients
in the universal semifield
$\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})\subset\mathcal{F}_{Y}$. Here, $x$\- and
$y$-variables are on an equal footing and independent of each other. In
particular, $y$-variables do not serve as coefficients for $x$-variables.

For any seed $\Sigma=(\mathbf{x},\mathbf{y},B)$, we attach an $n$-tuple
$\hat{\mathbf{y}}=(\hat{y}_{1},\dots,\hat{y}_{n})$ of elements in the ambient
field $\mathcal{F}_{X}$ defined by

(2.3) $\displaystyle\hat{y}_{i}=\prod_{j=1}^{n}x_{j}^{b_{ji}}.$

They play an important role in cluster algebra theory. We call them
$\hat{y}$-variables. They are algebraically independent if and only if $B$ is
nonsingular.

For any integer $a$, we define

(2.4) $\displaystyle[a]_{+}:=\max(a,0).$

We have a useful identity:

(2.5) $\displaystyle a=[a]_{+}-[-a]_{-}.$

######  Definition 2.5 (Seed mutation).

For any seed $\Sigma=(\mathbf{x},\mathbf{y},B)$ and $k\in\\{1,\dots,n\\}$, we
define a new seed
$\Sigma^{\prime}=(\mathbf{x}^{\prime},\mathbf{y}^{\prime},B^{\prime})$ by the
following rule:

(2.6) $\displaystyle x^{\prime}_{i}$ $\displaystyle=\begin{cases}\displaystyle
x_{k}^{-1}\left(\prod_{j=1}^{n}x_{j}^{[-b_{jk}]_{+}}\right)(1+\hat{y}_{k})&i=k,\\\
x_{i}&i\neq k,\end{cases}$ (2.7) $\displaystyle y^{\prime}_{i}$
$\displaystyle=\begin{cases}\displaystyle y_{k}^{-1}&i=k,\\\
y_{i}y_{k}^{[b_{ki}]_{+}}(1+y_{k})^{-b_{ki}}&i\neq k,\end{cases}$ (2.8)
$\displaystyle b^{\prime}_{ij}$
$\displaystyle=\begin{cases}-b_{ij}&\text{$i=k$ or $j=k$,}\\\
b_{ij}+b_{ik}[b_{kj}]_{+}+[-b_{ik}]_{+}b_{kj}&i,j\neq k,\end{cases}$

where $\hat{y}_{k}$ in (2.6) is the $\hat{y}$-variable in (2.3). The seed
$\Sigma^{\prime}$ is called the mutation of $\Sigma$ at direction $k$, and
denoted by $\mu_{k}(\Sigma)=\mu_{k}(\mathbf{x},\mathbf{y},B)$.

The following facts ensure that
$(\mathbf{x}^{\prime},\mathbf{y}^{\prime},B^{\prime})$ is indeed a seed.

  * (a).

If $D$ is a skew-symmetrizer of $B$, then it is also a skew-symmetrizer of
$B^{\prime}$. In particular, the matrix $B^{\prime}$ is skew-symmetrizable.

  * (b).

Regard the mutation $\mu_{k}$ as a map from the set of seeds to itself. Then,
$\mu_{k}$ is an involution. In particular, $\mathbf{x}^{\prime}$ and
$\mathbf{y}^{\prime}$ are free generating elements in $\mathcal{F}_{X}$ and
$\mathcal{F}_{Y}$, respectively.

Also, one can easily verify that $|B^{\prime}|=|B|$, so that $B^{\prime}$ is
nonsingular if and only if $B$ is nonsingular.

The following fact can be easily verified.

######  Proposition 2.6 ([FZ07, Prop. 3.9]).

The $\hat{y}$-variables mutate as

(2.9)
$\displaystyle\hat{y}^{\prime}_{i}=\begin{cases}\displaystyle\hat{y}_{k}^{-1}&i=k,\\\
\hat{y}_{i}\hat{y}_{k}^{[b_{ki}]_{+}}(1+\hat{y}_{k})^{-b_{ki}}&i\neq
k\end{cases}$

in the ambient field $\mathcal{F}_{X}$ by the same rule as (2.7).

There is some flexibility to write the mutation formulas (2.6)–(2.8) as
follows. (The choice $\varepsilon=1$ reduces to the one in (2.6)–(2.8).)

######  Proposition 2.7 ([Nak12, Eqs. (2.9), (2.10)], [BFZ05, Eq. (3.1)]).

Let $\varepsilon\in\\{1,-1\\}$. Then, the right hand sides of the following
equalities do not depend on the choice of $\varepsilon$:

(2.10) $\displaystyle x^{\prime}_{i}$
$\displaystyle=\begin{cases}\displaystyle
x_{k}^{-1}\left(\prod_{j=1}^{n}x_{j}^{[-\varepsilon
b_{jk}]_{+}}\right)(1+\hat{y}_{k}^{\varepsilon})&i=k,\\\ x_{i}&i\neq
k,\end{cases}$ (2.11) $\displaystyle y^{\prime}_{i}$
$\displaystyle=\begin{cases}\displaystyle y_{k}^{-1}&i=k,\\\
y_{i}y_{k}^{[\varepsilon b_{ki}]_{+}}(1+y_{k}^{\varepsilon})^{-b_{ki}}&i\neq
k,\end{cases}$ (2.12) $\displaystyle b^{\prime}_{ij}$
$\displaystyle=\begin{cases}-b_{ij}&\text{$i=k$ or $j=k$,}\\\
b_{ij}+b_{ik}[\varepsilon b_{kj}]_{+}+[-\varepsilon b_{ik}]_{+}b_{kj}&i,j\neq
k.\end{cases}$

The claim can be easily checked with the identity (2.5). We call the above
formulas the $\varepsilon$-expressions of mutations.

###  2.2. Cluster patterns

Let $\mathbb{T}_{n}$ be the $n$-regular tree graph, where the edges are
labeled by $1$,…, $n$ such that the edges attached to each vertex have
different labels. By abusing the notation, the set of vertices of
$\mathbb{T}_{n}$ is also denoted by $\mathbb{T}_{n}$.

######  Definition 2.8 (Cluster pattern/$B$ pattern).

A collection of seeds
$\mathbf{\Sigma}=\\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\\}_{t\in\mathbb{T}_{n}}$
in $(\mathcal{F}_{X},\mathcal{F}_{Y})$ indexed by $\mathbb{T}_{n}$ is called a
cluster pattern if, for any vertices $t,t^{\prime}\in\mathbb{T}_{n}$ which are
connected with an edge labeled by $k$, the equality
$\Sigma_{t^{\prime}}=\mu_{k}(\Sigma_{t})$ holds. The collection of exchange
matrices $\mathbf{B}=\\{B_{t}\\}_{t\in\mathbb{T}_{n}}$ extracted from a
cluster pattern $\mathbf{\Sigma}$ is called the $B$-pattern of
$\mathbf{\Sigma}$.

We arbitrary choose a distinguished vertex (the initial vertex) $t_{0}$ in
$\mathbb{T}_{n}$. Any cluster pattern
$\mathbf{\Sigma}=\\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\\}_{t\in\mathbb{T}_{n}}$
is uniquely determined from the initial seed $\Sigma_{t_{0}}$ at the initial
vertex $t_{0}$ by repeating mutations.

Note that one can take a common skew-symmetrizer $D$ for all $B_{t}$ in
$\mathbf{B}$. By (2.2), we have

(2.13) $\displaystyle DB_{t}D^{-1}=-B_{t}^{T}.$

In particular, if $B_{t}$ is skew-symmetric for some $t$, it is skew-symmetric
for any $t$. Such a cluster pattern or a $B$-pattern is said to be skew-
symmetric. Similarly, if $B_{t}$ is nonsingular for some $t$, it is
nonsingular for any $t$. Such a cluster pattern or a $B$-pattern is said to be
nonsingular.

For a seed $\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})$
($t\in\mathbb{T}_{n}$) in a cluster pattern $\mathbf{\Sigma}$, we use the
notation

(2.14) $\displaystyle\mathbf{x}_{t}=(x_{1;t},\dots,x_{n;t}),\
\mathbf{y}_{t}=(y_{1;t},\dots,y_{n;t}),\ B_{t}=(b_{ij;t})_{i,j=1}^{n}.$

For the initial seed
$\Sigma_{t_{0}}=(\mathbf{x}_{t_{0}},\mathbf{y}_{t_{0}},B_{t_{0}})$, we often
drop the indices $t_{0}$ as

(2.15) $\displaystyle\mathbf{x}_{t_{0}}=\mathbf{x}=(x_{1},\dots,x_{n}),\
\mathbf{y}_{t_{0}}=\mathbf{y}=(y_{1},\dots,y_{n}),\
B_{t_{0}}=B=(b_{ij})_{i,j=1}^{n},$

if there is no confusion. We also use similar notations for
$\hat{y}$-variables $\hat{\mathbf{y}}_{t}=(\hat{y}_{1;t},\dots,\hat{y}_{n;t})$
and the initial $\hat{y}$-variables
$\hat{\mathbf{y}}_{t_{0}}=\hat{\mathbf{y}}=(\hat{y}_{1},\dots,\hat{y}_{n})$.

We recall the definition of cluster algebras, though the focus of this article
is to study the properties of cluster patterns rather than cluster algebras.

######  Definition 2.9 (Cluster algebra).

For a cluster pattern $\mathbf{\Sigma}$, the associated cluster algebra
$\mathcal{A}(\mathbf{\Sigma})$ is the $\mathbb{Z}$-subalgebra of the ambient
field $\mathcal{F}_{X}$ generated by all $x$-variables (cluster variables)
$x_{i,t}$ ($i=1,\dots,n;t\in\mathbb{T}_{n}$).

###  2.3. $C$-matrices, $G$-matrices, and $F$-polynomials

Following [FZ07], we introduce $C$-matrices, $G$-matrices, and
$F$-polynomials. They play a crucial role in the study of cluster patterns.

####  2.3.1. Matrix notations

We start by introducing the following matrix notations (see, e.g., [NZ12]).
For an $n\times n$ matrix $A$, let $[A]_{+}$ denote the matrix obtained from
$A$ by replacing the entry $a_{ij}$ with $[a_{ij}]_{+}$. Let $A^{\bullet k}$
denote the matrix obtained from $A$ by replacing the entry $a_{ij}$ ($j\neq
k$) with $0$. Similarly, let $A^{k\bullet}$ denote the matrix obtained from
$A$ by replacing the entry $a_{ij}$ ($i\neq k$) with 0.

For any $n\times n$ matrices, the following facts are easily checked, where
for the last two equalities we use (2.5).

(2.16) $\displaystyle AB^{\bullet k}=(AB)^{\bullet k},\quad
A^{k\bullet}B=(AB)^{k\bullet},$ (2.17) $\displaystyle A^{\bullet
k}B=A^{\bullet k}B^{k\bullet}=AB^{k\bullet},$ (2.18) $\displaystyle
A[B]_{+}+[-A]_{+}B=A[-B]_{+}+[A]_{+}B,$ (2.19) If $AB=CD$, then
$A[B]_{+}-C[D]_{+}=A[-B]_{+}-C[-D]_{+}$.

Let $J_{k}$ be the $n\times n$ diagonal matrix obtained from the identity
matrix by replacing the $k$th diagonal entry with $-1$. The following
properties will be repeatedly used.

######  Lemma 2.10.

Let $A$ be any $n\times n$ matrix whose diagonal entries are 0. Let
$P=J_{k}+A^{k\bullet}$ or $J_{k}+A^{\bullet k}$. Then, we have

(2.20) $\displaystyle A^{\bullet k}J_{k}=-A^{\bullet k},\quad J_{k}A^{\bullet
k}=A^{\bullet k},\quad A^{k\bullet}J_{k}=A^{k\bullet},\quad
J_{k}A^{k\bullet}=-A^{k\bullet},$ (2.21) $\displaystyle P^{2}=I,$ (2.22)
$\displaystyle|P|=-1.$

###### Proof.

Let us prove (2.21) for $P=J_{k}+A^{k\bullet}$. By $a_{kk}=0$, we have

(2.23)
$\displaystyle\begin{split}P^{2}&=J_{k}^{2}+J_{k}A^{k\bullet}+A^{k\bullet}J_{k}+A^{k\bullet}A^{k\bullet}\\\
&=I-A^{k\bullet}+A^{k\bullet}=I.\end{split}$

Also, we have $|P|=|J_{k}|=-1$. ∎

######  Example 2.11.

The first case of the matrix mutation (2.8) is written as

(2.24)
$\displaystyle(B^{\prime})^{k\bullet}=-B^{k\bullet},\quad(B^{\prime})^{\bullet
k}=-B^{\bullet k}.$

Also, the entire mutation (2.8) is written as [BFZ05, Eq. (3.2)]

(2.25) $\displaystyle B^{\prime}=(J_{k}+[-B]_{+}^{\bullet
k})B(J_{k}+[B]_{+}^{k\bullet}).$

By (2.21), we have the relation

(2.26) $\displaystyle(J_{k}+[-B]_{+}^{\bullet
k})B^{\prime}=B(J_{k}+[B]_{+}^{k\bullet}).$

####  2.3.2. $C$-matrices

Let $\mathbf{\Sigma}$ be any cluster pattern, and let $\mathbf{B}$ be the
$B$-pattern of $\mathbf{\Sigma}$. Let $t_{0}\in\mathbb{T}_{n}$ be a given
initial vertex.

We introduce a collection of matrices
$\mathbf{C}^{t_{0}}=\mathbf{C}=\\{C_{t}\\}_{t\in\mathbb{T}_{n}}$ called a
$C$-pattern, which are uniquely determined from $\mathbf{B}$ and $t_{0}$,
therefore, eventually only from $t_{0}$ and $B_{t_{0}}$.

######  Definition 2.12 ($C$-matrices).

The $C$-matrices $C_{t}=(c_{ij;t})_{i,j=1}^{n}$ ($t\in\mathbb{T}_{n}$) of a
cluster pattern $\mathbf{\Sigma}$ (or a $B$-pattern $\mathbf{B}$) with an
initial vertex $t_{0}$ are $n\times n$ integer matrices which are uniquely
determined by the following initial condition and the mutation rule:

(2.27) $\displaystyle C_{t_{0}}$ $\displaystyle=I,$ (2.28) $\displaystyle
C_{t^{\prime}}$
$\displaystyle=C_{t}J_{k}+C_{t}[B_{t}]_{+}^{k\bullet}+[-C_{t}]_{+}^{\bullet
k}B_{t},$

or, more explicitly,

(2.29) $\displaystyle c_{ij;t^{\prime}}$
$\displaystyle=\begin{cases}-c_{ik;t}&j=k,\\\
c_{ij;t}+c_{ik;t}[b_{kj;t}]_{+}+[-c_{ik;t}]_{+}b_{kj;t}&j\neq k,\end{cases}$

where $t$ and $t^{\prime}$ are connected with an edge labeled by $k$. Each
column vector $\mathbf{c}_{i;t}=(c_{ji;t})_{j=1}^{n}$ of a matrix $C_{t}$ is
called a $c$-vector.

######  Remark 2.13.

The $C$-matrices appeared implicitly as the lower halves of the extended
exchange matrices for the principal coefficients in [FZ07, Def. 3.1].

The mutation (2.28), in particular, implies

(2.30) $\displaystyle\mathbf{c}_{k;t^{\prime}}=-\mathbf{c}_{k;t},$

or, equivalently,

(2.31) $\displaystyle(C)_{t^{\prime}}^{\bullet k}=-C_{t}^{\bullet k}.$

One can easily check that the mutation (2.29) is involutive using (2.5),
(2.8), and (2.30). Alternatively, in the matrix form, by multiplying $J_{k}$
from the right to (2.28), we have

(2.32) $\displaystyle C_{t^{\prime}}J_{k}$ $\displaystyle=C_{t}+C_{t}^{\bullet
k}[B_{t}]_{+}^{k\bullet}+[-C_{t}]_{+}^{\bullet k}B_{t}^{k\bullet}.$

By (2.24) and (2.31), this can be written as

(2.33) $\displaystyle C_{t}$
$\displaystyle=C_{t^{\prime}}J_{k}+C_{t^{\prime}}[-B_{t^{\prime}}]_{+}^{k\bullet}+[C_{t^{\prime}}]_{+}^{\bullet
k}B_{t^{\prime}}.$

Then, by (2.18), we have

(2.34) $\displaystyle C_{t}$
$\displaystyle=C_{t^{\prime}}J_{k}+C_{t^{\prime}}[B_{t^{\prime}}]_{+}^{k\bullet}+[-C_{t^{\prime}}]_{+}^{\bullet
k}B_{t^{\prime}},$

which is the desired result. Moreover, by (2.18) again, we have the following
$\varepsilon$-expression of (2.28):

######  Proposition 2.14 ([NZ12, Eq. (2.4)]).

Let $\varepsilon\in\\{1,-1\\}$. Then, the right hand side of the following
equality does not depend on the choice of $\varepsilon$:

(2.35) $\displaystyle C_{t^{\prime}}$
$\displaystyle=C_{t}J_{k}+C_{t}[\varepsilon
B_{t}]_{+}^{k\bullet}+[-\varepsilon C_{t}]_{+}^{\bullet k}B_{t}.$

####  2.3.3. $G$-matrices

Under the same assumption of $C$-matrices, we introduce another collection of
matrices $\mathbf{G}^{t_{0}}=\mathbf{G}=\\{G_{t}\\}_{t\in\mathbb{T}_{n}}$
called a $G$-pattern, which are uniquely determined from $\mathbf{B}$,
$t_{0}$, and $\mathbf{C}^{t_{0}}$ defined above.

######  Definition 2.15 ($G$-matrices).

The $G$-matrices $G_{t}=(g_{ij;t})_{i,j=1}^{n}$ ($t\in\mathbb{T}_{n}$) of a
cluster pattern $\mathbf{\Sigma}$ (or a $B$-pattern $\mathbf{B}$) with an
initial vertex $t_{0}$ are $n\times n$ integer matrices which are uniquely
determined by the following initial condition and the mutation rule:

(2.36) $\displaystyle G_{t_{0}}$ $\displaystyle=I,$ (2.37) $\displaystyle
G_{t^{\prime}}$ $\displaystyle=G_{t}J_{k}+G_{t}[-B_{t}]_{+}^{\bullet
k}-B_{t_{0}}[-C_{t}]_{+}^{\bullet k},$

or, more explicitly,

(2.38) $\displaystyle g_{ij;t^{\prime}}$
$\displaystyle=\begin{cases}\displaystyle-
g_{ik;t}+\sum_{\ell=1}^{n}g_{i\ell;t}[-b_{\ell
k;t}]_{+}-\sum_{\ell=1}^{n}b_{i\ell;t_{0}}[-c_{\ell k;t}]_{+}&j=k,\\\
g_{ij;t}&j\neq k,\end{cases}$

where $t$ and $t^{\prime}$ are connected with an edge labeled by $k$. Each
column vector $\mathbf{g}_{i;t}=(g_{ji;t})_{j=1}^{n}$ of a matrix $G_{t}$ is
called a $g$-vector.

######  Remark 2.16.

The $g$-vectors were originally introduced as the degree vectors of cluster
varialbes with principal coefficients in [FZ07].

The first nontrivial property so far is the following relation between $C$\-
and $G$-matrices.

######  Proposition 2.17 ([FZ07, Eq. (6.14)]).

The following equality holds for any $t\in\mathbb{T}_{n}$.

(First duality.)

(2.39) $\displaystyle G_{t}B_{t}=B_{t_{0}}C_{t}.$

The proof in [FZ07, Eq. (6.14)] relied on the fact that $g$-vectors are the
degree vectors of cluster variables with principal coefficients. Here we give
an alternative proof based on the definitions of $C$\- and $G$-matrices here.
Let $d(t,t^{\prime})$ denote the distance in $\mathbb{T}_{n}$ between two
vertices $t,t^{\prime}\in\mathbb{T}_{n}$.

###### Proof.

We prove it by the induction on $t$ along $\mathbb{T}_{n}$ from $t_{0}$. For
$t=t_{0}$, the equality (2.39) holds because $C_{t_{0}}=G_{t_{0}}=I$. Suppose
that the equality (2.39) holds for $t\in\mathbb{T}_{n}$ such that
$d(t_{0},t)=d$. Let $t^{\prime}$ be a vertex which is connected to $t$ with an
edge labeled by $k$ such that $d(t_{0},t^{\prime})=d+1$. Then, we have

(2.40)
$\displaystyle\begin{split}G_{t^{\prime}}B_{t^{\prime}}&=G_{t}(J_{k}+[-B_{t}]_{+}^{\bullet
k})B_{t^{\prime}}-B_{t_{0}}[-C_{t}]^{\bullet k}B_{t^{\prime}}^{k\bullet}\\\
&=G_{t}B_{t}(J_{k}+[B_{t}]_{+}^{k\bullet})+B_{t_{0}}[-C_{t}]^{\bullet
k}B_{t}^{k\bullet}\\\
&=B_{t_{0}}C_{t}(J_{k}+[B_{t}]_{+}^{k\bullet})+B_{t_{0}}[-C_{t}]^{\bullet
k}B_{t}\\\ &=B_{t_{0}}C_{t^{\prime}},\end{split}$

where in the second equality we used (2.24) and (2.26). ∎

######  Remark 2.18.

There has been no name on this equality so far in the literature; however, due
to its importance, we name it as above, where the duality means a relation
between $C$\- and $G$-matrices. (Later we will have the second and the third
dualities, which are more nontrivial.)

Thanks to (2.19) and (2.39), one can easily check that the mutation (2.37) is
involutive as before. Moreover, by (2.19) and (2.39) again, we have the
following $\varepsilon$-expression of (2.37).

######  Proposition 2.19 ([FZ07, Eqs. (6.12), (6.13)]).

Let $\varepsilon\in\\{1,-1\\}$. Then, the right hand side of the following
equality does not depend on the choice of $\varepsilon$:

(2.41) $\displaystyle G_{t^{\prime}}$
$\displaystyle=G_{t}J_{k}+G_{t}[-\varepsilon B_{t}]_{+}^{\bullet
k}-B_{t_{0}}[-\varepsilon C_{t}]^{\bullet k}.$

####  2.3.4. $F$-polynomials

Let $\mathbf{y}=(y_{1},\dots,y_{n})$ be an $n$-tuple of formal variables.
Although it is not necessary at this moment, it is natural to identify them
with the initial $y$-variables $\mathbf{y}_{t_{0}}$ of $\mathbf{\Sigma}$ as we
will see soon. Under the same assumption of $C$-matrices, we introduce a
collection of rational functions
$\mathbf{F}^{t_{0}}=\mathbf{F}=\\{F_{i;t}(\mathbf{y})\\}_{i=1,\dots,n;t\in\mathbb{T}_{n}}$
in $\mathbf{y}$ called a $F$-pattern, which are uniquely determined from
$\mathbf{B}$, $t_{0}$, and $\mathbf{C}^{t_{0}}$ as follows.

######  Definition 2.20 ($F$-polynomials).

The $F$-polynomials $F_{i;t}(\mathbf{y})$ ($i=1,\dots,n$;
$t\in\mathbb{T}_{n}$) of a cluster pattern $\mathbf{\Sigma}$ (or a $B$-pattern
$\mathbf{B}$) with an initial vertex $t_{0}$ are rational functions in
$\mathbf{y}$ with coefficients in $\mathbb{Q}$ which are uniquely determined
by the following initial condition and the mutation rule:

(2.42) $\displaystyle F_{i;t_{0}}(\mathbf{y})$ $\displaystyle=1,$ (2.43)
$\displaystyle F_{i;t^{\prime}}(\mathbf{y})$
$\displaystyle=\begin{cases}\frac{\displaystyle
M_{k;t}(\mathbf{y})}{\displaystyle F_{k;t}(\mathbf{y})}&i=k,\\\
F_{i;t}(\mathbf{y})&i\neq k,\end{cases}$

where $t$ and $t^{\prime}$ are connected with an edge labeled by $k$, and
$M_{k,t}(\mathbf{y})$ is a rational function in $\mathbf{y}$ defined as
follows:

(2.44) $\displaystyle M_{k;t}(\mathbf{y})$
$\displaystyle=\prod_{j=1}^{n}y_{j}^{[c_{jk;t}]_{+}}\prod_{j=1}^{n}F_{j;t}(\mathbf{y})^{[b_{jk;t}]_{+}}+\prod_{j=1}^{n}y_{j}^{[-c_{jk;t}]_{+}}\prod_{j=1}^{n}F_{j;t}(\mathbf{y})^{[-b_{jk;t}]_{+}}.$

In the situation in (2.43), we have

(2.45) $\displaystyle M_{k;t}(\mathbf{y})=M_{k;t^{\prime}}(\mathbf{y}).$

Therefore, the mutation (2.43) is involutive. We do not discuss the
$\varepsilon$-expression for (2.43), because we do not use it here.

The $F$-polynomials were originally introduced as the specialized
$x$-variables with principal coefficients under the specialization
$x_{1}=\dots=x_{n}=1$ in [FZ07]. The Laurent phenomenon applied to them yields
the following important fact, which justifies the name $F$-polynomials.

######  Theorem 2.21 ([FZ07, Prop. 3.6]).

For any $i=1,\dots,n$ and $t\in\mathbb{T}_{n}$, the rational function
$F_{i;t}(\mathbf{y})$ is a polynomial in $\mathbf{y}$ with coefficients in
$\mathbb{Z}$.

######  Remark 2.22.

In case we need to clarify the choice of the initial vertex $t_{0}$, we write
$C$-matrices and others as, $C^{t_{0}}_{t}=(c^{t_{0}}_{ij;t})_{i,j=1}^{n}$,
$\mathbf{c}_{i;t}^{t_{0}}$, $G^{t_{0}}_{t}=(g^{t_{0}}_{ij;t})_{i,j=1}^{n}$,
$\mathbf{g}_{i;t}^{t_{0}}$, $F_{i;t}^{t_{0}}(\mathbf{y})$. Conversely, if
there is no confusion, we suppress the superscript $t_{0}$ as above for
simplicity.

###  2.4. Separation formulas and tropicalization

Let us present one of the most fundamental results on cluster patterns in
[FZ07].

######  Theorem 2.23 (Separation Formulas [FZ07, Prop. 3.13, Cor. 6.3]).

Let $\mathbf{\Sigma}$ be any cluster pattern, and let $t_{0}\in\mathbb{T}_{n}$
be a given initial vertex. Let $\mathbf{C}$, $\mathbf{G}$, $\mathbf{F}$ be the
$C$-, $G$-, and $F$-patterns of $\mathbf{\Sigma}$ with the initial vertex
$t_{0}$. Let $\mathbf{x}$, $\mathbf{y}$, and $\hat{\mathbf{y}}$ be the initial
$x$-, $y$-variables, and $\hat{y}$-variables for the initial seed
$\Sigma_{t_{0}}$. Then, the following formulas hold.

(2.46) $\displaystyle x_{i;t}$
$\displaystyle=\left(\prod_{j=1}^{n}x_{j}^{g_{ji;t}}\right)F_{i;t}(\hat{\mathbf{y}}),$
(2.47) $\displaystyle y_{i;t}$
$\displaystyle=\left(\prod_{j=1}^{n}y_{j}^{c_{ji;t}}\right)\prod_{j=1}^{n}F_{j;t}(\mathbf{y})^{b_{ji;t}}.$

###### Proof.

Here is an alternative proof to the one in [FZ07]. Namely, using the mutations
of $C$\- and $G$-matrices, and $F$-polynomials, one can easily verify (2.46)
and (2.47) by the induction on $t$ along $\mathbb{T}_{n}$ from $t_{0}$ with
the help of (2.39). We leave the details as an exercise to the readers. ∎

Since $\hat{y}$-variables mutate exactly in the same manner as $y$-variables,
a parallel formula holds for them:

(2.48) $\displaystyle\hat{y}_{i;t}$
$\displaystyle=\left(\prod_{j=1}^{n}\hat{y}_{j}^{c_{ji;t}}\right)\prod_{j=1}^{n}F_{j;t}(\hat{\mathbf{y}})^{b_{ji;t}}.$

Alternatively, one can derive it from (2.46) with the help of the duality
(2.39).

Thank to the separation formulas, the study of cluster patterns reduces to the
study of $C$\- and $G$-matrices, and $F$-polynomials. Moreover, the formulas
play a central role in the interplay between the cluster algebraic structure
and various subjects.

The original meaning of the separation formula in [FZ07, Cor. 6.3] is that a
general version of the formula (2.46) for $x$-variables with coefficients
naturally “separates” the additions of $x$-variables and coefficients.
Unfortunately, we cannot observe it in (2.46), because our $x$-variables are
without coefficients. Below we attach an additional meaning of “separation” to
the formulas in view of the tropicalization.

First, we briefly recall the notion of tropicalization in the context of
semifields [FZ07].

######  Definition 2.24.

A semifield $\mathbb{P}$ is a multiplicative abelian group equipped with a
binary operation $\oplus$ on $\mathbb{P}$ which is commutative, associative,
and distributive, i.e., $(a\oplus b)c=ac\oplus bc$. The operation $\oplus$ is
called the addition in $\mathbb{P}$.

Here we exclusively use the following two semifields.

######  Example 2.25.

(a). Universal semifield $\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$. Let
$\mathbf{u}=(u_{1},\dots,u_{n})$ be an $n$-tuple of commutative variables. We
say that a rational function $f(\mathbf{u})\in\mathbb{Q}(\mathbf{u})$ in $u$
has a subtraction-free expression if it is expressed as
$f(\mathbf{u})=p(\mathbf{u})/q(\mathbf{u})$, where $p(\mathbf{u})$ and
$q(\mathbf{u})$ are nonzero polynomials in $\mathbf{u}$ whose coefficients are
nonnegative. Let $\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$ be the set of all
rational functions in $\mathbf{u}$ which have subtraction-free expressions.
Then, $\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$ is a semifield by the usual
multiplication and addition in $\mathbb{Q}(\mathbf{u})$.

(b). Tropical semifield $\mathrm{Trop}(\mathbf{u})$. Let
$\mathbf{u}=(u_{1},\dots,u_{n})$ be an $n$-tuple of commutative variables. Let
$\mathrm{Trop}(\mathbf{u})$ be the set of all Laurent monomials of
$\mathbf{u}$ with coefficient 1, which is a multiplicative abelian group by
the usual multiplication. We define the addition $\oplus$ by

(2.49)
$\displaystyle\prod_{i=1}^{n}u_{i}^{a_{i}}\oplus\prod_{i=1}^{n}u_{i}^{b_{i}}:=\prod_{i=1}^{n}u_{i}^{\min(a_{i},b_{i})}.$

Then, $\mathrm{Trop}(\mathbf{u})$ becomes a semifield. The addition $\oplus$
is called the tropical sum.

Consider $\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$ and
$\mathrm{Trop}(\mathbf{u})$ with common generating variables
$\mathbf{u}=(u_{1},\dots,u_{n})$. Then, we have a unique semifield
homomorphism

(2.50)
$\displaystyle\pi_{\mathrm{trop}}:\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})\rightarrow\mathrm{Trop}(\mathbf{u})$

such that $\pi_{\mathrm{trop}}(u_{i})=u_{i}$ for any $i=1,\dots,n$. We call it
the tropicalization homomorphism. Roughly speaking, it extracts the “leading
monomial” of a rational function $f(\mathbf{u})$ in
$\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$.

######  Example 2.26.

For $\mathbf{u}=(u_{1},u_{2},u_{3})$,

(2.51)
$\displaystyle\pi_{\mathrm{trop}}\left(\frac{3u_{1}u_{2}^{2}u_{3}^{2}+2u_{1}^{2}u_{2}u_{3}}{3u_{2}^{2}+u_{1}^{2}u_{2}^{2}+u_{1}u_{2}^{3}u_{3}}\right)=\frac{u_{1}u_{2}u_{3}}{u_{2}^{2}}=u_{1}u_{2}^{-1}u_{3}.$

Now let $\mathbf{\Sigma}$ be any cluster pattern, and let
$t_{0}\in\mathbb{T}_{n}$ be a given initial vertex. Let
$\mathbf{y}_{t_{0}}=\mathbf{y}$ be the initial $y$-variables in the ambient
field $\mathcal{F}_{Y}$. Then, the semifield generated by $\mathbf{y}$ in
$\mathcal{F}_{Y}$ (by the usual multiplication and addition) is identified
with the universal semifield $\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})$. Moreover,
any $y$-variable $y_{i;t}$ belongs to $\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})$
because the mutation (2.7) is a subtraction-free operation. Let us apply the
tropicalization homomorphism
$\pi_{\mathrm{trop}}^{t_{0}}:\mathbb{Q}_{\mathrm{sf}}(\mathbf{y})\rightarrow\mathrm{Trop}(\mathbf{y})$
to $y$-variables.

######  Definition 2.27 (Tropical $y$-variable).

We call the image $\pi_{\mathrm{trop}}^{t_{0}}(y_{i;t})$ a tropical
$y$-variable with respect to the initial vertex $t_{0}$.

The following fact can be easily deduced from (2.43) by the induction on $t$
along $\mathbb{T}_{n}$ from $t_{0}$.

######  Proposition 2.28 ([FZ07, Eq. (5.5)]).

(2.52) $\displaystyle\pi_{\mathrm{trop}}^{t_{0}}(F_{i;t}(\mathbf{y}))=1.$

######  Remark 2.29.

The equality (2.52) does not mean the polynomial $F_{i;t}(\mathbf{y})$ has a
nonzero constant term. For example, $\pi_{\mathrm{trop}}(y_{1}+y_{2})=1$.

Tropical $y$-variables are naturally identified with $c$-vectors as follows:

######  Proposition 2.30 ([FZ07, Eq. (3.14)]).

We have

(2.53)
$\displaystyle\pi_{\mathrm{trop}}^{t_{0}}(y_{i;t})=y^{\mathbf{c}_{i;t}}:=\prod_{j=1}^{n}y_{j}^{c_{ji;t}}.$

###### Proof.

This follows from the separation formula (2.47) and (2.52). ∎

######  Remark 2.31.

In fact, (2.53) is the definition of $c$-vectors in [FZ07, Eq. (5.8)], where
the tropical $y$-variables are identified with principal coefficients.

Now looking back the separation formula (2.47) for $y$-variables, we see that
it naturally “separates” the tropical and the nontropical parts of
$y$-variables.

On the other hand, the tropicalization of $x$-variables with respect to the
initial $x$-variables $\mathbf{x}$ does not yield $g$-vectors; rather, it
yield the so-called $d$-vectors (denominator vectors in [FZ07]), which are the
leading exponents of the Laurent polynomial expression of $x$-variables.
However, if we regard the operation of setting all $F$-polynomials
$F_{i;t}(\mathbf{y})$ to 1 also as “a kind of tropicalization” for
$x$-variables, then the separation formula (2.46) for $x$-variables can be
viewed again as separating the “tropical” and the “nontropical” parts of
$x$-variables, where a $g$-vector $\mathbf{g}_{i;t}=(g_{ji;t})_{j=1}^{n}$ is
identified with a tropical $x$-variable defined by

(2.54) $\displaystyle x^{\mathbf{g}_{i;t}}:=\prod_{j=1}^{n}x_{j}^{g_{ji;t}}.$

In the same token, we call

(2.55)
$\displaystyle\hat{y}^{\mathbf{c}_{i;t}}:=\prod_{j=1}^{n}\hat{y}_{j}^{c_{ji;t}}$

a tropical $\hat{y}$-variable with respect to the initial vertex $t_{0}$.
Thanks to the duality (2.39), we see that the notion is compatible with a
tropical $x$-variable as

(2.56)
$\displaystyle\hat{y}^{\mathbf{c}_{i;t}}=\prod_{j=1}^{n}(x^{\mathbf{g}_{j;t}})^{b_{ji;t}}.$

Alternatively, based on the duality (2.39), we introduce a $\hat{c}$-vector
$\hat{\mathbf{c}}_{i;t}$ by

(2.57) $\displaystyle\hat{\mathbf{c}}_{i;t}$
$\displaystyle=B_{t_{0}}\mathbf{c}_{i;t}=G_{t}\mathbf{b}_{i;t},$

where $\mathbf{b}_{i;t}$ is the $i$th column vector of $B_{t}$. In other
words, it is the $i$th column vector of a $\hat{C}$-matrix defined by

(2.58) $\displaystyle\hat{C}_{t}:=B_{t_{0}}C_{t}=G_{t}B_{t}.$

Then, the tropical $\hat{y}$-variable in (2.56) is expressed also as

(2.59) $\displaystyle\hat{y}^{\mathbf{c}_{i;t}}={x}^{\hat{\mathbf{c}}_{i;t}}.$

These expressions and notions play a major role in the scattering diagram
method later.

###  2.5. Sign-coherence of $C$-matrices and Laurent positivity

In the paper [FZ07] several conjectures on the properties of cluster patterns
were proposed. Here we recall especially two (three, precisely speaking)
conjectures among them, which are the theme of this article.

Let $\mathbf{C}$ and $\mathbf{F}$ be the $C$\- and $F$-patterns of any cluster
pattern $\mathbf{\Sigma}$ with a given initial point $t_{0}\in\mathbb{T}_{n}$.

The first conjecture is the following:

######  Conjecture 2.32 ([FZ07, Conj. 5.4]).

Every $F$-polynomial $F_{i;t}(\mathbf{y})$ has constant term 1.

This seemingly innocent conjecture turned out to be difficult to prove simply
by the induction on $t$ along $\mathbb{T}_{n}$. However, it was noticed by
[FZ07] that Conjecture 2.32 is equivalent to another remarkable conjecture on
$C$-matrices and $c$-vectors.

######  Definition 2.33 (Row/Column sign-coherence).

We say that a vector $\mathbf{v}\in\mathbb{Z}^{n}$ is positive (resp.
negative) if it is nonzero vector and its all components are nonnegative
(resp. nonpositive). We say that a matrix $M$ is column sign-coherent (resp.
row sign-coherent) if each column vector (resp. row vector) of $M$ is either
positive or negative.

######  Conjecture 2.34 (Column sign-coherence of $C$-matrices [FZ07, Prop.
5.6]).

Every $C$-matrix $C_{t}$ is column sign-coherent. Equivalently, every
$c$-vector $\mathbf{c}_{i;t}$ is either positive or negative.

The equivalence between Conjectures 2.32 and 2.34 easily follows from the
mutation of $F$-polynomials (2.43) [FZ07, Prop. 5.6].

The sign-coherence conjecture was proved in the skew-symmetric case by [DWZ10,
Pla11, Nag13] with the representation/categorical methods, and in general by
[GHKK18] with the scattering diagram method.

######  Theorem 2.35 ([GHKK18, Cor. 5.5]).

Conjectures 2.32 and 2.34 hold for any $\mathbf{\Sigma}$ and $t_{0}$.

The second conjecture we consider is the following:

######  Conjecture 2.36 (Laurent positivity [FZ07, Section 3], [FZ02, Section
3]).

Every $F$-polynomial $F_{i;t}(\mathbf{y})\in\mathbb{Z}[\mathbf{y}]$ has no
negative coefficients.

Through the separation formula, this is equivalent to the earlier conjecture
in [FZ02] on the positivity of the coefficients in the Laurent polynomial
expressions for $x$-variables. This conjecture was proved for surface type by
[MSW11], for acyclic type by [KQ14], in the skew-symmetric case by [LS15,
Dav18], with various methods, and in general by [GHKK18] with the scattering
diagram method.

######  Theorem 2.37 ([GHKK18, Theorem 4.10]).

Conjecture 2.36 holds for any $\mathbf{\Sigma}$ and $t_{0}$.

From these conjectures (now theorems) many other conjectures in [FZ07] and new
results follow (e.g., [NZ12, CHL20, CL20, Nak20]). Therefore, they are
considered to be in the heart of cluster algebra theory.

As mentioned in Section 1, the purpose of the article is to digest and
reconstruct the proofs of these conjectures in [GHKK18]. Therefore, we
temporally forget Theorems 2.35 and 2.37, and keep Conjectures 2.32, 2.34, and
2.36 as conjectures until we prove them again.

######  Remark 2.38.

As for Theorem 2.35, it is Conjecture 2.32 that was proved in [GHKK18] with
some toric geometrical setting. In contrast, we prove Conjecture 2.34 without
any toric geometrical setting.

##  3\. More about $C$\- and $G$-matrices

In this section we study more about $C$\- and $G$-matrices, namely, the
tropical part of a cluster pattern. In this section we only consider a
$B$-pattern $\mathbf{B}$, and we do not refer to a cluster pattern
$\mathbf{\Sigma}$. Therefore, we regard that $C$\- and $G$-matrices are
directly associated with a $B$-pattern $\mathbf{B}$.

###  3.1. Mutations and second duality

Let $\mathbf{B}$ be any $B$-pattern, and let $\mathbf{C}^{t_{0}}=\mathbf{C}$
and $\mathbf{G}^{t_{0}}=\mathbf{G}$ are the $C$\- and $G$-patterns of
$\mathbf{B}$ with a given initial vertex $t_{0}$. In view of Proposition 2.30,
we introduce the following notion.

######  Definition 3.1 (Tropical sign).

Suppose that a $c$-vector $\mathbf{c}_{i;t}^{t_{0}}=\mathbf{c}_{i;t}$ is
either positive or negative. Then, the common sign
$\varepsilon_{i;t}^{t_{0}}=\varepsilon_{i;t}\in\\{1,-1\\}$ of all nonzero
components of $\mathbf{c}_{i;t}$ is called the tropical sign of
$\mathbf{c}_{i;t}$.

######  Example 3.2.

(a). For the initial vertex $t_{0}$, we have $C_{t_{0}}=I$. Therefore, we have

(3.1) $\displaystyle\varepsilon_{i;t_{0}}=1,\quad(i=1,\dots,n).$

(b). Let $t,t^{\prime}\in\mathbb{T}_{n}$ be vertices which are connected with
an edge labeled by $k$. Then, by (2.30), we have

(3.2) $\displaystyle\varepsilon_{k;t^{\prime}}=-\varepsilon_{k;t}.$

######  Proposition 3.3.

Suppose that the sign-coherence conjecture holds. Then, the following facts
hold.

(a). For any $t\in\mathbb{T}_{n}$, we have

(Unimodularity [NZ12, Prop. 4.2].)

(3.3) $\displaystyle|C_{t}|=|G_{t}|\in\\{1,-1\\},$

(Second duality [NZ12, Eq. (3.11)].)

(3.4) $\displaystyle D^{-1}G_{t}^{T}DC_{t}=I,$

where $D$ is a common skew-symmetrizer of $\mathbf{B}$.

(b). For any $t,t^{\prime}\in\mathbb{T}_{n}$ which are connected with an edge
labeled by $k$, we have

(Mutations [NZ12, Prop. 1.3].)

(3.5) $\displaystyle C_{t^{\prime}}$
$\displaystyle=C_{t}(J_{k}+[\varepsilon_{k;t}B_{t}]_{+}^{k\bullet}),$ (3.6)
$\displaystyle G_{t^{\prime}}$
$\displaystyle=G_{t}(J_{k}+[-\varepsilon_{k;t}B_{t}]_{+}^{\bullet k}).$

Since we need some details of the proof later, we present it below.

###### Proof of Proposition 3.3.

We first prove (b). By the column sign-coherence of $C_{t}$, the tropical sign
$\varepsilon_{k;t}$ is defined for any $k$. Then, we have

(3.7) $\displaystyle[-\varepsilon_{k;t}C_{t}]_{+}^{\bullet k}=O.$

We set $\varepsilon=\varepsilon_{k;t}$ in the $\varepsilon$-expressions of
mutations (2.35) and (2.41). The resulting relations are (3.5) and (3.6).

We prove (a) by the induction on $t$ along $\mathbb{T}_{n}$ starting from
$t_{0}$. Let us introduce the following statement:

  * $(a)_{d}$.

The claim (a) holds for any $t\in\mathbb{T}_{n}$ such that $d(t_{0},t)=d$.

First, $(a)_{0}$ holds because $C_{t_{0}}=G_{t_{0}}=I$. Next, suppose that
$(a)_{d}$ holds for some $d$. Let $t,t^{\prime}\in\mathbb{T}_{n}$ be vertices
which are connected with an edge labeled by $k$ such that $d(t_{0},t)=d$ and
$d(t_{0},t^{\prime})=d+1$. Let us temporally set the matrices in (3.5) and
(3.6) as

$\displaystyle P=J_{k}+[\varepsilon_{k;t}B_{t}]_{+}^{k\bullet},\quad
Q=J_{k}+[-\varepsilon_{k;t}B_{t}]_{+}^{\bullet k}.$

Then, by Lemma 2.10, we have

(3.8) $\displaystyle P^{2}=I,\quad|P|=|Q|=-1.$

Thus, by the induction assumption $|C_{t}|=|G_{t}|=\pm 1$, we have
$|C_{t^{\prime}}|=|G_{t^{\prime}}|=\mp 1$. Also, by (2.13), we have

(3.9) $\displaystyle DPD^{-1}=Q^{T}.$

Using the induction assumption $D^{-1}(G_{t})^{T}DC_{t}=I$, we have

(3.10) $\displaystyle
D^{-1}G_{t^{\prime}}^{T}DC_{t^{\prime}}=D^{-1}Q^{T}G_{t}^{T}DC_{t}P=P(D^{-1}G_{t}^{T}DC_{t})P=P^{2}=I.$

Therefore, $(a)_{d+1}$ holds. ∎

######  Remark 3.4.

For the later use, we record the following facts in the above proof.

  * •

To prove (b), we only use the sign-coherence of $C_{t}$ for $t$ therein.

  * •

To prove $(a)_{d}\Longrightarrow(a)_{d+1}$, we only use the sign-coherence of
$C_{t}$ such that $d(t_{0},t)=d$.

The following expressions of formulas (3.5) and (3.6) by $c$\- and $g$-vectors
are also useful.

######  Proposition 3.5.

Suppose that the sign-coherence conjecture holds. Then, we have

(3.11) $\displaystyle\mathbf{c}_{i;t^{\prime}}$
$\displaystyle=\begin{cases}\displaystyle-\mathbf{c}_{k;t}&i=k,\\\
\mathbf{c}_{i;t}+[\varepsilon_{k;t}b_{ki;t}]_{+}\mathbf{c}_{k;t}&i\neq
k,\end{cases}$ (3.12) $\displaystyle\mathbf{g}_{i;t^{\prime}}$
$\displaystyle=\begin{cases}\displaystyle-\mathbf{g}_{k;t}+\sum_{j=1}^{n}[-\varepsilon_{k;t}b_{jk;t}]_{+}\mathbf{g}_{j;t}&i=k,\\\
\mathbf{g}_{i;t}&i\neq k.\end{cases}$

We note that the matrices $C_{t}$ and $B_{t_{0}}$ in turn determine $B_{t}$.

######  Proposition 3.6 ([NZ12, Eq. (2.9)]).

Suppose that the sign-coherence conjecture holds. Then, the following equality
holds:

(3.13) $\displaystyle C_{t}^{T}(DB_{t_{0}})C_{t}=DB_{t}.$

###### Proof.

By the dualities (2.39) and (3.4), we have

(3.14) $\displaystyle
C_{t}^{T}DB_{t_{0}}C_{t}=C_{t}^{T}DG_{t}B_{t}=C_{t}^{T}DG_{t}D^{-1}DB_{t}=DB_{t}.$

∎

###  3.2. Dual mutations and third duality

Let $\mathbf{B}=\\{B_{t}\\}_{t\in\mathbb{T}_{n}}$ be any $B$-pattern, and let
$t_{0}$ be a given initial vertex. Since we now vary the initial vertex below,
we write the corresponding $C$\- and $G$-matrices as $C_{t}^{t_{0}}$ and
$G_{t}^{t_{0}}$ as in Remark 2.22.

We note that the mutation (2.12) is compatible with the matrix transpose.
Namely, if $B^{\prime}=\mu_{k}(B)$, then $B^{\prime}{}^{T}=\mu_{k}(B^{T})$.
Therefore, ${\mathbf{B}}^{T}:=\\{B_{t}^{T}\\}_{t\in\mathbb{T}_{n}}$ is also a
$B$-pattern, which is called the transpose of $\mathbf{B}$. We temporarily
write the associated $C$\- and $G$-matrices as $\tilde{C}_{t}^{t_{0}}$,
$\tilde{G}_{t}^{t_{0}}$.

We borrow the following notation from [FG19]: For any matrix $M$,
$\varepsilon_{k\bullet}(M)$ (resp. $\varepsilon_{\bullet k}(M)$) stands for
the common sign of the $k$th row (resp, $k$th column) assuming that it is
either positive or negative. For example, the tropical sign
$\varepsilon_{i;t}$ in Definition 3.1 is written as $\varepsilon_{\bullet
i}(C^{t_{0}}_{t})$.

######  Proposition 3.7.

Suppose that the sign-coherence conjecture holds. Then, the following facts
hold:

(a). For any $t_{0},t\in\mathbb{T}_{n}$, we have

(Third duality [NZ12, Eq. (1.13)].)

(3.15) $\displaystyle C_{t}^{t_{0}}$
$\displaystyle=(\tilde{G}_{t_{0}}^{t})^{T},$ (3.16) $\displaystyle
G_{t}^{t_{0}}$ $\displaystyle=(\tilde{C}_{t_{0}}^{t})^{T}.$

In particular, each $G$-matrix $G_{t}^{t_{0}}$ is row sign-coherent, and the
equality $\varepsilon_{k\bullet}(G_{t}^{t_{0}})=\varepsilon_{\bullet
k}(\tilde{C}_{t_{0}}^{t})$ holds.

(b). For any $t_{0},t_{1},t\in\mathbb{T}_{n}$ such that $t_{0}$ and $t_{1}$
are connected with an edge labeled by $k$, we have

(Dual mutations [NZ12, Prop. 1.4, Eq. (4.2)], [FG19, Prop. 3.6])

(3.17) $\displaystyle C^{t_{1}}_{t}$
$\displaystyle=(J_{k}+[-{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t},$
(3.18) $\displaystyle G^{t_{1}}_{t}$
$\displaystyle=(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{\bullet
k})G^{t_{0}}_{t}.$

Here, ${\varepsilon}_{k\bullet}(G^{t_{0}}_{t})$ is well-defined thanks to (a).

Again, we need some details of the proof later. Furthermore, the statement and
the proof in [NZ12] are a little indirect in the present context. Thus, we
give a more transparent proof, still in the spirit of [NZ12].

###### Proof.

We prove the claims by the induction on the distance $d(t_{0},t)$ in
$\mathbb{T}_{n}$, where we vary $t_{0}$ for (a) and $t$ for (b).

Let us introduce the following claims:

  * $(a)_{d}$.

The claim (a) holds for any $t_{0},t\in\mathbb{T}_{n}$ such that
$d(t_{0},t)=d$.

  * $(b)_{d}$.

The claim (b) holds for any $t_{0},t_{1},t\in\mathbb{T}_{n}$ such that
$d(t_{0},t)=d$ and $t_{0}$ and $t_{1}$ are connected with an edge labeled by
$k$.

We prove the claims in the following order,

(3.19)
$\displaystyle(a)_{0}\Longrightarrow(b)_{0}\Longrightarrow(a)_{1}\Longrightarrow(b)_{1}\Longrightarrow(a)_{2}\Longrightarrow\cdots,$

assuming all preceding claims.

First, we show that $(a)_{0}$ and $(b)_{0}$ hold. The claim $(a)_{0}$, where
$t_{0}=t$, trivially holds, because all relevant matrices are the identity
matrix $I$. Let us prove $(b)_{0}$, where $t_{0}=t$. By assumption, $t_{0}$
and $t_{1}$ are connected with an edge labeled by $k$. Then, by Proposition
3.3 (b) and (2.24), we have

(3.20) $\displaystyle C^{t_{1}}_{t_{0}}$
$\displaystyle=C^{t_{1}}_{t_{1}}(J_{n}+[B_{t_{1}}]^{k\bullet}_{+})=(J_{n}+[-B_{t_{0}}]^{k\bullet}_{+})C^{t_{0}}_{t_{0}},$
(3.21) $\displaystyle G^{t_{1}}_{t_{0}}$
$\displaystyle=G^{t_{1}}_{t_{1}}(J_{n}+[-B_{t_{1}}]^{\bullet
k}_{+})=(J_{n}+[B_{t_{0}}]^{\bullet k}_{+})G^{t_{1}}_{t_{1}}.$

Thus, $(b)_{0}$ holds.

Next, assuming the claims in (3.19) up to $(b)_{d}$, we show $(a)_{d+1}$.
Assume $d(t_{0},t)=d$ and $d(t_{1},t)=d+1$ in (3.17). Take the transpose of
(3.17). Then, we apply (3.15), which is valid thanks to $(a)_{d}$, to its
right hand side. We have

(3.22)
$\displaystyle(C^{t_{1}}_{t})^{T}=\tilde{G}^{t}_{t_{0}}(J_{k}+[-\varepsilon_{\bullet
k}(\tilde{C}^{t}_{t_{0}})B_{t_{0}}^{T}]_{+}^{\bullet k}]).$

The right hand side is $\tilde{G}^{t}_{t_{1}}$ by (3.6). Thus, we have (3.17).
The other case is similar.

Finally, assuming the claims in (3.19) up to $(a)_{d+1}$, we show $(b)_{d+1}$.
This is the nontrivial part. It is enough to concentrate on the following
situation. Let $t_{0}$, $t_{1}$, $t$, $t^{\prime}\in\mathbb{T}_{n}$ be
vertices such that $d(t_{0},t)=d$, $d(t_{0},t^{\prime})=d+1$, where $t_{1}$
and $t_{0}$ are connected with an edge labeled by $k$, while $t$ and
$t^{\prime}$ are connected with an edge labeled by $\ell$. They are depicted
as follows, where $t_{1}$ could be between $t_{0}$ and $t$.

By (3.5) and (3.17) with $(b)_{d}$, we obtain

(3.23)
$\displaystyle\begin{split}C^{t_{1}}_{t^{\prime}}&=C^{t_{1}}_{t}(J_{k}+[\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})B_{t}]_{+}^{\ell\bullet})\\\
&=(J_{k}+[-{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t}(J_{\ell}+[\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})B_{t}]_{+}^{\ell\bullet}).\end{split}$

By $(a)_{d+1}$, the sign ${\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})$ is
well-defined. Let us temporally assume the identity:

(3.24) $\displaystyle\begin{split}&\
(J_{k}+[-{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t}(J_{\ell}+[\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})B_{t}]_{+}^{\ell\bullet})\\\
=&\
(J_{k}+[-{\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t}(J_{\ell}+[\varepsilon_{\bullet\ell}(C^{t_{0}}_{t})B_{t}]_{+}^{\ell\bullet}).\end{split}$

Then, the right hand side of (3.23) reduces to
$(J_{k}+[-{\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t^{\prime}}$,
which is the desired result for (3.17) of $(b)_{d+1}$. Similarly, from the
identity

(3.25) $\displaystyle\begin{split}&\
(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{\bullet
k})G^{t_{0}}_{t}(J_{\ell}+[-\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})B_{t}]_{+}^{\bullet\ell})\\\
=&\
(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})B_{t_{0}}]_{+}^{\bullet
k})G^{t_{0}}_{t}(J_{\ell}+[-\varepsilon_{\bullet\ell}(C^{t_{0}}_{t})B_{t}]_{+}^{\bullet\ell}),\end{split}$

we obtain (3.18) of $(b)_{d+1}$. It remains to prove (3.24) and (3.25). To do
it, we need to know a precise relation between signs
${\varepsilon}_{k\bullet}(G^{t_{0}}_{t})$ and
${\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})$, and also
$\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})$ and
$\varepsilon_{\bullet\ell}(C^{t_{0}}_{t})$.

The following facts hold:

  * (i).

By (3.6), $G^{t_{0}}_{t}$ and $G^{t_{0}}_{t^{\prime}}$ differ only in the
$\ell$th column.

  * (ii).

By $(b)_{d}$, $C^{t_{0}}_{t}$ and $C^{t_{1}}_{t}$ differ only in the $k$th
row.

  * (iii).

By the duality (3.4), the $k$th row of $G^{t_{0}}_{t}$ has only a unique
component $\varepsilon$ at $\ell$th column if and only the $\ell$th column of
$C^{t_{0}}_{t}$ has a unique nonzero component $\varepsilon^{\prime}$ at $k$th
row. Furthermore, if it happens,
$\varepsilon=\varepsilon^{\prime}\in\\{1,-1\\}$, thanks to the unimodularity
(3.3).

Now we consider two cases.

Case 1. Suppose that the $k$th row of $G^{t_{0}}_{t}$ has a nonzero component
at some column other than the $\ell$th column. Then, from Facts (i)–(iii)
above and the row and column sign-coherence of $G$\- and $C$-matrices in
$(a)_{d+1}$, we conclude that

(3.26)
$\displaystyle{\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})={\varepsilon}_{k\bullet}(G^{t_{0}}_{t}),\quad\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})=\varepsilon_{\bullet\ell}(C^{t_{0}}_{t}).$

Therefore, (3.24) and (3.25) hold trivially.

Case 2. Suppose that the $k$th row of $G^{t_{0}}_{t}$ has only one nonzero
component which is at the $\ell$th column. Let us write this component as
$\varepsilon\in\\{1,-1\\}$. As mentioned in Fact (iii), $\varepsilon$
coincides with the only one nonzero component in the $\ell$th column of
$C^{t_{0}}_{t}$ which is at the $k$th row. Then, by (3.6) and $(b)_{d}$, we
have

(3.27)
$\displaystyle{\varepsilon}_{k\bullet}(G^{t_{0}}_{t^{\prime}})=-\varepsilon,\quad{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})=\varepsilon,\quad\varepsilon_{\bullet\ell}(C^{t_{1}}_{t})=-\varepsilon,\quad\varepsilon_{\bullet\ell}(C^{t_{0}}_{t})=\varepsilon.$

Let us prove (3.24), which is now written as

(3.28) $\displaystyle\begin{split}&\
(J_{k}+[-{\varepsilon}B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t}(J_{\ell}+[-{\varepsilon}B_{t}]_{+}^{\ell\bullet})\\\
=&\
(J_{k}+[{\varepsilon}B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t}(J_{\ell}+[\varepsilon
B_{t}]_{+}^{\ell\bullet}).\end{split}$

By (2.21), it is rewritten as

(3.29) $\displaystyle\begin{split}&\
(J_{k}+[{\varepsilon}B_{t_{0}}]_{+}^{k\bullet})(J_{k}+[-{\varepsilon}B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t}\\\
=&\ C^{t_{0}}_{t}(J_{\ell}+[\varepsilon
B_{t}]_{+}^{\ell\bullet})(J_{\ell}+[-{\varepsilon}B_{t}]_{+}^{\ell\bullet}).\end{split}$

After some manipulation, it reduces to

(3.30)
$\displaystyle(B_{t_{0}}C_{t}^{t_{0}})^{k\bullet}=C_{t}^{t_{0}}(B_{t})^{\ell\bullet}.$

By the assumption, we have
$(C_{t}^{t_{0}})^{\bullet\ell}=(G_{t}^{t_{0}})^{k\bullet}=\varepsilon E_{kl}$,
where $E_{kl}$ is the matrix whose entries are zero except for the
$(k,\ell)$-entry which is 1. Then, the right hand side of (3.30) can be
rewritten as

(3.31) $\displaystyle(C_{t}^{t_{0}})^{\bullet\ell}B_{t}=\varepsilon
E_{kl}B_{t}=(G_{t}^{t_{0}})^{k\bullet}B_{t}=(G_{t}^{t_{0}}B_{t})^{k\bullet}.$

Thus, (3.30) reduces to the duality (2.39). Therefore, (3.24) is proved.
Similarly, skipping details a little, (3.25) reduces to

(3.32) $\displaystyle(B_{t_{0}})^{\bullet
k}G_{t}^{t_{0}}=(G_{t}^{t_{0}}B_{t})^{\bullet\ell}.$

Then, by repeating a similar argument, again (3.32) reduces to the duality
(2.39). Therefore, (3.25) is proved. ∎

######  Remark 3.8.

For the later use we record the following facts in the above proof.

  * •

To prove $(a)_{d}$, $(b)_{d}$ $\Longrightarrow(a)_{d+1}$, we only use the
sign-coherence of $\tilde{C}_{t_{0}}^{t}$ such that $d(t_{0},t)=d$.

  * •

To prove $(b)_{d}$, $(a)_{d+1}$ $\Longrightarrow(b)_{d+1}$, we only use the
sign-coherence of $C_{t}^{t_{0}}$, $C_{t}^{t_{1}}$, $\tilde{C}_{t_{0}}^{t}$,
$\tilde{C}_{t_{0}}^{t^{\prime}}$ such that $d(t_{0},t)=d$,
$d(t_{0},t^{\prime})=d+1$, $d(t_{1},t)=d\pm 1$.

######  Remark 3.9.

Combining three dualities (2.39), (3.4), (3.15), we have various relations
among $C$\- and $G$-matrices therein. For example, by (2.39) and (3.15), we
have

(3.33) $\displaystyle
B_{t_{0}}C_{t}^{t_{0}}=(B_{t}^{T}\tilde{C}_{t_{0}}^{t})^{T},\quad
G_{t}^{t_{0}}B_{t_{0}}=(\tilde{G}_{t_{0}}^{t}B_{t}^{T})^{T}.$

The $\hat{C}$-matrices in (2.58) mutate as a “hybrid” of $C$\- and
$G$-matrices in the following sense.

######  Proposition 3.10.

Suppose that the sign-coherence conjecture holds. For any
$t_{0},t_{1},t,t^{\prime}\in\mathbb{T}_{n}$ such that $t_{0}$ and $t_{1}$, and
also, $t$ and $t^{\prime}$ are connected with an edge labeled by $k$,
respectively, we have

(3.34) $\displaystyle\hat{C}_{t^{\prime}}^{t_{0}}$
$\displaystyle=\hat{C}_{t}^{t_{0}}(J_{k}+[{\varepsilon}_{\bullet
k}(C^{t_{0}}_{t})B_{t}]_{+}^{k\bullet}),$ (3.35)
$\displaystyle\hat{C}^{t_{1}}_{t}$
$\displaystyle=(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{\bullet
k})\hat{C}^{t_{0}}_{t}.$

###### Proof.

The first formula is obtained from (2.58) by multiplying $B_{t_{0}}$ from the
left. The second one is obtained from (3.18) by multiplying $B_{t}$ from the
right. ∎

###  3.3. Principal extension of $B$-patterns

For any $B$-pattern $\mathbf{B}$ of rank $n$ and a given vertex
$t_{0}\in\mathbb{T}_{n}$, we introduce a $2n\times 2n$ skew-symmetrizable
matrix

(3.36) $\displaystyle\overline{B}_{t_{0}}=\begin{pmatrix}B_{t_{0}}&-I\\\
I&O\\\ \end{pmatrix},$

which is called the principal extension of $B_{t_{0}}$. It is important that
$\overline{B}_{t_{0}}$ is nonsingular. Choose any point
$\overline{t}_{0}\in\mathbb{T}_{2n}$. We naturally identify $\mathbb{T}_{n}$
as a subtree of $\mathbb{T}_{2n}$ by identifying $t_{0}$ with
$\overline{t}_{0}$, and also the edges with labels $k=1,\dots,n$ in
$\mathbb{T}_{n}$ and $\mathbb{T}_{2n}$. Consider a $B$-pattern
$\overline{\mathbf{B}}=\\{\overline{B}_{t}\\}_{t\in\mathbb{T}_{2n}}$ of rank
$2n$ generated by $\overline{B}_{t_{0}}$, which is called the principal
extension of $\mathbf{B}$ with the initial vertex $t_{0}$.

The following stability properties have repeatedly appeared in the literatures
in various forms. Here, we quote from [FG19].

######  Proposition 3.11 (e.g., [FG19, Theorem 4.2, Remark 4.4]).

Let $C_{t}$, $G_{t}$, $F_{i;t}(\mathbf{y})$ be $C$\- and $G$-matrices, and
$F$-polynomials of $\mathbf{B}$ with the initial point
$t_{0}\in\mathbb{T}_{n}$. Let $\overline{C}_{t}$, $\overline{G}_{t}$,
$\overline{F}_{i;t}(\overline{\mathbf{y}})$ be $C$\- and $G$-matrices, and
$F$-polynomials of $\overline{\mathbf{B}}$ with the same initial point
$t_{0}\in\mathbb{T}_{n}\subset\mathbb{T}_{2n}$, where
$\overline{\mathbf{y}}=(y_{1},\dots,y_{2n})$. Then, the following relations
hold for any $t\in\mathbb{T}_{n}\subset\mathbb{T}_{2n}$:

(3.37) $\displaystyle\overline{C}_{t}$
$\displaystyle=\begin{pmatrix}C_{t}&Z_{t}\\\ O&I\end{pmatrix},$ (3.38)
$\displaystyle\overline{G}_{t}$ $\displaystyle=\begin{pmatrix}G_{t}&O\\\
O&I\end{pmatrix},$ (3.39)
$\displaystyle\overline{F}_{i;t}(\overline{\mathbf{y}})$
$\displaystyle=\begin{cases}F_{i;t}(\mathbf{y})&i=1,\dots,n,\\\
1&i=n+1,\dots,2n,\end{cases}$

where $Z_{t}$ is a certain matrix. Moreover, $Z_{t}=O$ if the sign-coherence
of $C$-matrices for $\mathbf{B}$ hold.

These equalities can be proved by the induction on $t$ along $\mathbb{T}_{n}$
from $t_{0}$ [FG19, Theorem 4.2].

By Proposition 3.11, any information on $C_{t}$, $G_{t}$,
$F_{i;t}(\mathbf{y})$ can be extracted from the one on $\overline{C}_{t}$,
$\overline{G}_{t}$, $\overline{F}_{i;t}(\overline{\mathbf{y}})$.

###  3.4. $G$-cones and $G$-fans

From now on, we study the geometric aspect of $G$-matrices.

We recall some notions from convex geometry.

######  Definition 3.12 (Cone).

Let $N\simeq\mathbb{Z}^{n}$ be a lattice of rank $n$. Let
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}\simeq\mathbb{R}^{n}$ be the
extension to an $n$-dimensional vector space over $\mathbb{R}$. Let
$M=\mathrm{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ and
$M_{\mathbb{R}}=\mathrm{Hom}_{\mathbb{R}}(M_{\mathbb{R}},\mathbb{Z})$ be their
duals. Let $\langle\ ,\ \rangle$ be the canonical paring $N_{\mathbb{R}}\times
M_{\mathbb{R}}\rightarrow\mathbb{Z}$.

  * •

For given elements $a_{1},\dots,a_{s}\in N\subset N_{\mathbb{R}}$, the subset
of $N_{\mathbb{R}}$ defined by

(3.40) $\displaystyle\sigma(a_{1},\dots,a_{r}):=\mathbb{R}_{\geq
0}a_{1}+\cdots\mathbb{R}_{\geq 0}a_{r}$

is called a convex rational polyhedral cone, or simply, a cone generated by
$a_{1},\dots,a_{r}$. The set $\sigma(\emptyset):=\\{0\\}$ is also a cone.

  * •

A cone $\sigma$ is said to be strongly convex if
$\sigma\cap(-\sigma)=\\{0\\}$.

  * •

A cone is simplicial if it is generated by $\mathbb{Z}$-linearly independent
elements in $N$. Thus, a simplicial cone is strongly convex.

  * •

A cone is nonsingular if it is generated by a subset of a $\mathbb{Z}$-basis
in $N$. Thus, a nonsingular cone is simplicial.

  * •

The dimension of a cone $\sigma$ is the dimension of the minimal subspace of
$V$ which contains $\sigma$.

  * •

The dual cone $\sigma^{\vee}$ of a cone $\sigma$ is defined by

$\sigma^{\vee}=\\{u\in M_{\mathbb{R}}\mid\langle a,u\rangle\geq 0\ \text{for
any $a\in\sigma$})\\},$

which is a cone in $M_{\mathbb{R}}$.

  * •

A subset $\tau$ of a cone $\sigma$ is called a face of $\sigma$ if there is
some $u\in\sigma^{\vee}$ such that $\tau=\sigma\cap u^{\perp}$, where
$u^{\perp}=\\{a\in N_{\mathbb{R}}\mid\langle a,u\rangle=0\\}$. In particular,
for a simplicial cone $\sigma$ with a $\mathbb{Z}$-linearly independent
generating set $S$, any face of $\sigma$ is a cone generated by a subset of
$S$.

######  Example 3.13.

For $N=\mathbb{Z}^{2}$ and $N_{\mathbb{R}}=\mathbb{R}^{2}$,
$\sigma(\mathbf{e}_{1})$ and $\sigma(\mathbf{e}_{1},\mathbf{e}_{2})$ are
strongly convex cones, while
$\sigma(\mathbf{e}_{1},-\mathbf{e}_{1})=\mathbb{R}\mathbf{e}_{1}$ and
$\sigma(\mathbf{e}_{1},-\mathbf{e}_{1},\mathbf{e}_{2},-\mathbf{e}_{2})=\mathbb{R}^{2}$
are cones which are not strongly convex. The cone
$\sigma=\sigma(\mathbf{e}_{1},\mathbf{e}_{2})$ has four faces $\sigma$,
$\sigma(\mathbf{e}_{1})$, $\sigma(\mathbf{e}_{2})$, $\\{0\\}$, while the cone
$\sigma=\sigma(\mathbf{e}_{1},-\mathbf{e}_{1})$ has the unique face which is
$\sigma$.

######  Definition 3.14 (Fan).

  * •

A fan $\Delta$ in $N_{\mathbb{R}}\simeq\mathbb{R}^{n}$ is a nonempty (possibly
infinite) set of strongly convex cones in $N_{\mathbb{R}}$ satisfying the
following conditions:

    * (i).

If $\tau$ is a face of $\sigma\in\Delta$, then $\tau\in\Delta$.

    * (ii).

If $\sigma,\tau\in\Delta$, then $\sigma\cap\tau\in\Delta$.

  * •

A subset $|\Delta|=\bigcup_{\sigma\in\Delta}\sigma$ of $\mathbb{R}^{n}$ is
called the support of a fan $\Delta$.

  * •

A fan $\Delta$ is simplicial if all cones of $\Delta$ are simplicial.

  * •

A fan $\Delta$ is nonsingular if all cones of $\Delta$ are nonsingular. Thus,
a nonsingular fan is simplicial.

  * •

A fan $\Delta$ is complete if $|\Delta|=N_{\mathbb{R}}$.

We apply these geometric notions to $G$-matrices following [RS16, Rea14]. In
the rest of this section we temporarily assume the sign-coherence of
$C$-matrices and study some basic properties of $G$-cones.

Let $\mathbf{B}$ be any $B$-pattern, and let $\mathbf{G}^{t_{0}}=\mathbf{G}$
be the $G$-pattern of $\mathbf{B}$ with a given initial point $t_{0}$.

######  Definition 3.15 ($G$-cone).

For each $G$-matrix $G_{t}$, the cone spanned by its $g$-vectors in
$\mathbb{R}^{n}$

(3.41)
$\displaystyle\sigma(G_{t}):=\sigma(\mathbf{g}_{1;t},\dots,\mathbf{g}_{n;t})$

is called a $G$-cone. The interior of $\sigma(G_{t})$ is denoted by
$\sigma^{\circ}(G_{t})$. Each face of $\sigma(G_{t})$ spanned by
$\mathbf{g}_{j;t}$’s excluding the $i$th $g$-vector $\mathbf{g}_{i;t}$ is
denoted by $\sigma_{i}(G_{t})$ for $i=1,\dots,n$.

By the unimodularity (3.3), every $G$-cone $\sigma(G_{t})$ is a nonsingular
cone, and its faces $\sigma_{i}(G_{t})$ ($i=1,\dots,n$) are cones of
codimension one.

From now on, let us especially take a common skew-symmetrizer $D$ of the
$B$-pattern $\mathbf{B}$ in the following form:

(3.42) $\displaystyle D=\mathrm{diag}({d}_{1}^{-1}\dots,{d}_{n}^{-1}),$

where $d_{1}$, …, $d_{n}$ are positive integers. (This seemingly awkward
choice is suitable to the formulation of scattering diagrams in [GHKK18] we
use later. See Example 6.6.) Accordingly, we introduce the following
nonstandard inner product $(\ ,\ )_{D}$ in $\mathbb{R}^{n}$:

(3.43)
$\displaystyle(\mathbf{u},\mathbf{v})_{D}=\mathbf{u}^{T}D\mathbf{v}\quad(\mathbf{u},\mathbf{v}\in\mathbb{R}^{n}).$

Below, for a given $B$-pattern $\mathbf{B}$, we always assume this inner
product $(\cdot,\cdot)_{D}$ in $\mathbb{R}^{n}$. For
$\mathbf{v}\neq\mathbf{0}\in\mathbb{R}^{n}$, $\mathbf{v}^{\perp}$ is a
hyperplane defined by

(3.44)
$\displaystyle\mathbf{v}^{\perp}:=\\{\mathbf{u}\in\mathbb{R}^{n}\mid(\mathbf{u},\mathbf{v})_{D}=0\\}.$

We say that a normal vector $\mathbf{n}$ of a face $\sigma_{i}(G_{t})$ of
$\sigma(G_{t})$ is inward for the cone $\sigma(G_{t})$ if
$(\mathbf{n},\mathbf{g}_{i;t})_{D}>0$.

In this geometrical setting, the duality (3.4) can be rephrased in the
following manner.

######  Proposition 3.16.

Suppose that the sign-coherence conjecture holds. Then, the following facts
hold:

(a). The $g$-vectors $\mathbf{g}_{1;t}$, $\dots$, $\mathbf{g}_{n;t}$ and the
vectors ${d}_{1}\mathbf{c}_{1;t}$, $\dots$, ${d}_{n}\mathbf{c}_{n;t}$ are dual
to each other with respect to the inner product $(\cdot,\cdot)_{D}$.

(b). In particular, each $c$-vector $\mathbf{c}_{i;t}$ is a normal vector of
the face $\sigma_{i}(G_{t})$ of $\sigma(G_{t})$ with respect to the inner
product $(\cdot,\cdot)_{D}$. Moreover, it is inward for the cone
$\sigma(G_{t})$.

Let $\Delta(\mathbf{G}^{t_{0}})=\Delta(\mathbf{G})$ be the set of all faces of
all $G$-cones $\sigma(G_{t})$ ($t\in\mathbb{T}_{n}$) in the $G$-pattern
$\mathbf{G}^{t_{0}}$. Let $t,t^{\prime}\in\mathbb{T}_{n}$ be vertices which
are connected with an edge labeled by $k$. Then, by (3.12), two cones
$\sigma(G_{t})$ and $\sigma(G_{t^{\prime}})$ intersect each other in their
common face $\sigma_{k}(G_{t})=\sigma_{k}(G{{}_{t^{\prime}}})$ with
codimension one. However, for a cone $\sigma(G_{t^{\prime\prime}})$ obtained
from $\sigma(G_{t})$ after several mutations, it is not clear at all that
$\sigma(G_{t^{\prime\prime}})$ intersects $\sigma(G_{t})$ only in their common
face so that the set $\Delta(\mathbf{G})$ forms a fan. It was conjectured by
[RS16, Conj. 3.14] that this always happens. Then, it was shown to be true by
[Rea14] under the sign-coherence conjecture.

######  Theorem 3.17 ([Rea14, Theorem 8.7]).

Suppose that the sign-coherence conjecture holds. Then, For any $B$-pattern
$\mathbf{B}$ and a given initial vertex $t_{0}$, the set of cones
$\Delta(\mathbf{G}^{t_{0}})$ is a fan in $\mathbb{R}^{n}$.

In [Rea14] this was proved in a more general setting of mutation fans therein.
We provide a self-contained proof of Theorem 3.17 in Section 3.7 without
referring to mutation fans, because the tools and techniques we use therein
are also relevant to our main purpose.

The resulting fan $\Delta(\mathbf{G}^{t_{0}})$, which is nonsingular by (3.3),
is called the $g$-vector fan in [RS16]. To simplify a little, we call it the
$G$-fan of $\mathbf{B}$ (or $\mathbf{\Sigma}$) with the initial vertex
$t_{0}$.

Let $\\#\mathbf{G}^{t_{0}}$ be the number of distinct $G$-matrices in
$\mathbf{G}^{t_{0}}$, which can be finite or infinite.

######  Proposition 3.18.

Suppose that the sign-coherence conjecture holds. Then, the $G$-fan
$\Delta(\mathbf{G}^{t_{0}})$ is complete if and only if
$\\#\mathbf{G}^{t_{0}}$ is finite.

###### Proof.

This is true by the following topological reason. For each subset $T$ of
$\mathbb{T}_{n}$, we attach a subset $\Delta_{T}$ of
$\Delta(\mathbf{G}^{t_{0}})$ consisting of the faces of all $\sigma(G_{t})$
with $t\in T$. Let $|\Delta_{T}|^{c}$ be the complement of the support
$|\Delta_{T}|$ in $\mathbb{R}^{n}$. Let $S^{n-1}$ be the unit sphere in
$\mathbb{R}^{n}$. Let $D_{T}=|\Delta_{T}|^{c}\cap S^{n-1}$. For
$T=\\{t_{0}\\}$, $D_{T}$ is topologically a $(n-1)$-dimensional open disk. We
consider an increasing sequence of subsets of $\mathbb{T}_{n}$,
$T_{0}=\\{t_{0}\\}\subset T_{1}\subset T_{2}\subset\cdots$. Correspondingly,
we have a decreasing sequence of subsets of $S^{n-1}$, $D_{T_{0}}\supset
D_{T_{1}}\supset D_{T_{2}}\supset\cdots$. Then, $D_{T_{s}}$ is a disjoint
union of a finite number of open disks, such that at each process of
increasing $T_{s}$, one of the followings occurs (by adding a $G$-cone):

  * •

An open disk shrinks.

  * •

An open disk splits into two open disks.

  * •

An open disk disappears.

If $\\#\mathbf{G}^{t_{0}}$ is finite, the process should terminate at some
$T_{s}$. This occurs only when $D_{T_{s}}=\emptyset$. Thus,
$\Delta(\mathbf{G}^{t_{0}})$ is complete. If $\\#\mathbf{G}^{t_{0}}$ is
infinite, the process never terminates. Then, when $T_{s}$ converges to
$\mathbb{T}_{n}$, $D_{T_{s}}$ converges to a (possibly infinite) disjoint
union of open disks and points. ∎

######  Remark 3.19.

We say that a cluster pattern $\mathbf{\Sigma}$ is of finite type if there are
only finitely many distinct seeds in $\mathbf{\Sigma}$. It is known, for
example, by [Nak20, Theorem 5.2], that a cluster pattern $\mathbf{\Sigma}$ is
of finite type if and only if $\\#\mathbf{G}^{t_{0}}$ is finite for
$\mathbf{G}^{t_{0}}$ of $\mathbf{\Sigma}$. Then, combining it with Proposition
3.18, we have that a cluster pattern $\mathbf{\Sigma}$ is of finite type if
and only if the $G$-fan $\Delta(\mathbf{G}^{t_{0}})$ is complete for
$\mathbf{G}^{t_{0}}$ of $\mathbf{\Sigma}$. Also, we note that the proof of
[Nak20, Theorem 5.2] is based on the sign-coherence of $C$-matrices and the
Laurent positivity.

###  3.5. Rank 2 examples of $G$-fans

In the rank 2 case, one can confirm Theorem 3.17 and Proposition 3.18 by
explicitly calculating $G$-matrices.

Let us arrange a $B$-pattern of rank 2 in the following way:

(3.45) $\displaystyle\cdots\ \mathop{\leftrightarrow}^{\mu_{2}}\ B_{t_{-2}}\
\mathop{\leftrightarrow}^{\mu_{1}}\ B_{t_{-1}}\
\mathop{\leftrightarrow}^{\mu_{2}}\ B_{t_{0}}\
\mathop{\leftrightarrow}^{\mu_{1}}\ B_{t_{1}}\
\mathop{\leftrightarrow}^{\mu_{2}}\ B_{t_{2}}\
\mathop{\leftrightarrow}^{\mu_{1}}\ \cdots.$

We take $B_{t_{0}}$ as the initial exchange matrix. Let us present the
associated $G$-fan $\Delta(\mathbf{G})$ in $\mathbb{R}^{2}$ explicitly. Below
the type of a $B$-pattern refers to the type of the Cartan matrix
$A(B)=(a_{ij})$ associated with $B=B_{t_{0}}$ in the convention of [Kac90],
where

(3.46) $\displaystyle a_{ij}=\begin{cases}2&i=j,\\\ -|b_{ij}|&i\neq
j.\end{cases}$

(I). Finite type. The $G$-fan $\Delta(\mathbf{G})$ is complete as stated in
Proposition 3.18.

(a). Type $A_{2}$: Let

(3.47) $\displaystyle B_{t_{0}}=\begin{pmatrix}0&-1\\\ 1&0\\\ \end{pmatrix}.$

Along the mutation sequence $\mu_{1}$, $\mu_{2}$, $\cdots$, the $G$-matrices
are explicitly calculated as

(3.48) $\displaystyle\begin{split}G_{t_{0}}&=\begin{pmatrix}1&0\\\ 0&1\\\
\end{pmatrix},\quad G_{t_{1}}=\begin{pmatrix}-1&0\\\ 0&1\\\
\end{pmatrix},\quad G_{t_{2}}=\begin{pmatrix}-1&0\\\ 0&-1\\\ \end{pmatrix},\\\
G_{t_{3}}&=\begin{pmatrix}1&0\\\ -1&-1\\\ \end{pmatrix},\quad
G_{t_{4}}=\begin{pmatrix}1&1\\\ -1&0\\\ \end{pmatrix},\quad
G_{t_{5}}=\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}.\end{split}$

Both $G_{t_{5}}$ and $G_{t_{0}}$ define the same cone, corresponding to the
celebrated pentagon periodicity. The $G$-fan $\Delta(\mathbf{G})$ is depicted
in Figure 1 (a).

(b). Type $B_{2}$: Let

(3.49) $\displaystyle B_{t_{0}}=\begin{pmatrix}0&-1\\\ 2&0\\\ \end{pmatrix}.$

We start from the initial $G$-matrix $G_{t_{0}}=I$. Along the mutation
sequence $\mu_{1}$, $\mu_{2}$, $\cdots$, the following $g$-vectors appear in
this order, showing periodicity:

(3.50) $\displaystyle\begin{pmatrix}-1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ -1\\\ \end{pmatrix},\begin{pmatrix}1\\\
-2\\\ \end{pmatrix},\begin{pmatrix}1\\\ -1\\\
\end{pmatrix},\begin{pmatrix}1\\\ 0\\\ \end{pmatrix},\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}.$

The $G$-fan $\Delta(\mathbf{G})$ is depicted in Figure 1 (b).

(c). Type $G_{2}$: Let

(3.51) $\displaystyle B_{t_{0}}=\begin{pmatrix}0&-1\\\ 3&0\\\ \end{pmatrix}.$

We start from the initial $G$-matrix $G_{t_{0}}=I$. Along the mutation
sequence $\mu_{1}$, $\mu_{2}$, $\cdots$, the following $g$-vectors appear in
this order, showing periodicity:

(3.52) $\displaystyle\begin{pmatrix}-1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ -1\\\ \end{pmatrix},\begin{pmatrix}1\\\
-3\\\ \end{pmatrix},\begin{pmatrix}1\\\ -2\\\
\end{pmatrix},\begin{pmatrix}2\\\ -3\\\ \end{pmatrix},\begin{pmatrix}1\\\
-1\\\ \end{pmatrix},\begin{pmatrix}1\\\ 0\\\ \end{pmatrix},\begin{pmatrix}0\\\
1\\\ \end{pmatrix}.$

The $G$-fan $\Delta(\mathbf{G})$ is depicted in Figure 1 (c).

(a) $A_{2}$$G_{t_{0}}$$G_{t_{1}}$$G_{t_{2}}$$G_{t_{3}}$$G_{t_{4}}$ (b) $B_{2}$
(c) $G_{2}$

(d) $A_{1}^{(1)}$ (e) $A_{2}^{(2)}$ (f) Others

Figure 1. $G$-fans of rank 2.

(II). Infinite type. The $G$-fan $\Delta(\mathbf{G})$ is not complete.

(d). Type $A_{1}^{(1)}$: Let

(3.53) $\displaystyle B_{t_{0}}=\begin{pmatrix}0&-2\\\ 2&0\\\ \end{pmatrix}.$

We start from the initial $G$-matrix $G_{t_{0}}=I$. Along the mutation
sequence $\mu_{1}$, $\mu_{2}$, $\cdots$, the following $g$-vectors appear in
this order:

(3.54) $\displaystyle\begin{pmatrix}-1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ -1\\\ \end{pmatrix},\begin{pmatrix}1\\\
-2\\\ \end{pmatrix},\begin{pmatrix}2\\\ -3\\\
\end{pmatrix},\begin{pmatrix}3\\\ -4\\\ \end{pmatrix},\cdots.$

On the other hand, along the mutation sequence $\mu_{2}$, $\mu_{1}$, $\cdots$,
the following $g$-vectors appear in this order:

(3.55) $\displaystyle\begin{pmatrix}2\\\ -1\\\
\end{pmatrix},\begin{pmatrix}3\\\ -2\\\ \end{pmatrix},\begin{pmatrix}4\\\
-3\\\ \end{pmatrix},\begin{pmatrix}5\\\ -4\\\ \end{pmatrix},\cdots.$

The $G$-fan $\Delta(\mathbf{G})$ is depicted in Figure 1 (d). It covers the
region excluding the half line $\mathbb{R}_{+}(1,-1)$.

(e). Type $A_{2}^{(2)}$: Let

(3.56) $\displaystyle B_{t_{0}}=\begin{pmatrix}0&-1\\\ 4&0\\\ \end{pmatrix}.$

We start from the initial $G$-matrix $G_{t_{0}}=I$. Along the mutation
sequence $\mu_{1}$, $\mu_{2}$, $\cdots$, the following $g$-vectors appear in
this order:

(3.57) $\displaystyle\begin{pmatrix}-1\\\ 0\\\
\end{pmatrix},\begin{pmatrix}0\\\ -1\\\ \end{pmatrix},\begin{pmatrix}1\\\
-4\\\ \end{pmatrix},\begin{pmatrix}1\\\ -3\\\
\end{pmatrix},\begin{pmatrix}3\\\ -8\\\ \end{pmatrix},\begin{pmatrix}2\\\
-5\\\ \end{pmatrix},\begin{pmatrix}5\\\ -12\\\
\end{pmatrix},\begin{pmatrix}3\\\ -7\\\ \end{pmatrix},\cdots.$

On the other hand, along the mutation sequence $\mu_{2}$, $\mu_{1}$, $\cdots$,
the following $g$-vectors appear in this order:

(3.58) $\displaystyle\begin{pmatrix}1\\\ -1\\\
\end{pmatrix},\begin{pmatrix}3\\\ -4\\\ \end{pmatrix},\begin{pmatrix}2\\\
-3\\\ \end{pmatrix},\begin{pmatrix}5\\\ -8\\\
\end{pmatrix},\begin{pmatrix}3\\\ -5\\\ \end{pmatrix},\begin{pmatrix}7\\\
-12\\\ \end{pmatrix},\cdots.$

The $G$-fan $\Delta(\mathbf{G})$ is depicted in Figure 1 (e). It covers the
region excluding the half line $\mathbb{R}_{+}(1,-2)$.

(f). Non-affine type: Let

(3.59) $\displaystyle B_{t_{0}}=\begin{pmatrix}0&-c\\\ b&0\\\
\end{pmatrix},\quad(b,c>0,\ bc\geq 5).$

We start from the initial $G$-matrix $G_{t_{0}}=I$. Along the mutation
sequence $\mu_{1}$, $\mu_{2}$, $\cdots$, the following $g$-vectors appear in
this order:

(3.60) $\displaystyle\mathbf{g}_{1}=\begin{pmatrix}-1\\\ 0\\\ \end{pmatrix},\
\mathbf{g}_{2}=\begin{pmatrix}0\\\ -1\\\ \end{pmatrix},\ \mathbf{g}_{3},\
\mathbf{g}_{4},\cdots,$

where $\mathbf{g}_{i}$’s obey the recursion,

(3.61)
$\displaystyle\mathbf{g}_{i+2}=\begin{cases}-\mathbf{g}_{i}+b\mathbf{g}_{i+1}&\text{$i$
odd},\\\ -\mathbf{g}_{i}+c\mathbf{g}_{i+1}&\text{$i$ even}.\end{cases}$

On the other hand, along the mutation sequence $\mu_{2}$, $\mu_{1}$, $\cdots$
from the initial $G$-matrix, the following $g$-vectors appear in this order:

(3.62) $\displaystyle\mathbf{g}^{\prime}_{1}=\begin{pmatrix}c\\\ -1\\\
\end{pmatrix},\ \mathbf{g}^{\prime}_{2}=\begin{pmatrix}bc-1\\\ -b\\\
\end{pmatrix},\ \mathbf{g}^{\prime}_{3},\ \mathbf{g}^{\prime}_{4},\cdots,$

where $\mathbf{g}^{\prime}_{i}$’s obey the recursion,

(3.63)
$\displaystyle\mathbf{g}^{\prime}_{i+2}=\begin{cases}-\mathbf{g}^{\prime}_{i}+c\mathbf{g}^{\prime}_{i+1}&\text{$i$
odd},\\\ -\mathbf{g}^{\prime}_{i}+b\mathbf{g}^{\prime}_{i+1}&\text{$i$
even}.\end{cases}$

It is known [Rea14, Eqs. (9.11), (9.12)] that, in the limit
$i\rightarrow\infty$, the rays $\sigma(\mathbf{g}_{i})$ and
$\sigma(\mathbf{g}^{\prime}_{i})$ monotonically converge to
$\sigma(\mathbf{v})$ and $\sigma(\mathbf{v}^{\prime})$, respectively, where

(3.64) $\displaystyle\mathbf{v}=\begin{pmatrix}bc-\sqrt{bc(bc-4)}\\\ -2b\\\
\end{pmatrix},\quad\mathbf{v}^{\prime}=\begin{pmatrix}bc+\sqrt{bc(bc-4)}\\\
-2b\\\ \end{pmatrix}.$

The $G$-fan $\Delta(\mathbf{G})$ is depicted in Figure 1 (f). It covers the
region excluding the region
$\mathbb{R}_{+}\mathbf{v}+\mathbb{R}_{+}\mathbf{v}^{\prime}$ (the hatched
region in the figure).

###  3.6. Piecewise linear isomorphisms between $G$-fans

In this subsection we study a canonical bijection between the sets
$\Delta(\mathbf{G}^{t_{0}})$ and $\Delta(\mathbf{G}^{t_{1}})$ for any
$t_{0},t_{1}\in\mathbb{T}_{n}$. The idea comes from [GHKK18], but this is
independent of their results. Also, the result of this and the next
subsections has considerable overlap with the results in [Rea14, Rea20b], in
which $G$-fans were studied as a part of mutation fans.

Let $\mathbf{B}$ be any $B$ pattern. Let us start with the following
consequence of the sign-coherence.

######  Proposition 3.20.

Suppose that the sign-coherence conjecture holds. Then, for any
$t_{0},t\in\mathbb{T}_{n}$, the cone $\sigma(G_{t}^{t_{0}})$ intersects the
hyperplanes $\mathbf{e}_{i}^{\perp}$ $(i=1,\dots,n)$ only in the boundary of
$\sigma(G_{t}^{t_{0}})$.

###### Proof.

Suppose that a cone $\sigma(G_{t}^{t_{0}})$ intersects with
$\mathbf{e}_{i}^{\perp}$ for some $i$ in its interior
$\sigma^{\circ}(G_{t}^{t_{0}})$. This means that the $i$th row of the
$G$-matrix $G_{t}^{t_{0}}$ has both positive and negative components. This
contradicts the row sign-coherence of $G_{t}^{t_{0}}$ in Proposition 3.7. ∎

Let $t_{0},t_{1}\in\mathbb{T}_{n}$ be vertices which are connected with an
edge labeled by $k$. Recall the formula (3.18), which is valid under the
assumption of the sign-coherence of $C$-matrices. It is written in the vector
form as follows.

(3.65)
$\displaystyle\mathbf{g}_{i;t}^{t_{1}}=(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{\bullet
k})\mathbf{g}^{t_{0}}_{i;t},\quad(i=1,\dots,n).$

Let us introduce the half spaces of $\mathbb{R}^{n}$ as

(3.66) $\displaystyle\mathbb{R}^{n}_{k,+}=\\{\mathbf{v}\in{\mathbb{R}}^{n}\mid
v_{k}\geq 0\\},\quad\mathbb{R}^{n}_{k,-}=\\{\mathbf{v}\in{\mathbb{R}}^{n}\mid
v_{k}\leq 0\\}.$

The matrix
$(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{\bullet k})$
in (3.65) is regarded as a piecewise linear map

(3.67)
$\displaystyle\begin{matrix}\varphi_{t_{0}}^{t_{1}}:&\mathbb{R}^{n}&\rightarrow&\mathbb{R}^{n}\\\
&\mathbf{v}&\mapsto&\begin{cases}(J_{k}+[B_{t_{0}}]_{+}^{\bullet
k})\mathbf{v}&\mathbf{v}\in\mathbb{R}^{n}_{k,+},\\\
(J_{k}+[-B_{t_{0}}]_{+}^{\bullet
k})\mathbf{v}&\mathbf{v}\in\mathbb{R}^{n}_{k,-},\end{cases}\end{matrix}$

where we note that $\varphi_{t_{0}}^{t_{1}}(\mathbf{v})=\mathbf{v}$ if
$v_{k}=0$. We have, by (3.65) and (3.67), for any $t\in\mathbb{T}_{n}$ and
$i=1,\dots,k$,

(3.68)
$\displaystyle\varphi_{t_{0}}^{t_{1}}(\mathbf{g}^{t_{0}}_{i;t})=\mathbf{g}_{i;t}^{t_{1}}.$

The map $\varphi_{t_{1}}^{t_{0}}$ is also defined in the same way by replacing
$B_{t_{0}}$ with $B_{t_{1}}$.

######  Proposition 3.21.

The following relation holds:

(3.69) $\displaystyle\varphi_{t_{1}}^{t_{0}}\circ\varphi_{t_{0}}^{t_{1}}$
$\displaystyle=\varphi_{t_{0}}^{t_{1}}\circ\varphi_{t_{1}}^{t_{0}}=\mathrm{id}.$

###### Proof.

Note that, for $\mathbf{v}^{\prime}=\varphi_{t_{0}}^{t_{1}}(\mathbf{v})$, we
have $v^{\prime}_{k}=-v_{k}$ due to $b_{kk}^{t_{0}}=0$. We also recall that
$(B_{t_{1}})^{\bullet k}=-(B_{t_{0}})^{\bullet k}$. Then, for
$\mathbf{v}\in\mathbb{R}^{n}_{k,+}$,

(3.70)
$\displaystyle\begin{split}\varphi_{t_{1}}^{t_{0}}\circ\varphi_{t_{0}}^{t_{1}}(\mathbf{v})&=(J_{k}+[-B_{t_{1}}]_{+}^{\bullet
k})(J_{k}+[B_{t_{0}}]_{+}^{\bullet k})\mathbf{v}\\\
&=(J_{k}+[B_{t_{0}}]_{+}^{\bullet k})^{2}\mathbf{v}=\mathbf{v}\end{split}$

thanks to Lemma 2.10. The other case is similar.

∎

######  Proposition 3.22 (cf. [Rea14, Prop. 8.13]).

Suppose that the sign-coherence conjecture holds. Then, we have

(3.71) $\displaystyle\varphi_{t_{0}}^{t_{1}}(\sigma(G_{t}^{t_{0}}))$
$\displaystyle={\sigma}(G_{t}^{t_{1}}),\quad(t\in\mathbb{T}_{n}).$

Therefore, the correspondence
$\varphi_{t_{0}}^{t_{1}}:\mathbf{g}^{t_{0}}_{i;t}\mapsto\mathbf{g}^{t_{1}}_{i;t}$
gives a bijection between ${\Delta}(\mathbf{G}^{t_{0}})$ and
${\Delta}(\mathbf{G}^{t_{1}})$ which preserves the intersection and the
inclusion of cones.

###### Proof.

We only need to prove (3.71). By Proposition 3.20, each cone
$\sigma(G_{t}^{t_{0}})$ belongs to either $\mathbb{R}^{n}_{k,+}$ or
$\mathbb{R}^{n}_{k,-}$, so that it is linearly transformed under
$\varphi_{t_{0}}^{t_{1}}$. Therefore, by (3.68), its image is
$\sigma(G_{t}^{t_{1}})$. ∎

######  Remark 3.23.

For the later use, we record the following fact in the above proof.

  * •

To prove (3.71), we only use the formula (3.65) for $t_{0}$, $t_{1}$ and $t$
therein, and the row sign-coherence of $G_{t}^{t_{0}}$ therein.

######  Example 3.24.

Let us clarify Proposition 3.22 explicitly for type $A_{2}$ based on the
convention in Section 3.5. The fans ${\Delta}(\mathbf{G}^{t_{-1}})$,
${\Delta}(\mathbf{G}^{t_{0}})$, ${\Delta}(\mathbf{G}^{t_{1}})$ are depicted in
Figure 2. Also, the piecewise linear maps $\varphi_{t_{0}}^{t_{1}}$ and
$\varphi_{t_{0}}^{t_{-1}}$ are given by combining the following linear maps:

(3.72) $\displaystyle J_{1}+[B_{t_{0}}]_{+}^{\bullet 1}=\begin{pmatrix}-1&0\\\
1&1\end{pmatrix},\quad(J_{1}+[-B_{t_{0}}]_{+}^{\bullet
1})=\begin{pmatrix}-1&0\\\ 0&1\end{pmatrix},$ (3.73) $\displaystyle
J_{2}+[B_{t_{0}}]_{+}^{\bullet 2}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\quad(J_{2}+[-B_{t_{0}}]_{+}^{\bullet
2})=\begin{pmatrix}1&1\\\ 0&-1\end{pmatrix}.$

Then, the correspondence in Proposition 3.22 can be easily confirmed in Figure
2.

$\varphi_{t_{0}}^{t_{-1}}$$\longleftarrow$${\Delta}(\mathbf{G}^{t_{-1}})$$G_{t_{4}}^{t_{-1}}=G_{t_{-1}}^{t_{-1}}$$G_{t_{0}}^{t_{-1}}$$G_{t_{1}}^{t_{-1}}$$G_{t_{3}}^{t_{-1}}$$G_{t_{2}}^{t_{-1}}$
$\varphi_{t_{0}}^{t_{1}}$$\longrightarrow$${\Delta}(\mathbf{G}^{t_{0}})$$G_{t_{0}}^{t_{0}}$$G_{t_{1}}^{t_{0}}$$G_{t_{2}}^{t_{0}}$$G_{t_{3}}^{t_{0}}$$G_{t_{4}}^{t_{0}}$
${\Delta}(\mathbf{G}^{t_{1}})$$G_{t_{1}}^{t_{1}}$$G_{t_{2}}^{t_{1}}$$G_{t_{3}}^{t_{1}}$$G_{t_{0}}^{t_{1}}$$G_{t_{4}}^{t_{1}}$
Figure 2. Bijections between $G$-fans for type $A_{2}$.

By applying the above isomorphism repeatedly along the tree $\mathbb{T}_{n}$,
we obtain the following important fact.

######  Proposition 3.25.

Suppose that the sign-coherence conjecture holds. Then, for any vertices
$t_{0},t_{1}\in\mathbb{T}_{n}$, which are not necessarily connected with an
edge, the correspondence
$\mathbf{g}^{t_{0}}_{i;t}\mapsto\mathbf{g}^{t_{1}}_{i;t}$ gives a bijection
between ${\Delta}(\mathbf{G}^{t_{0}})$ and ${\Delta}(\mathbf{G}^{t_{1}})$
which preserves the intersection and the inclusion of cones.

For the later use, we also describe the relation between
$\varphi_{t_{0}}^{t_{1}}$ here and a certain piecewise linear isomorphism in
[GHKK18].

Again, let $t_{0},t_{1}\in\mathbb{T}_{n}$ be vertices which are connected by
an edge labeled by $k$. First, for $B_{t_{0}}$ and $k$, we introduce a basis
$\mathbf{a}_{1},\dots,\mathbf{a}_{n}$ of $\mathbb{R}^{n}$ as

(3.74)
$\displaystyle\mathbf{a}_{i}=\begin{cases}\displaystyle-\mathbf{e}_{i}+\sum_{j=1}^{n}[-b_{jk;t_{0}}]_{+}\mathbf{e}_{j}&i=k,\\\
\mathbf{e}_{i}&i\neq k.\end{cases}$

In fact, by specializing the formula in (3.12) with $t=t_{0}$, and applying
the fact $\varepsilon_{k;t_{0}}=1$ therein, we see that

(3.75) $\displaystyle\mathbf{a}_{i}=\mathbf{g}_{i;t_{1}}^{t_{0}}.$

The change of a basis of $\mathbb{R}^{n}$ from $\mathbf{e}_{1}$, …,
$\mathbf{e}_{n}$ to $\mathbf{a}_{1}$, …, $\mathbf{a}_{n}$ induces the
following linear isomorphism

(3.76)
$\displaystyle\begin{matrix}\eta_{t_{0}}^{t_{1}}:&\mathbb{R}^{n}&\rightarrow&\mathbb{R}^{n}\\\
&\mathbf{v}&\mapsto&(J_{k}+[-B_{t_{0}}]_{+}^{\bullet
k})\mathbf{v}\end{matrix}$

such that $\eta_{t_{0}}^{t_{1}}(\mathbf{a}_{i})=\mathbf{e}_{i}$.

Next, for the same $B_{t_{0}}$ and $k$, we define a piecewise linear map
$T_{k;t_{0}}$ on $\mathbb{R}^{N}$ following [GHKK18] as

(3.77)
$\displaystyle\begin{matrix}T_{k;t_{0}}:&\mathbb{R}^{n}&\rightarrow&\mathbb{R}^{n}\\\
&\mathbf{v}&\mapsto&\begin{cases}(I+(B_{t_{0}})^{\bullet
k})\mathbf{v}&\mathbf{v}\in\mathbb{R}^{n}_{k,+},\\\
\mathbf{v}&\mathbf{v}\in\mathbb{R}^{n}_{k,-}.\end{cases}\end{matrix}$

The following fact appeared in [Mul16, Rea20b].

######  Proposition 3.26 ([Mul16, Lemma 5.2.1], [Rea20b, Cor. 4.4]).

The above maps are related as follows:

(3.78) $\displaystyle\varphi_{t_{0}}^{t_{1}}=\eta_{t_{0}}^{t_{1}}\circ
T_{k;t_{0}}.$

###### Proof.

By the definitions of these maps,

(3.79) $\displaystyle\eta_{t_{0}}^{t_{1}}\circ T_{k;t_{0}}(\mathbf{v})$
$\displaystyle=\begin{cases}(J_{k}+[-B_{t_{0}}]_{+}^{\bullet
k})(I+(B_{t_{0}})^{\bullet k})\mathbf{v}&\mathbf{v}\in\mathbb{R}^{n}_{k,+},\\\
(J_{k}+[-B_{t_{0}}]_{+}^{\bullet
k})\mathbf{v}&\mathbf{v}\in\mathbb{R}^{n}_{k,-}.\end{cases}$

Thanks to (2.5), we have

$\displaystyle(J_{k}+[-B_{t_{0}}]_{+}^{\bullet k})(I+(B_{t_{0}})^{\bullet
k})=J_{k}+(B_{t_{0}})^{\bullet k}+[-B_{t_{0}}]_{+}^{\bullet
k}=J_{k}+[B_{t_{0}}]_{+}^{\bullet k}.$

∎

###  3.7. Proof of Theorem 3.17

Let us prove Theorem 3.17 under the assumption of the sign-coherence of
$C$-matrices.

We recall that the condition for $\Delta(\mathbf{G}^{t_{0}})$ to form a fan is
stated as follows:

######  Condition 3.27.

For any $t,t^{\prime}\in\mathbb{T}_{n}$, the cones $\sigma(G_{t}^{t_{0}})$ and
$\sigma(G_{t^{\prime}}^{t_{0}})$ intersect in their common face.

Thanks to Proposition 3.22, the condition for $\Delta(\mathbf{G}^{t_{0}})$ to
form a fan is rephrased as follows, where we used the bijection between
$\Delta(\mathbf{G}^{t_{0}})$ and $\Delta(\mathbf{G}^{t})$ therein.

######  Condition 3.28.

For any $t,t^{\prime}\in\mathbb{T}_{n}$, the cones
$\sigma(G_{t^{\prime}}^{t})$ and $\sigma(I)$ intersect in their common face,
where $\sigma(I)=\sigma(\mathbf{e}_{1},\dots,\mathbf{e}_{n})$.

By changing the notation, we may prove Condition 3.28 for
$\sigma(G_{t}^{t_{0}})$ instead of $\sigma(G_{t^{\prime}}^{t})$ therein. Then,
the condition can be further rephrased as follows:

Claim. For any $t_{0},t\in\mathbb{T}_{n}$, if a $g$-vector
$\mathbf{g}_{i;t}^{t_{0}}$ intersects $\sigma(I)$ nontrivially, then
$\mathbf{g}_{i;t}^{t_{0}}=\mathbf{e}_{\ell}$ for some $\ell$.

Let us prove the claim. For $J\subset\\{1,\dots,n\\}$, let
$J^{c}=\\{1,\dots,n\\}\setminus J$ be its compliment, and let
$\sigma_{J}(G_{t}^{t_{0}})$ be the face of $\sigma(G_{t}^{t_{0}})$ generated
by $\mathbf{g}_{j;t}^{t_{0}}$’s ($j\in J^{c}$). For example, we have
$\sigma_{\\{1,\dots,n\\}}(G_{t}^{t_{0}})=\\{0\\}$,
$\sigma_{\\{i\\}^{c}}(G_{t}^{t_{0}})=\sigma(\mathbf{g}_{i;t}^{t_{0}})$.

Thanks to Proposition 3.16, we have

(3.80)
$\displaystyle\sigma_{J}(G_{t}^{t_{0}})=\sigma(G_{t}^{t_{0}})\cap\left(\bigcap_{j\in
J}({\mathbf{c}}_{j;t}^{t_{0}})^{\perp}\right).$

On the other hand, since ${\mathbf{c}}_{j;t}^{t_{0}}$ is either positive or
negative, we have

(3.81)
$\displaystyle\sigma(I)\cap({\mathbf{c}}_{j;t}^{t_{0}})^{\perp}=\sigma(I)\cap\bigcap_{k\in
K[j]}{(\mathbf{e}_{k})}^{\perp}=\sigma_{K[j]}(I),$

where $K[j]$ is the set of $k$ such that the $k$th component of
${\mathbf{c}}_{j;t}^{t_{0}}$ is nonzero. Therefore, we have

(3.82)
$\displaystyle\sigma_{J}(G_{t}^{t_{0}})\cap\sigma(I)=\sigma(G_{t}^{t_{0}})\cap\left(\bigcap_{j\in
J}\sigma_{K[j]}(I)\right)=\sigma(G_{t}^{t_{0}})\cap\sigma_{K[J]}(I),$

where $K[J]$ is the set of $k$ such that the $k$th component of
${\mathbf{c}}_{j;t}^{t_{0}}$ is nonzero for some $j\in J$. Now we set
$J=\\{i\\}^{c}$. Then, we have
$\sigma_{\\{i\\}^{c}}(G_{t}^{t_{0}})=\sigma(\mathbf{g}_{i;t}^{t_{0}})$ and
$K[J]=\\{1,\dots,n\\}$ or $\\{\ell\\}^{c}$ for some $\ell$. In the former
case, we see from (3.82) that $\sigma(\mathbf{g}_{i;t}^{t_{0}})$ intersects
$\sigma(I)$ trivially. In the latter case, we have

(3.83)
$\displaystyle\sigma(\mathbf{g}_{i;t}^{t_{0}})\cap\sigma(I)=\sigma(G_{t}^{t_{0}})\cap\sigma(\mathbf{e}_{\ell}).$

Therefore, if $\sigma(\mathbf{g}_{i;t}^{t_{0}})\cap\sigma(I)\neq\\{0\\}$, we
have $\sigma(\mathbf{g}_{i;t}^{t_{0}})=\sigma(\mathbf{e}_{\ell})$. Then, by
the unimodularity (3.3), we have $\mathbf{g}_{i;t}^{t_{0}}=\mathbf{e}_{\ell}$
as desired. This completes the proof of Theorem 3.17.

###  3.8. Properties of $\hat{c}$-vectors

In this subsection we observe some properties of $\hat{c}$-vectors which are
relevant to the scattering diagram method.

The following properties do not depend on the sign-coherence conjecture.

######  Proposition 3.29.

The following facts hold:

(a). Any $\hat{c}$-vector $\hat{\mathbf{c}}_{i;t}$ is orthogonal to the
${c}$-vector ${\mathbf{c}}_{i;t}$.

(b). Any $\hat{c}$-vector $\hat{\mathbf{c}}_{i;t}$ is in the subspace in
$\mathbb{R}^{n}$ spanned by $\sigma_{i}(G_{t})$.

###### Proof.

(a). Since $DB_{t_{0}}$ is skew-symmetric, we have

(3.84)
$\displaystyle(\hat{\mathbf{c}}_{i;t},{\mathbf{c}}_{i;t})_{D}=({\mathbf{c}}_{i;t},\hat{\mathbf{c}}_{i;t})_{D}=({\mathbf{c}}_{i;t})^{T}DB_{t_{0}}{\mathbf{c}}_{i;t}=0.$

(b). By the second expression of $\hat{c}$-vectors in (2.57), we have

(3.85)
$\displaystyle\hat{\mathbf{c}}_{i;t}=\sum_{j=1}^{n}b_{ji;t}\mathbf{g}_{j;t},$

where we note that $b_{ii;t}=0$. ∎

######  Definition 3.30.

Suppose that the sign-coherence conjecture holds. Let $\mathbf{c}_{i;t}$ be a
$c$-vector, and let $\varepsilon_{i;t}$ be its tropical sign in Definition
3.1. We call the vectors
$\mathbf{c}_{i;t}^{+}=\varepsilon_{i;t}\mathbf{c}_{i;t}$ and
$\hat{\mathbf{c}}_{i;t}^{+}=\varepsilon_{i;t}\hat{\mathbf{c}}_{i;t}$ a
$c^{+}$-vector and a $\hat{c}^{+}$-vector, respectively.

######  Remark 3.31.

Any $c^{+}$-vector $\mathbf{c}_{i;t}^{+}$ is positive due to the sign-
coherence assumption, while a $\hat{c}^{+}$-vector
$\hat{\mathbf{c}}_{i;t}^{+}$ is not so at all in general.

The following fact is related to the incoming/outgoing property of walls
appearing later in Definition 6.16.

######  Proposition 3.32.

Suppose that the sign-coherence conjecture holds. Then, if

(3.86) $\displaystyle\hat{\mathbf{c}}_{i;t}^{+}\in\sigma_{i}(G_{t}),$

then ${\mathbf{c}}_{i;t}^{+}=\mathbf{e}_{\ell}$ for some $\ell$.

###### Proof.

In view of (3.85), the condition (3.86) is equivalent to the following
condition:

(3.87) The vector $\varepsilon_{i;t}\mathbf{b}_{i;t}$ is either positive or
zero.

Suppose that (3.87) happens, where we replace $i$ with $k$ therein for the
sake of convenience. Let $t^{\prime}$ be the vertex which is connected to $t$
with an edge labeled by $k$. Then, the mutation (3.12) is simplified by (3.87)
as

(3.88) $\displaystyle\mathbf{g}_{i;t^{\prime}}$
$\displaystyle=\begin{cases}\displaystyle-\mathbf{g}_{k;t}&i=k,\\\
\mathbf{g}_{i;t}&i\neq k.\end{cases}$

Now we recall the row sign-coherence of $G$-matrices in Proposition 3.7. Then,
(3.88) happens only when the following condition is satisfied:

  * •

For each $\ell=1,\dots,n$, if the $\ell$th element of $\mathbf{g}_{k;t}$ is
nonzero, then the $\ell$th element of other $\mathbf{g}_{i;t}$ ($i\neq k$)
vanishes.

Thanks to the unimodularity (3.3) of $G_{t}$, one can sharpen the condition as
follows:

  * •

There is some $\ell$ such that $\mathbf{g}_{k;t}=\pm\mathbf{e}_{\ell}$, and
the $\ell$th element of other $\mathbf{g}_{i;t}$ ($i\neq k$) vanishes.

Then, by the duality (3.4), $\mathbf{c}_{k;t}=\pm\mathbf{e}_{\ell}$. Thus,
$\mathbf{c}_{k;t}^{+}=\mathbf{e}_{\ell}$. ∎

##  4\. Scattering diagrams and $C$\- and $G$-matrices

In this section we present a proof of the sign-coherence conjecture based on
the results on scattering diagrams in [GHKK18]. At the same time we also
establish the correspondence between a $G$-fan and the support of a scattering
diagram.

###  4.1. Supports of scattering diagrams

To write down a precise definition of scattering diagrams in [GHKK18] requires
considerable pages, and it will be postponed in Section 6. Here we summarize
the minimal information and results [GHKK18] which are needed to prove the
sign-coherence conjecture.

Informally speaking, for a certain vector space
$M_{\mathbb{R}}\simeq\mathbb{R}^{n}$, a scattering diagram
$\mathfrak{D}=\\{(\mathfrak{d}_{\lambda},f_{\mathfrak{d}_{\lambda}})\\}_{\lambda\in\Lambda}$
in $M_{\mathbb{R}}$ is a finite or countably infinite collection of walls,
where $\Lambda$ is an index set, and each wall
$(\mathfrak{d}_{\lambda},f_{\mathfrak{d}_{\lambda}})$ is a pair such that
$\mathfrak{d}_{\lambda}$ is a (not necessarily strongly convex) cone of
codimension one in $M_{\mathbb{R}}$ called the support of a wall, and
$f_{\mathfrak{d}_{\lambda}}$ is a certain formal power series called a wall
function. The union of the supports of walls

(4.1)
$\displaystyle\mathrm{Supp}(\mathfrak{D})=\bigcup_{\lambda\in\Lambda}\mathfrak{d}_{\lambda}\subset
M_{\mathbb{R}}$

is called the support of a scattering diagram $\mathfrak{D}$.

For any nonsingular skew-symmetrizable matrix $B$, one can construct a
scattering diagram $\mathfrak{D}_{\mathfrak{s}}$, where $\mathfrak{s}$ is a
seed data in [GHKK18] corresponding to $B$. There is a canonical isomorphism
$\phi_{\mathfrak{s}}:M_{\mathbb{R}}\buildrel{\sim}\over{\rightarrow}\mathbb{R}^{n}$
such that the image of $\mathrm{Supp}(\mathfrak{D}_{\mathfrak{s}})$ in
$\mathbb{R}^{n}$ only depends on $B$. We write this image as
$\mathfrak{S}(B)$, and we call it the support diagram for $B$, for simplicity.
A connected component of $\mathbb{R}^{n}\setminus\mathfrak{S}(B)$ is called a
chamber of $\mathfrak{S}(B)$.

From the viewpoint of cluster patterns, the supports and the wall functions of
walls contain the information of the tropical and the nontropical part of a
cluster pattern, respectively. Since at this moment we are interested only in
the tropical part, i.e., $C$\- and $G$-matrices, we may safely forget wall
functions, and concentrate on the support diagram $\mathfrak{S}(B)$.

Let $D$ be a skew-symmetrizer of $B$ in the form (3.42), and let $(\ ,\ )_{D}$
be the inner product in $\mathbb{R}^{n}$ defined in (3.43). The following
facts are direct consequences of the definition/theorem of the scattering
diagram $\mathfrak{D}_{\mathfrak{s}}$ in [GHKK18, Theorems 1.12 &1.28], which
will appear later as Theorem 6.29.

######  Proposition 4.1 ([GHKK18, Theorems 1.12 &1.28]).

For any nonsingular skew-symmetrizable matrix $B$, the support diagram
$\mathfrak{S}(B)$ has the following properties.

(a). The support $\mathfrak{d}_{\lambda}$ of each wall of $\mathfrak{S}(B)$
has a positive normal vector $\mathbf{n}_{\lambda}$ with respect to the inner
product $(\ ,\ )_{D}$.

(b). Each hyperplane $\mathbf{e}_{i}^{\perp}$ ($i=1,\dots,n$) with respect to
the inner product $(\ ,\ )_{D}$ is a subset of $\mathfrak{S}(B)$.

(c). The interior $\sigma^{\circ}(I)$ of the cone
$\sigma(I)=\sigma(\mathbf{e}_{1},\dots,\mathbf{e}_{n})$ is a chamber of
$\mathfrak{S}(B)$. In other words, $\sigma^{\circ}(I)$ does not intersect
$\mathfrak{S}(B)$, while its boundary belongs to $\mathfrak{S}(B)$. (This is a
consequence of (a) and (b).)

Also, the following property of the support diagram $\mathfrak{S}(B)$ is
obtained from [GHKK18, Theorem 1.24], which will appear later as Theorem 7.7,
together with Proposition 3.26. See the explanation after Theorem 7.7 for
details.

######  Proposition 4.2 ([GHKK18, Theorem 1.24], [Mul16, Lemma 5.2.1],[Rea20b,
Cor. 4.4]).

Let $\mathbf{B}$ be any nonsingular $B$-pattern. Let
$t_{0},t_{1}\in\mathbb{T}_{n}$ be vertices which are connected with an edge
labeled by $k$. Let $\varphi_{t_{0}}^{t_{1}}$ be the map defined in (3.69).
Then, the support diagrams $\mathfrak{S}(B_{t_{0}})$ and
$\mathfrak{S}(B_{t_{1}})$ are related by

(4.2) $\displaystyle\varphi_{t_{0}}^{t_{1}}(\mathfrak{S}(B_{t_{0}}))$
$\displaystyle=\mathfrak{S}(B_{t_{1}}).$

This is clearly a parallel result for $G$-fans in Proposition 3.22, which was
proved under the assumption of the sign-coherence of $C$-matrices.

###  4.2. Proof of sign-coherence conjecture

Let us present a proof of Conjecture 2.34 relying on Propositions 4.1 and 4.2.
Our strategy is to establish both

  * •

the identification of the interior $\sigma^{\circ}(G^{t_{0}}_{t})$ of each
$G$-cone as a chamber in $\mathfrak{S}(B_{t_{0}})$,

  * •

the column sign-coherence of $C$-matrices $C_{t}^{t_{0}}$,

simultaneously by the induction on the distance $d(t_{0},t)$ in
$\mathbb{T}_{n}$, where both $t_{0}$ and $t$ vary.

Now we prove the following theorem. See [Rea20b, Theorem 4.10] for a closely
related result for mutation fans.

######  Theorem 4.3.

Let $\mathbf{B}=\\{B_{t}\\}_{t\in\mathbb{T}_{n}}$ be any nonsingular
$B$-pattern. Then, for any $t_{0},t\in\mathbb{T}_{n}$, the following facts
hold:

(a). The set $\sigma^{\circ}(G_{t}^{t_{0}})$ is a chamber of
$\mathfrak{S}(B_{t_{0}})$.

(b). Each $c$-vector $\mathbf{c}_{i;t}^{t_{0}}$ is a normal vector of the
support of some wall of $\mathfrak{S}(B_{t_{0}})$.

(c). The column sign-coherence holds for the $C$-matrix $C_{t}^{t_{0}}$.

###### Proof.

As in Section 3.2, we introduce the transpose
$\mathbf{B}^{T}=\\{(B_{t})^{T}\\}_{t\in\mathbb{T}_{n}}$ of $\mathbf{B}$. Let
$\tilde{C}_{t}^{t_{0}}$, $\tilde{G}_{t}^{t_{0}}$ be $C$\- and $G$-matrices of
$\mathbf{B}^{T}$.

Consider the following claims $(a)_{d}$, $(b)_{d}$, $(c)_{d}$ for $d=0,1,2$,
….

$(a)_{d}$. For any $t_{0},t\in\mathbb{T}_{n}$ such that $d(t_{0},t)=d$, the
following facts hold:

(4.3) $\displaystyle|C_{t}^{t_{0}}|$
$\displaystyle=|G_{t}^{t_{0}}|\in\\{1,-1\\},$ (4.4) $\displaystyle
D^{-1}(G_{t}^{t_{0}})^{T}DC_{t}^{t_{0}}$ $\displaystyle=I,$ (4.5)
$\displaystyle C_{t}^{t_{0}}$ $\displaystyle=(\tilde{G}_{t_{0}}^{t})^{T},$
(4.6) $\displaystyle G_{t}^{t_{0}}$
$\displaystyle=(\tilde{C}_{t_{0}}^{t})^{T}.$

$(b)_{d}$. For any $t_{0},t\in\mathbb{T}_{n}$ such that $d(t_{0},t)=d$, the
following facts hold:

  * (i).

The set $\sigma^{\circ}(G_{t}^{t_{0}})$ is a chamber of
$\mathfrak{S}(B_{t_{0}})$.

  * (ii).

Each $c$-vector $\mathbf{c}_{i;t}^{t_{0}}$ is a normal vector of the support
of a wall of $\mathfrak{S}(B_{t_{0}})$.

  * (iii).

The column sign-coherence holds for $C_{t}^{t_{0}}$.

  * (iv).

The row sign-coherence holds for $G_{t}^{t_{0}}$.

$(c)_{d}$. For any $t_{0},t_{1},t,t^{\prime}\in\mathbb{T}_{n}$ such that

  * •

$d(t_{0},t)=d$,

  * •

$t_{0}$ and $t_{1}$ are connected with an edge labeled by $k$,

  * •

$t$ and $t^{\prime}$ are connected with an edge labeled by $k$,

the following facts hold:

(4.7) $\displaystyle C_{t^{\prime}}^{t_{0}}$
$\displaystyle=C_{t}^{t_{0}}(J_{k}+[{\varepsilon}_{\bullet
k}(C^{t_{0}}_{t})B_{t}]_{+}^{k\bullet}),$ (4.8) $\displaystyle
G_{t^{\prime}}^{t_{0}}$
$\displaystyle=G_{t}^{t_{0}}(J_{k}+[-{\varepsilon}_{\bullet
k}(C^{t_{0}}_{t})B_{t}]_{+}^{\bullet k}),$ (4.9) $\displaystyle C^{t_{1}}_{t}$
$\displaystyle=(J_{k}+[-{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{k\bullet})C^{t_{0}}_{t},$
(4.10) $\displaystyle G^{t_{1}}_{t}$
$\displaystyle=(J_{k}+[{\varepsilon}_{k\bullet}(G^{t_{0}}_{t})B_{t_{0}}]_{+}^{\bullet
k})G^{t_{0}}_{t}.$

We prove the claims in the following order,

(4.11)
$\displaystyle(a)_{0}\Longrightarrow(b)_{0}\Longrightarrow(c)_{0}\Longrightarrow(a)_{1}\Longrightarrow(b)_{1}\Longrightarrow(c)_{1}\Longrightarrow(a)_{2}\Longrightarrow\cdots,$

assuming all preceding claims. Moreover, we run the same induction procedure
for $C$\- and $G$-matrices $\tilde{C}_{t}^{t_{0}}$, $\tilde{G}_{t}^{t_{0}}$ of
$\mathbf{B}^{T}$ in the background.

We first prove $(a)_{0}$, $(b)_{0}$, and $(c)_{0}$. The claims $(a)_{0}$ and
$(c)_{0}$ were proved in Propositions 3.3 and 3.7. The claim $(b)_{0}$ holds
because $C_{t_{0}}^{t_{0}}=G_{t_{0}}^{t_{0}}=I$, and Proposition 4.1 (c),
where we have already received an important input from scattering diagrams.

Now, assuming the claims in (4.11) up to $(c)_{d}$, we prove $(a)_{d+1}$. This
was already proved in Propositions 3.3 and 3.7 by taking account of Remarks
3.4 and 3.8.

Next, assuming the claims in (4.11) up to $(a)_{d+1}$, we prove $(b)_{d+1}$.
(This is the highlights of the proof of the sign-coherence conjecture.) Let
$t_{0},t_{1},t\in\mathbb{T}_{n}$ be vertices such that $d(t_{0},t)=d$,
$d(t_{1},t)=d+1$, and $t_{0}$ and $t_{1}$ are connected with an edge labeled
by $k$. By $(b)_{d}$, $\sigma^{\circ}(G_{t}^{t_{0}})$ is a chamber in
$\mathfrak{S}(B_{t_{0}})$. By $(a)_{d}$, $(c)_{d}$, Proposition 3.22, and
Remark 3.23, we have
$\sigma(G_{t}^{t_{1}})=\varphi_{t_{0}}^{t_{1}}(\sigma(G_{t}^{t_{0}}))$.
Meanwhile, by Proposition 4.2, the piecewise linear isomorphism
$\varphi_{t_{0}}^{t_{1}}$ maps a chamber in $\mathfrak{S}(B_{t_{0}})$ to a
chamber in $\mathfrak{S}(B_{t_{1}})$. Therefore, $\sigma(G_{t}^{t_{1}})$ is a
chamber in $\mathfrak{S}(B_{t_{1}})$. This proves (i). By (i), each face
$\sigma_{i}(G_{t}^{t_{1}})$ ($i=1,\dots,n$), which is a cone of codimension
one by $(a)_{d+1}$, is a subset of the union of the supports of some walls of
$\mathfrak{S}(B_{t_{1}})$. Meanwhile, by $(a)_{d+1}$, each $c$-vector
$\mathbf{c}_{i;t}^{t_{1}}$ is a normal vector of the face
$\sigma_{i}(G_{t}^{t_{1}})$ with respect to the inner product $(\ ,\ )_{D}$.
Therefore, each $c$-vector $\mathbf{c}_{i;t}^{t_{1}}$ is a normal vector of
the support of some wall of $\mathfrak{S}(B_{t_{1}})$. This proves (ii). Then,
by Proposition 4.1 (a), the $c$-vector $\mathbf{c}_{i;t}^{t_{1}}$ is either
positive or negative. This proves (iii), i.e., the column sign-coherence of
the $C$-matrix $C_{t}^{t_{1}}$. The same result is proved in the background
for the $C$-matrix $\tilde{C}_{t_{1}}^{t}$ of the $B$-pattern
$\mathbf{B}^{T}$. Thus, by $(a)_{d+1}$, the $G$-matrix $G_{t}^{t_{1}}$ is row
sign-coherent. This proves (iv).

Finally, assuming the claims in (4.11) up to $(b)_{d+1}$, we prove
$(c)_{d+1}$. This was already proved in Propositions 3.3 and 3.7 by taking
account of Remarks 3.4 and 3.8. ∎

Our last task is to remove the assumption of nonsingularity for $B$-patterns
in Theorem 4.3. By Proposition 3.11, Theorem 4.3 is adapted for possibly
nonsingular $B$-patterns as follows:

######  Theorem 4.4.

Let $\mathbf{B}=\\{B_{t}\\}_{t\in\mathbb{T}_{n}}$ be any (possibly singular)
$B$-pattern. Then, for any $t_{0},t\in\mathbb{T}_{n}$, the following facts
hold:

(a). The set $\sigma^{\circ}(G_{t}^{t_{0}})$ is a chamber of
$\mathfrak{S}(\overline{B}_{t_{0}})\cap\langle\mathbf{e}_{1},\dots,\mathbf{e}_{n}\rangle$,
where $\overline{B}_{t_{0}}$ is the matrix in (3.36) and
$\langle\mathbf{e}_{1},\dots,\mathbf{e}_{n}\rangle$ is the subspace of
$\mathbb{R}^{2n}$ spanned by $\mathbf{e}_{1},\dots,\mathbf{e}_{n}$.

(b). Each $c$-vector $\mathbf{c}_{i;t}^{t_{0}}$ is a normal vector of
$\overline{\mathfrak{d}}\cap\langle\mathbf{e}_{1},\dots,\mathbf{e}_{n}\rangle$,
where $\overline{\mathfrak{d}}$ is the support of some wall of
$\mathfrak{S}(\overline{B}_{t_{0}})$.

(c). The column sign-coherence holds for the $C$-matrix $C_{t}^{t_{0}}$.

###### Proof.

Let $\overline{C}_{t}$ and $\overline{G}_{t}$ be $C$\- and $G$-matrices of the
principal extension $\overline{\mathbf{B}}$ of $\mathbf{B}$ with the initial
vertex $t_{0}$ in Section 3.3.

(c). The matrix $\overline{C}_{t}$ is column sign-coherent by Theorem 4.3 (c).
Then, the claim follows from (3.37).

(a). By (3.38),
$\sigma(G_{t}^{t_{0}})=\sigma(\overline{G}_{t}^{t_{0}})\cap\langle\mathbf{e}_{1},\dots,\mathbf{e}_{n}\rangle$.
Then, the claim follows from Theorem 4.3 (a).

(b). This follows from (3.37) and Theorem 4.3 (b). ∎

This completes the proof of Conjecture 2.34.

From now on, we use the column sign-coherence of $C$-matrices not as an
assumption but as a theorem.

##  5\. More about $F$-polynomials

Let us turn to study $F$-polynomials, namely, the nontropical part of a
cluster pattern.

###  5.1. Fock-Goncharov decomposition

Let $\mathbf{\Sigma}$ be any cluster pattern. We come back to the mutations of
$x$\- and $y$-variables, especially in the $\varepsilon$-expression (2.10) and
(2.11). Let $t,t^{\prime}\in\mathbb{T}_{n}$ be vertices which are connected
with an edge labeled by $k$. One can regard mutations (2.10) and (2.11) as
isomorphisms between rational function fields as follows, where
$\mathbf{x}_{t}$, $\mathbf{x}_{t^{\prime}}$, $\mathbf{y}_{t}$,
$\mathbf{y}_{t^{\prime}}$ are $n$-tuple of formal variables:

(5.1)
$\displaystyle\mu_{k;t}:\mathbb{Q}(\mathbf{x}_{t^{\prime}})\rightarrow\mathbb{Q}(\mathbf{x}_{t}),\
\mathbb{Q}(\mathbf{y}_{t^{\prime}})\rightarrow\mathbb{Q}(\mathbf{y}_{t}),$
(5.2) $\displaystyle\mu_{k;t}(x_{i;t^{\prime}})=\begin{cases}\displaystyle
x_{k;t}^{-1}\left(\prod_{j=1}^{n}x_{j;t}^{[-\varepsilon
b_{jk;t}]_{+}}\right)(1+\hat{y}_{k;t}^{\varepsilon})&i=k,\\\ x_{i;t}&i\neq
k,\end{cases}$ (5.3)
$\displaystyle\mu_{k;t}(y_{i;t^{\prime}})=\begin{cases}\displaystyle
y_{k;t}^{-1}&i=k,\\\ y_{i;t}y_{k;t}^{[\varepsilon
b_{ki;t}]_{+}}(1+y_{k;t}^{\varepsilon})^{-b_{ki;t}}&i\neq k.\end{cases}$

Note that the map is in the opposite direction, namely, from $t^{\prime}$ to
$t$. Here and below, we use common symbols for maps for $x$\- and
$y$-variables as above in view of the parallelism between them.

To make use of the flexibility of the choice of sign $\varepsilon$, we first
choose a given initial vertex $t_{0}$. Since we have already established the
sign-coherence of $C$-matrices $C_{t}^{t_{0}}=C_{t}$, the tropical sign
$\varepsilon_{i;t}^{t_{0}}=\varepsilon_{i;t}$ in Definition 3.1 is defined for
any $t$ and $i$. Now we set $\varepsilon=\varepsilon_{k;t}$ in (5.2) and
(5.3).

Then, modifying the idea of [FG09] with tropical signs, we consider the
following decompositions of the maps $\mu_{k;t}$:

(5.4) $\displaystyle\mu_{k;t}=\rho_{k;t}\circ\tau_{k;t}.$

Here, $\tau_{k;t}^{t_{0}}=\tau_{k;t}$ are the following isomorphisms,

(5.5)
$\displaystyle\tau_{k;t}:\mathbb{Q}(\mathbf{x}_{t^{\prime}})\rightarrow\mathbb{Q}(\mathbf{x}_{t}),\
\mathbb{Q}(\mathbf{y}_{t^{\prime}})\rightarrow\mathbb{Q}(\mathbf{y}_{t}),$
(5.6) $\displaystyle\tau_{k;t}(x_{i;t^{\prime}})=\begin{cases}\displaystyle
x_{k;t}^{-1}\prod_{j=1}^{n}x_{j;t}^{[-\varepsilon_{k;t}b_{jk;t}]_{+}}&i=k,\\\
x_{i;t}&i\neq k,\end{cases}$ (5.7)
$\displaystyle\tau_{k;t}(y_{i;t^{\prime}})=\begin{cases}\displaystyle
y_{k;t}^{-1}&i=k,\\\ y_{i;t}y_{k;t}^{[\varepsilon_{k;t}b_{ki;t}]_{+}}&i\neq
k,\end{cases}$

while $\rho_{k;t}^{t_{0}}=\rho_{k;t}$ are the following automorphisms,

(5.8)
$\displaystyle\rho_{k;t}:\mathbb{Q}(\mathbf{x}_{t})\rightarrow\mathbb{Q}(\mathbf{x}_{t}),\
\mathbb{Q}(\mathbf{y}_{t})\rightarrow\mathbb{Q}(\mathbf{y}_{t}),$ (5.9)
$\displaystyle\rho_{k;t}(x_{i;t})=x_{i;t}(1+\hat{y}_{k;t}^{\varepsilon_{k;t}})^{-\delta_{ik}},$
(5.10)
$\displaystyle\rho_{k;t}(y_{i;t}s)=y_{i;t}(1+y_{k;t}^{\varepsilon_{k;t}})^{-b_{ki;t}}.$

Note that the maps $\tau_{k;t}$ and $\rho_{k;t}$ depend on the choice of the
initial vertex $t_{0}$ through tropical signs. Also, they are defined only
after establishing the sign-coherence of $C$-matrices. We call the
decomposition (5.4) the Fock-Goncharov decomposition of a mutation
${\mu}_{k;t}$ (with tropical sign) with respect to the initial vertex $t_{0}$.
We also call $\tau_{k;t}$ and $\rho_{k;t}$ the tropical part and the
nontropical part of ${\mu}_{k;t}$, respectively. Indeed, it is clear that
(5.6) and (5.7) are the exponential form of the mutations of $g$-vectors
(3.12) and $c$-vectors (3.11), respectively.

######  Remark 5.1.

The decomposition (5.4) was introduced in [FG09] for $\varepsilon=1$. The
signed version here were used, for example, in the application to the Stokes
automorphisms [IN14] and dilogarithm identities [GNR17].

######  Proposition 5.2.

Let $t,t^{\prime}\in\mathbb{T}_{n}$ be vertices which are connected with an
edge labeled by $k$. Then, the following relations hold.

(5.11) $\displaystyle\mu_{k;t^{\prime}}\circ\mu_{k;t}$
$\displaystyle=\mathrm{id},$ (5.12)
$\displaystyle\tau_{k;t^{\prime}}\circ\tau_{k;t}$ $\displaystyle=\mathrm{id}.$

###### Proof.

The equality (5.11) is nothing but the involution of the mutations of $x$\-
and $y$-variables. The equality (5.12) for both $x$\- and $y$-variables can be
verified by the fact,

(5.13) $\displaystyle\varepsilon_{k;t^{\prime}}=-\varepsilon_{k;t},\quad
b_{ki;t^{\prime}}=-b_{ki;t},\quad b_{jk;t^{\prime}}=-b_{jk;t}.$

∎

For any $t\in\mathbb{T}_{n}$, consider a sequence of vertices $t_{0}$,
$t_{1}$, …, $t_{r+1}=t\in\mathbb{T}_{n}$ such that they are sequentially
connected with edges labeled by $k_{0}$, …, $k_{r}$. Then, we define
isomorphisms

(5.14) $\displaystyle\mu_{t}^{t_{0}}$
$\displaystyle:=\mu_{k_{0};t_{0}}\circ\mu_{k_{1};t_{1}}\circ\dots\circ\mu_{k_{r};t_{r}}:\mathbb{Q}(\mathbf{x}_{t})\rightarrow\mathbb{Q}(\mathbf{x}),\
\mathbb{Q}(\mathbf{y}_{t})\rightarrow\mathbb{Q}(\mathbf{y}),$ (5.15)
$\displaystyle\tau_{t}^{t_{0}}$
$\displaystyle:=\tau_{k_{0};t_{0}}\circ\tau_{k_{1};t_{1}}\circ\dots\circ\tau_{k_{r};t_{r}}:\mathbb{Q}(\mathbf{x}_{t})\rightarrow\mathbb{Q}(\mathbf{x}),\
\mathbb{Q}(\mathbf{y}_{t})\rightarrow\mathbb{Q}(\mathbf{y}),$

where we set $\mathbf{x}_{t_{0}}=\mathbf{x}$, $\mathbf{y}_{t_{0}}=\mathbf{y}$.
Thanks to Proposition 5.2, $\mu_{t}^{t_{0}}$ and $\tau_{t}^{t_{0}}$ depend
only on $t_{0}$ and $t$. Namely, we do not have to care about the redundancy
$k_{s+1}=k_{s}$ in the sequence $k_{0},\dots,k_{r}$.

Recall the notions of tropical $x$\- and $y$-variables in (2.54) and (2.53).
The following proposition tells that the tropical parts $\tau_{t}^{t_{0}}$ are
nothing but the mutations of tropical $x$\- and $y$-variables.

######  Proposition 5.3.

The following formulas hold:

(5.16) $\displaystyle\mu_{t}^{t_{0}}(x_{i;t})$
$\displaystyle=x^{\mathbf{g}_{i;t}}F_{i;t}(\hat{\mathbf{y}}),$ (5.17)
$\displaystyle\mu_{t}^{t_{0}}(y_{i;t})$
$\displaystyle=y^{\mathbf{c}_{i;t}}\prod_{j=1}^{n}F_{j;t}(\mathbf{y})^{b_{ji;t}},$
(5.18) $\displaystyle\tau_{t}^{t_{0}}(x_{i;t})$
$\displaystyle=x^{\mathbf{g}_{i;t}},$ (5.19)
$\displaystyle\tau_{t}^{t_{0}}(y_{i;t})$ $\displaystyle=y^{\mathbf{c}_{i;t}}.$

###### Proof.

The first two equalities are the separation formulas in Theorem 2.23. Let us
prove the equality (5.18) for $x$-variables by the induction on $t$ along
$\mathbb{T}_{n}$ starting from $t_{0}$. For $t=t_{0}$,
$x^{\mathbf{g}_{i;t_{0}}}=x^{\mathbf{e}_{i}}=x_{i}$. Therefore, (5.18) holds.
Suppose that it is true for $t$ with $d(t_{0},t)=d$. Let $t^{\prime}$ be a
vertex which is connected to $t$ with an edge labeled by $k$ such that
$d(t_{0},t^{\prime})=d+1$. Then, we have

(5.20) $\displaystyle\begin{split}\quad\
\tau_{t^{\prime}}^{t_{0}}(x_{i;t^{\prime}})&=(\tau_{t}^{t_{0}}\circ\tau_{k;t})(x_{i;t^{\prime}})\\\
&=\begin{cases}\displaystyle\tau_{t}^{t_{0}}(x_{k;t}^{-1}\prod_{j=1}^{n}x_{j;t}^{[-\varepsilon_{k;t}b_{jk;t}]_{+}})=x^{-\mathbf{g}_{k;t}+\sum_{j=1}^{n}[-\varepsilon_{k;t}b_{jk;t}]_{+}\mathbf{g}_{j;t}}&i=k,\\\
\tau_{t}^{t_{0}}(x_{i;t})=x^{\mathbf{g}_{i;t}}&i\neq k\end{cases}\\\
&=x^{\mathbf{g}_{i;t^{\prime}}},\end{split}$

where we used (3.12) in the last equality. Therefore, (5.18) holds for
$t^{\prime}$. Similarly, the equality (5.19) for $y$-variables follows from
(3.11). ∎

Under these maps, $\hat{y}$-variables transform in the same way as
$y$-variables as expected.

######  Proposition 5.4.

The following formulas hold:

(5.21) $\displaystyle\tau_{k;t}(\hat{y}_{i;t^{\prime}})$
$\displaystyle=\begin{cases}\displaystyle\hat{y}_{k;t}^{-1}&i=k,\\\
\hat{y}_{i;t}\hat{y}_{k;t}^{[\varepsilon_{k;t}b_{ki;t}]_{+}}&i\neq
k,\end{cases}$ (5.22) $\displaystyle\rho_{k;t}(\hat{y}_{i;t^{\prime}})$
$\displaystyle=\hat{y}_{i;t}(1+\hat{y}_{k;t}^{\varepsilon_{k;t}})^{-b_{ki;t}},$
(5.23) $\displaystyle\tau_{t}^{t_{0}}(\hat{y}_{i;t})$
$\displaystyle=\hat{y}^{\mathbf{c}_{i;t}}=x^{\hat{\mathbf{c}}_{i;t}},$

where $\hat{\mathbf{c}}_{i;t}=B_{t_{0}}\mathbf{c}_{i;t}$ is a $\hat{c}$-vector
in (2.57).

###### Proof.

The equality (5.21) follows from (5.6) and (2.12). The equality (5.22) follows
from (5.9). The equality (5.23) follows from (5.18) and (2.56),

∎

###  5.2. Nontropical parts and $F$-polynomials

Observing Proposition 5.3, it is clear that the nontropical parts $\rho_{k;t}$
are responsible to generate and mutate $F$-polynomials. Let us make this
statement more manifest.

Let us introduce the following automorphisms $\mathfrak{q}_{k;t}$ for the
initial $x$\- and $y$-variables,

(5.24) $\displaystyle\mathfrak{q}_{k;t}:\mathbb{Q}(\mathbf{x})$
$\displaystyle\rightarrow\mathbb{Q}(\mathbf{x}),\
\mathbb{Q}(\mathbf{y})\rightarrow\mathbb{Q}(\mathbf{y}),$ (5.25)
$\displaystyle\mathfrak{q}_{k;t}(x^{\mathbf{m}})$
$\displaystyle=x^{\mathbf{m}}(1+\hat{y}^{\mathbf{c}_{k;t}^{+}})^{-(\mathbf{m},d_{k}\mathbf{c}_{k;t})_{D}}$
(5.26)
$\displaystyle=x^{\mathbf{m}}(1+x^{\hat{\mathbf{c}}_{k;t}^{+}})^{-(\mathbf{m},d_{k}\mathbf{c}_{k;t})_{D}},$
(5.27) $\displaystyle\mathfrak{q}_{k;t}(y^{\mathbf{n}})$
$\displaystyle=y^{\mathbf{n}}(1+y^{{\mathbf{c}}_{k;t}^{+}})^{(\mathbf{n},d_{k}\hat{\mathbf{c}}_{k;t})_{D}},$

where $\mathbf{m},\mathbf{n}\in\mathbb{Z}^{n}$, and
$\mathbf{c}_{k;t}^{+}=\varepsilon_{k;t}\mathbf{c}_{k;t}$ and
$\hat{\mathbf{c}}_{k;t}^{+}=\varepsilon_{k;t}\hat{\mathbf{c}}_{k;t}$ are a
$c^{+}$-vector and a $\hat{c}^{+}$-vector, respectively, in Definition 3.30.

The seemingly asymmetric definitions in (5.25) and (5.27) are justified by the
following properties.

######  Proposition 5.5.

The following facts hold:

(a). We have the formulas

(5.28)
$\displaystyle\mathfrak{q}_{k;t}(x^{\mathbf{g}_{i;t}})=x^{\mathbf{g}_{i;t}}(1+\hat{y}^{\mathbf{c}_{k;t}^{+}})^{-\delta_{ik}},$
(5.29)
$\displaystyle\mathfrak{q}_{k;t}(y^{\mathbf{c}_{i;t}})=y^{\mathbf{c}_{i;t}}(1+y^{{\mathbf{c}}_{k;t}^{+}})^{-b_{ki;t}}.$

(b). The following relation holds:

(5.30)
$\displaystyle\tau^{t_{0}}_{t}\circ\rho_{k;t}=\mathfrak{q}_{k;t}\circ\tau^{t_{0}}_{t}.$

(c). If $t^{\prime}$ and $t$ are connected with an edge label by $k$, we have

(5.31)
$\displaystyle\mathfrak{q}_{k;t^{\prime}}\circ\mathfrak{q}_{k;t}=\mathrm{id}.$

###### Proof.

(a). The claim follows from the following equalities:

(5.32) $\displaystyle(\mathbf{g}_{i;t},d_{k}\mathbf{c}_{k;t})_{D}$
$\displaystyle=\delta_{ik},$ (5.33)
$\displaystyle(\mathbf{c}_{i;t},d_{k}\hat{\mathbf{c}}_{k;t})_{D}$
$\displaystyle=-b_{ki;t}.$

The equality (5.32) is the duality in (3.4). The equality (5.33) follows from
(2.13) and (3.13) as follows:

(5.34) $\displaystyle C_{t}^{T}DB_{t_{0}}C_{t}D^{-1}=DB_{t}D^{-1}=-B_{t}^{T}.$

(b). Thanks to (5.28) and Propositions 5.3 and 5.4, we have

(5.35) $\displaystyle x_{i;t}\ {\buildrel\rho_{k;t}\over{\mapsto}}\
x_{i;t}(1+\hat{y}_{k;t}^{\varepsilon_{k;t}})^{-\delta_{ik}}\
{\buildrel\tau_{t}^{t_{0}}\over{\mapsto}}\
x^{\mathbf{g}_{i;t}}(1+\hat{y}^{\mathbf{c}_{k;t}^{+}})^{-\delta_{ik}},$ (5.36)
$\displaystyle x_{i;t}\ {\buildrel\tau_{t}^{t_{0}}\over{\mapsto}}\
x^{\mathbf{g}_{i;t}}\ {\buildrel\mathfrak{q}_{k;t}\over{\mapsto}}\
x^{\mathbf{g}_{i;t}}(1+\hat{y}^{\mathbf{c}_{k;t}^{+}})^{-\delta_{ik}}.$

Therefore, the equality (5.30) holds for $x$-variables. Similarly, we have

(5.37) $\displaystyle y_{i;t}\ {\buildrel\rho_{k;t}\over{\mapsto}}\
y_{i;t}(1+{y}_{k;t}^{\varepsilon_{k;t}})^{-b_{ki;t}}\
{\buildrel\tau_{t}^{t_{0}}\over{\mapsto}}\
y^{\mathbf{c}_{i;t}}(1+\hat{y}^{\mathbf{c}_{k;t}^{+}})^{-b_{ki;t}},$ (5.38)
$\displaystyle y_{i;t}\ {\buildrel\tau_{t}^{t_{0}}\over{\mapsto}}\
y^{\mathbf{c}_{i;t}}\ {\buildrel\mathfrak{q}_{k;t}\over{\mapsto}}\
y^{\mathbf{c}_{i;t}}(1+{y}^{\mathbf{c}_{k;t}^{+}})^{-b_{ki;t}}.$

Therefore, the equality (5.30) holds also for $y$-variables.

(c). This follows from $\mathbf{c}_{k;t^{\prime}}=-\mathbf{c}_{k;t}$. ∎

For the same $t_{0}$, $t_{1}$, …, $t_{r+1}=t\in\mathbb{T}_{n}$ for
$\mu_{t}^{t_{0}}$ and $\tau_{t}^{t_{0}}$ in (5.14) and (5.15), we define the
antomorphism

(5.39) $\displaystyle\mathfrak{q}_{t}^{t_{0}}$
$\displaystyle:=\mathfrak{q}_{k_{0};t_{0}}\circ\mathfrak{q}_{k_{1};t_{1}}\circ\dots\circ\mathfrak{q}_{k_{r};t_{r}}:\mathbb{Q}(\mathbf{x})\rightarrow\mathbb{Q}(\mathbf{x}),\
\mathbb{Q}(\mathbf{y})\rightarrow\mathbb{Q}(\mathbf{y}).$

Again, by Proposition 5.5 (c), it depends only on $t_{0}$ and $t$.

One can separate the tropical and the nontropical parts of $\mu^{t_{0}}_{t}$
as follows:

######  Proposition 5.6.

The following decomposition holds:

(5.40)
$\displaystyle\mu^{t_{0}}_{t}=\mathfrak{q}_{t}^{t_{0}}\circ\tau^{t_{0}}_{t}:\mathbb{Q}(\mathbf{x}_{t})\rightarrow\mathbb{Q}(\mathbf{x}),\
\mathbb{Q}(\mathbf{y}_{t})\rightarrow\mathbb{Q}(\mathbf{y}).$

###### Proof.

This can be proved by the successive application of Proposition 5.5 (b). For
simplicity, we omit the symbol $\circ$ for the composition of maps below.
First, we note that

(5.41)
$\displaystyle\rho_{k_{0};t_{0}}(x_{i})=x_{i}(1+\hat{y}_{k_{0}})^{-\delta_{ik_{0}}}=\mathfrak{q}_{k_{0};t_{0}}(x_{i}),$
(5.42)
$\displaystyle\rho_{k_{0};t_{0}}(y_{i})=y_{i}(1+y_{k_{0}})^{-b_{k_{0}i;t_{0}}}=\mathfrak{q}_{k_{0};t_{0}}(y_{i}).$

Then, by Proposition 5.5 we have

$\displaystyle\mu_{t}^{t_{0}}$
$\displaystyle=\mu_{k_{0};t_{0}}\mu_{k_{1};t_{1}}\dots\mu_{k_{r};t_{r}}$
$\displaystyle=\rho_{k_{0};t_{0}}\tau_{k_{0};t_{0}}\rho_{k_{1};t_{1}}\tau_{k_{1};t_{1}}\dots\rho_{k_{r};t_{r}}\tau_{k_{r};t_{r}}$
$\displaystyle=\mathfrak{q}_{k_{0};t_{0}}\tau_{t_{1}}^{t_{0}}\rho_{k_{1};t_{1}}\tau_{k_{1};t_{1}}\dots\rho_{k_{r};t_{r}}\tau_{k_{r};t_{r}}$
$\displaystyle=\mathfrak{q}_{k_{0};t_{0}}\mathfrak{q}_{k_{1};t_{1}}\tau_{t_{1}}^{t_{0}}\tau_{k_{1};t_{1}}\dots\rho_{k_{r};t_{r}}\tau_{k_{r};t_{r}}$
$\displaystyle=\mathfrak{q}_{k_{0};t_{0}}\mathfrak{q}_{k_{1};t_{1}}\tau_{t_{2}}^{t_{0}}\rho_{k_{2};t_{2}}\dots\rho_{k_{r};t_{r}}\tau_{k_{r};t_{r}}$
$\displaystyle=\cdots$
$\displaystyle=\mathfrak{q}_{k_{0};t_{0}}\mathfrak{q}_{k_{1};t_{1}}\dots\mathfrak{q}_{k_{r};t_{r}}\tau_{t}^{t_{0}}$
$\displaystyle=\mathfrak{q}_{t}^{t_{0}}\tau^{t_{0}}_{t}.$

∎

We conclude that the automorphisms $\mathfrak{q}_{t}^{t_{0}}$ generate the
nontropical parts of $x$\- and $y$-variables in the following manner.

######  Theorem 5.7.

The following formulas hold:

(5.43) $\displaystyle\mathfrak{q}_{t}^{t_{0}}(x^{\mathbf{g}_{i;t}})$
$\displaystyle=x^{\mathbf{g}_{i;t}}F_{i;t}(\hat{\mathbf{y}}),$ (5.44)
$\displaystyle\mathfrak{q}_{t}^{t_{0}}(y^{\mathbf{c}_{i;t}})$
$\displaystyle=y^{\mathbf{c}_{i;t}}\prod_{j=1}^{n}F_{j;t}(\mathbf{y})^{b_{ji;t}},$
(5.45) $\displaystyle\mathfrak{q}_{t}^{t_{0}}(\hat{y}^{\mathbf{c}_{i;t}})$
$\displaystyle=\hat{y}^{\mathbf{c}_{i;t}}\prod_{j=1}^{n}F_{j;t}(\hat{\mathbf{y}})^{b_{ji;t}}.$

###### Proof.

The equalities (5.43) and (5.44) follow from Propositions 5.3 and 5.6. For
example, for $x$-variables,

(5.46)
$\displaystyle\mathfrak{q}_{t}^{t_{0}}(x^{\mathbf{g}_{i;t}})=\mathfrak{q}_{t}^{t_{0}}(\tau_{t}^{t_{0}}(x_{i;t}))=\mu_{t}^{t_{0}}(x_{i;t})=x^{\mathbf{g}_{i;t}}F_{i;t}(\hat{\mathbf{y}}).$

Similarly, the equality (5.45) follows from Propositions 5.4 and 5.6. ∎

###  5.3. Detropicalization

Let $\mathbf{\Sigma}$ be any cluster pattern of rank $n$, and let $t_{0}$ be a
given initial vertex. Let $S_{n}$ be the symmetric group of degree $n$. We
define the (left) action of a permutation $\nu\in S_{n}$ on $x$\- and
$y$-variables, etc., as follows.

(5.47) $\displaystyle\mathbf{x}^{\prime}$ $\displaystyle=\nu\mathbf{x}_{t},$
$\displaystyle\quad x^{\prime}_{i}$ $\displaystyle=x_{\nu^{-1}(i);t},$ (5.48)
$\displaystyle\mathbf{y}^{\prime}$ $\displaystyle=\nu\mathbf{x}_{t},$
$\displaystyle\quad y^{\prime}_{i}$ $\displaystyle=y_{\nu^{-1}(i);t},$ (5.49)
$\displaystyle B^{\prime}$ $\displaystyle=\nu B_{t},$ $\displaystyle\quad
b^{\prime}_{ij}$ $\displaystyle=b_{\nu^{-1}(i)\nu^{-1}(j);t},$ (5.50)
$\displaystyle C^{\prime}$ $\displaystyle=\nu C_{t},$ $\displaystyle\quad
c^{\prime}_{ij}$ $\displaystyle=c_{i\nu^{-1}(j);t},$ (5.51) $\displaystyle
G^{\prime}$ $\displaystyle=\nu G_{t},$ $\displaystyle\quad g^{\prime}_{ij}$
$\displaystyle=g_{i\nu^{-1}(j);t}.$

Note that these actions are compatible with the separation formulas in Theorem
2.23.

For any $t,t^{\prime}\in\mathbb{T}_{n}$ and any permutation $\nu\in S_{n}$, we
define the following isomorphisms.

(5.52)
$\displaystyle\nu_{t}^{t^{\prime}}:\mathbb{Q}(\mathbf{x}_{t})\rightarrow\mathbb{Q}(\mathbf{x}_{t^{\prime}}),\
\mathbb{Q}(\mathbf{y}_{t})\rightarrow\mathbb{Q}(\mathbf{y}_{t^{\prime}}),$
(5.53)
$\displaystyle\nu_{t}^{t^{\prime}}(x_{i;t})=x_{\nu^{-1}(i);t^{\prime}},$
(5.54)
$\displaystyle\nu_{t}^{t^{\prime}}(y_{i;t})=y_{\nu^{-1}(i);t^{\prime}}.$

This is now a right action of $S_{n}$. Then, by Proposition 5.3, one can
rephrase the periodicity condition for $G$-matrices and $x$-variables as
follows:

(5.55) $\displaystyle G_{t}^{t_{0}}=\nu G_{t^{\prime}}^{t_{0}}$
$\displaystyle\Longleftrightarrow\tau_{t}^{t_{0}}=\tau_{t^{\prime}}^{t_{0}}\circ\nu_{t}^{t^{\prime}}\
\mbox{(for $x$-variables)},$ (5.56)
$\displaystyle\mathbf{x}_{t}=\nu\mathbf{x}_{t^{\prime}}$
$\displaystyle\Longleftrightarrow\mu_{t}^{t_{0}}=\mu_{t^{\prime}}^{t_{0}}\circ\nu_{t}^{t^{\prime}}\
\mbox{(for $x$-variables)}.$

Also, the parallel statement holds for $C$-matrices and $y$-variables.

######  Proposition 5.8.

Let $t_{0},t,t^{\prime}$ be any vertices in $\mathbb{T}_{n}$. Suppose that

(5.57) $\displaystyle G_{t}^{t_{0}}=\nu G_{t^{\prime}}^{t_{0}}$

holds. Then, we have

(5.58) $\displaystyle\mathbf{x}_{t}=\nu\mathbf{x}_{t^{\prime}}$
$\displaystyle\Longleftrightarrow\mathfrak{q}_{t}^{t_{0}}=\mathfrak{q}_{t^{\prime}}^{t_{0}}\
\mbox{\rm(for $x$-variables)}.$

Also, the parallel statement holds for $C$-matrices and $y$-variables.

###### Proof.

The claim follows from (5.55), (5.56), and Proposition 5.6. ∎

Note that the permutation $\nu_{t}^{t^{\prime}}$ drops off in the condition
for $\mathfrak{q}_{t}^{t_{0}}$ in (5.58).

The properties

(5.59) $\displaystyle G_{t}^{t_{0}}=\nu G_{t^{\prime}}^{t_{0}}$
$\displaystyle\Longrightarrow\mathbf{x}_{t}=\nu\mathbf{x}_{t^{\prime}},$
(5.60) $\displaystyle C_{t}^{t_{0}}=\nu C_{t^{\prime}}^{t_{0}}$
$\displaystyle\Longrightarrow\mathbf{y}_{t}=\nu\mathbf{y}_{t^{\prime}}$

are called the detropicalization of $x$\- and $y$-variables, respectively.
They are the core of the synchronicity phenomenon occurring in cluster
patterns systematically studied in [Nak20].

######  Remark 5.9.

It is known that the detropicalizations (5.59) and (5.60) always hold [CHL20,
Lemma 2.4 & Theorem 2.5], [Nak20, Theorem 5.2]. However, the proofs therein
depend on the Laurent positivity in Theorem 2.37, which we temporarily
dismiss. We come back to the point later in Section 8.1.

##  6\. Scattering diagrams

In this section we present a precise formulation of scattering diagrams
following [GHK15, GHKK18]. To do this, we set the following guideline.

  * (a).

Since in our approach we do not need scattering diagrams with frozen variables
(corresponding to coefficients of geometrical type in [FZ07]), we skip the
relevant formulation. This makes a considerable simplification of the
formulation.

  * (b).

We omit the description of the underlying Lie algebra $\mathfrak{g}$ and Lie
group $G$ in [GHKK18] for scattering diagrams. They are crucial for the
construction of scattering diagrams, but we can skip them for our purpose.

  * (c).

We clarify and establish the connection between cluster patterns and
scattering diagrams in detail by step-by-step examples.

  * (d).

We mostly follow the notations and the conventions in [GHKK18]. However, we
take the transpose of the skew-symmetric forms in [GHKK18] to match the
convention of the exchange matrices in [FZ07].

###  6.1. Fixed data and seed

######  Definition 6.1 (Fixed data).

A fixed data $\Gamma$ consists of the following:

  * •

A lattice $N\simeq\mathbb{Z}^{n}$ of rank $n$ with a skew-symmetric bilinear
form

(6.1) $\displaystyle\\{\cdot,\cdot\\}:N\times N\rightarrow\mathbb{Q}.$

  * •

Mutually coprime positive integers $d_{1}$, …, $d_{n}$.

  * •

A sublattice $N^{\circ}\subset N$ of finite index such that

(6.2) $\displaystyle\\{N^{\circ},N\\}\subset\mathbb{Z}.$

  * •

$M=\mathrm{Hom}(N,\mathbb{Z})$,
$M^{\circ}=\mathrm{Hom}(N^{\circ},\mathbb{Z})$.

######  Remark 6.2.

In (6.2) we already took the transpose of the form in [GHKK18] as mentioned in
the above guideline (d).

Let $N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$ and