Duality between Final-Seed and Initial-Seed Mutations in Cluster Algebras

Duality between Final-Seed and Initial-Seed
Mutations in Cluster Algebras

Shogo FUJIWARA and Yasuaki GYODA

S. Fujiwara and Y. Gyoda Graduate School of Mathematics, Nagoya University,
Chikusa-ku, Nagoya, 464-8602 Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received October 16, 2018, in final form May 10, 2019; Published online May
15, 2019

We study the duality between the mutations and the initial-seed mutations in
cluster algebras, where the initial-seed mutations are the transformations of
rational expressions of cluster variables in terms of the initial cluster
under the change of the initial cluster. In particular, we define the maximal
degree matrices of the $F$-polynomials called the $F$-matrices and show that
the $F$-matrices have the self-duality which is analogous to the duality
between the $C$\- and $G$-matrices.

cluster algebra; mutation; duality

13F60

##  1 Introduction

_Cluster algebras_ are a class of commutative algebras introduced by [5],
which are generated by some distinguished elements called the _cluster
variables_. The cluster variables are given by applying the _mutations_
repeatedly starting from the initial cluster variables. Through the
description of mutations as transformations on quivers or triangulations, the
cluster algebras are applied in various areas of mathematics such that
representation theory of quivers [1], hyperbolic geometry [3, 4], etc.

Let us briefly recall the notion of mutations, which is essential in cluster
algebra theory. For simplicity, we consider here a cluster pattern _without
coefficients_ [5]. Let $\mathbb{T}_{n}$ be the $n$-regular tree. Then, for
each vertex $t\in\mathbb{T}_{n}$, cluster variables
$\mathbf{x}_{t}=(x_{1;t},\dots,x_{n;t})$ are attached such that for any
adjacent vertices $t$ and $t^{\prime}$ of $\mathbb{T}_{n}$, the variables
$\mathbf{x}_{t}$ and $\mathbf{x}^{\prime}_{t}$ are related by a rational
transformation called a _mutation_. Let $t_{0}\in\mathbb{T}_{n}$ be a given
point in $\mathbb{T}_{n}$ called the _initial point_. Then, repeating
mutations starting from the initial variables, we have the expression

$\displaystyle x_{i;t}=\mathcal{X}^{t_{0}}_{i;t}(\mathbf{x}_{t_{0}}),$

where the rational function $\mathcal{X}^{t_{0}}_{i;t}$ depends on $t$ and
$t_{0}$. For a vertex $t^{\prime}$ which is adjacent to $t$ in
$\mathbb{T}_{n}$, the mutation transforms the function
$\mathcal{X}^{t_{0}}_{i;t}$ to the function
$\mathcal{X}^{t_{0}}_{i;t^{\prime}}$. On the other hand, for a vertex $t_{1}$
which is adjacent to $t_{0}$ in $\mathbb{T}_{n}$, the mutation also transforms
the function $\mathcal{X}^{t_{0}}_{i;t}$ to the function
$\mathcal{X}^{t_{1}}_{i;t}$. We call this transformation the _initial-seed
mutations_. (In contrast to the initial-seed mutations, we call the ordinary
mutations also as the _final-seed mutations_ in this paper.)

Thanks to the _separation formulas_ of [6], the cluster variables and the
coefficients are described by the _$C$ -matrices_, the _$G$ -matrices_, and
the _$F$ -polynomials_, where the $C$\- and $G$-matrices are the “ _tropical
part_ ”, while the $F$-polynomials are the “ _nontropical part_ ”. The final-
seed and initial-seed mutations of the cluster variables and the coefficients
reduce to the ones for the $C$\- and $G$-matrices and the $F$-polynomials [6].
In [8], the duality between the $C$\- and $G$-matrices under the final-seed
and initial-seed mutations was established (under the assumption of the sign-
coherence of the $C$-matrices, which is established in [7]). In [9], an
analogous duality was established also for the _denominator vectors_ for some
classes of cluster algebras. In this paper, we treat the nontropical part of
the seeds, namely the $F$-polynomials. Since the $F$-polynomials are rather
complicated, we consider the “degree matrices” of $F$-polynomials (_F-
matrices_). Under applying the sign-coherence of the $C$-matrices, a column
vectors of the $F$-matrices correspond to the exponent vector of the unique
monomial with maximal degree of a $F$-polynomials. Then, we establish the
duality of those matrices under the final-seed and initial-seed mutations,
which is parallel to the ones in [8, 9]. This is the main result of the paper.
Also, in a process of proving the main result, we have a nice relation between
the $C$-, $G$-, $F$-matrices under the exchange of the initial matrix $B$ and
$-B$.

We also exhibit the final-seed and initial-seed mutation formulas of the $C$\-
and $G$-matrices systematically, some of which are new in the literature. We
stress that the duality of the final-seed and initial-seed mutations becomes
manifest only after applying the sign-coherence of the $C$-matrices [7].

The organisation of the paper is as follows: In Section 2, we start with
recalling the definitions of the seed mutations and the cluster patterns
according to [6] and give the final-seed mutations. Also, using the sign-
coherence of the $C$-matrices, we obtain a reduced form of the final-seed
mutations. The formulas involving the $F$-matrices are new results. In Section
3, the initial-seed mutation formulas are given using the _$H$ -matrices_ of
[6]. By sign-coherence of the $C$-matrices and the duality of the $C$\- and
$G$-matrices of [8], we prove the conjecture [6, Conjecture 6.10], where the
$H$-matrices are expressed by the $G$-matrices. Using this result, we obtain
the duality of the $F$-matrices. In Section 4, we consider the principal
extension of the exchange matrices and present some properties. We show that
these properties give alternative derivations of some equalities in previous
sections.

##  2 Final-seed mutations

###  2.1 Seed mutations and cluster patterns

We start with recalling the definitions of the seed mutations and the cluster
patterns according to [6]. See [6] for more information. A _semifield_
$\mathbb{P}$ is an abelian multiplicative group equipped with an addition
$\oplus$ which is distributive over the multiplication. We particularly make
use of the following two semifields.

Let $\mathbb{Q}_{\text{sf}}(u_{1},\dots,u_{\ell})$ be a set of rational
functions in $u_{1},\dots,u_{\ell}$ which have subtraction-free expressions.
Then, $\mathbb{Q}_{\text{sf}}(u_{1},u_{2},\dots,u_{\ell})$ is a semifield by
the usual multiplication and addition. We call it the _universal semifield_ of
$u_{1},\dots,u_{\ell}$ [6, Definition 2.1].

Let Trop$(u_{1},\dots,u_{\ell})$ be the abelian multiplicative group freely
generated by the elements $u_{1},\dots,u_{\ell}$. Then,
$\text{Trop}(u_{1},u_{2},\dots,u_{\ell})$ is a semifield by the following
addition:

$\displaystyle\prod_{j=1}^{\ell}u_{j}^{a_{j}}\oplus\prod_{j=1}^{\ell}u_{j}^{b_{j}}=\prod_{j=1}^{\ell}u_{j}^{\min(a_{j},b_{j})}.$

We call it the _tropical semifield_ of $u_{1},\dots,u_{\ell}$ [6, Definition
2.2]. We note that the universal semifield
$\mathbb{Q}_{\text{sf}}(u_{1},\dots,u_{\ell})$ has the following universality
property. For any semifield $\mathbb{P}$ and
$p_{1},\dots,p_{\ell}\in\mathbb{P}$, there exists an unique semifield
homomorphism $\pi$ such that

$\displaystyle\pi\colon\ \mathbb{Q}_{\text{sf}}(y_{1},\dots,y_{\ell})$
$\displaystyle\longrightarrow\mathbb{P},$ $\displaystyle y_{i}$
$\displaystyle\longmapsto p_{i}.$

For $F(y_{1},\dots,y_{\ell})\in\mathbb{Q}_{\text{sf}}(y_{1},\dots,y_{\ell})$,
we denote

$\displaystyle
F|_{\mathbb{P}}(p_{1},\dots,p_{\ell}):=\pi(F(y_{1},\dots,y_{\ell}))$

and we call it the _evaluation_ of $F$ at $p_{1},\dots,p_{\ell}$.

We fix a positive integer $n$ and a semifield $\mathbb{P}$. Let $\mathbb{ZP}$
be the group ring of $\mathbb{P}$ as a multiplicative group. Since
$\mathbb{ZP}$ is a domain [5, Section 5], its total quotient ring is a field
$\mathbb{Q}(\mathbb{P})$. Let $\mathcal{F}$ be the field of the rational
functions in $n$ indeterminates with coefficients in $\mathbb{Q}(\mathbb{P})$.

A _labeled seed with coefficients in $\mathbb{P}$_ is a triplet
$(\mathbf{x},\mathbf{y},B)$, where

  * •

$\mathbf{x}=(x_{1},\dots,x_{n})$ is an $n$-tuple of elements of $\mathcal{F}$
forming a free generating set of $\mathcal{F}$.

  * •

$\mathbf{y}=(y_{1},\dots,y_{n})$ is an $n$-tuple of elements of $\mathbb{P}$.

  * •

$B=(b_{ij})$ is an $n\times n$ integer matrix which is _skew-symmetrizable_ ,
that is, there exists a positive diagonal matrix $D$ such that $DB$ is skew-
symmetric. We call $D$ a _skew-symmetrizer_ of $B$.

We refer to $x_{i}$, $y_{i}$ and $B$ as the _cluster variables_ , the
_coefficients_ and the _exchange matrix_ , respectively.

Throughout the paper, for an integer $b$, we use the notation
$[b]_{+}=\max(b,0)$. We note that

$\displaystyle b=[b]_{+}-[-b]_{+}.$ (2.1)

Let $(\mathbf{x},\mathbf{y},B)$ be a labeled seed with coefficients in
$\mathbb{P}$, and let $k\in\\{1,\dots,n\\}$. The _seed mutation $\mu_{k}$ in
direction $k$_ transforms $(\mathbf{x},\mathbf{y},B)$ into another labeled
seed
$\mu_{k}(\mathbf{x},\mathbf{y},B)=(\mathbf{x^{\prime}},\mathbf{y^{\prime}},B^{\prime})$
defined as follows:

  * •

The entries of $B^{\prime}=(b^{\prime}_{ij})$ are given

$\displaystyle b^{\prime}_{ij}=\begin{cases}-b_{ij}&\text{if $i=k$ or
$j=k$,}\\\
b_{ij}+\left[b_{ik}\right]_{+}b_{kj}+b_{ik}\left[-b_{kj}\right]_{+}&\text{otherwise.}\end{cases}$
(2.2)

  * •

The coefficients $\mathbf{y^{\prime}}=(y^{\prime}_{1},\dots,y^{\prime}_{n})$
are given

$\displaystyle y^{\prime}_{j}=\begin{cases}y_{k}^{-1}&\text{if $j=k$,}\\\
y_{j}y_{k}^{[b_{kj}]_{+}}(y_{k}\oplus
1)^{-b_{kj}}&\text{otherwise.}\end{cases}$ (2.3)

  * •

The cluster variables
$\mathbf{x^{\prime}}=(x^{\prime}_{1},\dots,x^{\prime}_{n})$ are given

$\displaystyle
x^{\prime}_{j}=\begin{cases}\dfrac{y_{k}\mathop{\prod}\limits_{i=1}^{n}x_{i}^{[b_{ik}]_{+}}+\mathop{\prod}\limits_{i=1}^{n}x_{i}^{[-b_{ik}]_{+}}}{(y_{k}\oplus
1)x_{k}}&\text{if $j=k$,}\\\ x_{j}&\text{otherwise.}\end{cases}$ (2.4)

We remark that (2.3) can be also expressed as follows:

$\displaystyle y^{\prime}_{j}=\begin{cases}y_{k}^{-1}&\text{if $j=k$,}\\\
y_{j}y_{k}^{[-b_{kj}]_{+}}\big{(}{y_{k}}^{-1}\oplus
1\big{)}^{-b_{kj}}&\text{otherwise.}\end{cases}$ (2.5)

Let $\mathbb{T}_{n}$ be the _$n$ -regular tree_ whose edges are labeled by the
numbers $1,\dots,n$ such that the $n$ edges emanating from each vertex have
different labels. We write
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ to indicate that
vertices $t,t^{\prime}\in\mathbb{T}_{n}$ are joined by an edge labeled by $k$.
We fix an arbitrary vertex $t_{0}\in\mathbb{T}_{n}$, which is called the
_initial vertex_.

A _cluster pattern with coefficients in $\mathbb{P}$_ is an assignment of a
labeled seed $\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})$ with
coefficients in $\mathbb{P}$ to every vertex $t\in\mathbb{T}_{n}$ such that
the seeds $\Sigma_{t}$ and $\Sigma_{t^{\prime}}$ assigned to the endpoints of
any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ are obtained
from each other by the seed mutation in direction $k$. The elements of
$\Sigma_{t}$ are denoted as follows:

$\displaystyle\mathbf{x}_{t}=(x_{1;t},\dots,x_{n;t}),\qquad\mathbf{y}_{t}=(y_{1;t},\dots,y_{n;t}),\qquad
B_{t}=(b_{ij;t}).$ (2.6)

In particular, at $t_{0}$, we denote

$\displaystyle\mathbf{x}=\mathbf{x}_{t_{0}}=(x_{1},\dots,x_{n}),\qquad\mathbf{y}=\mathbf{y}_{t_{0}}=(y_{1},\dots,y_{n}),\qquad
B=B_{t_{0}}=(b_{ij}).$ (2.7)

We say that $n$ is _rank_ and the $\mathbb{P}$ is _coefficient semifield_ of
the cluster patterns.

If $\mathbb{P}=\text{Trop}(y_{1},\dots,y_{n})$ and
$\mathbf{y}_{t_{0}}=(y_{1},\dots,y_{n})$, we say that a cluster pattern
$\\{(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\\}_{t\in\mathbb{T}_{n}}$ has
_principal coefficients_ at the initial vertex $t_{0}$.

###  2.2 Final-seed mutations without sign-coherence of $\bm{C}$-matrices

In this subsection, we will define the $C$-matrices, the $G$-matrices and the
$F$-polynomials following [6]. We also introduce the $F$-matrices which are
new.

Throughout this paper, we use the following notations [8]. Let $J_{\ell}$
denote the $n\times n$ diagonal matrix obtained from the identity matrix
$I_{n}$ by replacing the $(\ell,\ell)$ entry with $-1$. For a $n\times n$
matrix $B=(b_{ij})$, let $[B]_{+}$ be the matrix obtained from $B$ by
replacing every entry $b_{ij}$ with $[b_{ij}]_{+}$. Also, let $B^{k\bullet}$
be the matrix obtained from $B$ by replacing all entries outside of the $k$th
row with zeros. Similarly, let $B^{\bullet k}$ be the matrix replacing all
entries outside of the $k$th column. Note that the maps $B\mapsto[B]_{+}$ and
$B\mapsto B^{k\bullet}$ commute with each other, and the same is true of
$B\mapsto[B]_{+}$ and $B\mapsto B^{\bullet k}$, so that the notations
$[B]^{k\bullet}_{+}$ and $[B]^{\bullet k}_{+}$ make sense. Also, we have
$AB^{\bullet k}=(AB)^{\bullet k}$ and $A^{k\bullet}B=(AB)^{k\bullet}$.

First, we recall the recursive definitions of the $C$-matrices, the
$G$-matrices and the $F$-polynomials, following [6]. For any initial exchange
matrix $B$ at $t_{0}$, let $\\{B_{t}\\}_{t\in\mathbb{T}_{n}}$ be the family of
the exchange matrices in (2.6) with $B_{t_{0}}=B$.

######  Definition 2.1 ([6, equation (5.9), Proposition 6.6]).

Let $B$ be any initial exchange matrix at $t_{0}$. Then, the families of
$n\times n$ integer matrices
$\big{\\{}C_{t}^{B;t_{0}}\big{\\}}_{t\in\mathbb{T}_{n}}$ and
$\big{\\{}G_{t}^{B;t_{0}}\big{\\}}_{t\in\mathbb{T}_{n}}$ are recursively
defined as follows:

  * (i)

We set the initial condition,

$\displaystyle C^{B;t_{0}}_{t_{0}}=I_{n},$

and for any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$
in $\mathbb{T}_{n}$ and $\varepsilon\in\\{\pm 1\\}$, we set the recurrence
relation,

$\displaystyle
C^{B;t_{0}}_{t^{\prime}}=C^{B;t_{0}}_{t}\big{(}J_{\ell}+[\varepsilon
B_{t}]^{\ell\bullet}_{+}\big{)}+\big{[}{-}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}B_{t}.$ (2.8)

(In (2.8), it does not make any difference whichever we choose,
$\varepsilon=1$ or $\varepsilon=-1$. See Remark 2.2.)

  * (ii)

We set the initial condition,

$\displaystyle G^{B;t_{0}}_{t_{0}}=I_{n},$

and for any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$
in $\mathbb{T}_{n}$ and $\varepsilon\in\\{\pm 1\\}$, we set the recurrence
relation,

$\displaystyle
G^{B;t_{0}}_{t^{\prime}}=G^{B;t_{0}}_{t}\big{(}J_{\ell}+[\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}-B\big{[}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}.$ (2.9)

The matrices $C^{B;t_{0}}_{t}$ and $G^{B;t_{0}}_{t}$ are called the _$C$
-matrix_ and the _$G$ -matrix_ at $t$.

######  Remark 2.2.

Because of (2.1), the right hand side of (2.8) does not depend on
$\varepsilon$. Meanwhile, the right hand side of (2.9) does not depend on
$\varepsilon$ due to the following equality [6, equation (6.14)]:

$\displaystyle G^{B;t_{0}}_{t}B_{t}=BC^{B;t_{0}}_{t}.$ (2.10)

We have two different expressions for the recursion (2.8) and (2.9) because
they will be useful in different situations.

######  Definition 2.3 ([6, Proposition 5.1]).

Let $B$ be any initial exchange matrix at $t_{0}$ and
$\mathbf{y}=(y_{1},\dots,y_{n})$ be formal variables. Then, the family of
polynomials $F^{B;t_{0}}_{j;t}(\mathbf{y})\in\mathbb{Z}[y_{1},\dots,y_{n}]$
indexed by $j\in\\{1,\dots,n\\}$ and $t\in\mathbb{T}_{n}$ are recursively
defined as follows: We set the initial condition,

$\displaystyle F^{B;t_{0}}_{j;t_{0}}(\mathbf{y})=1,\qquad j=1,\dots,n,$ (2.11)

and for any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$
in $\mathbb{T}_{n}$, we set the recurrence relation,

$\displaystyle
F^{B;t_{0}}_{j;t^{\prime}}(\mathbf{y})=\begin{cases}\displaystyle
F^{B;t_{0}}_{\ell;t}(\mathbf{y})^{-1}\left(\prod_{i=1}^{n}y_{i}^{[c^{B;t_{0}}_{i\ell;t}]_{+}}\prod_{i=1}^{n}F^{B;t_{0}}_{i;t}(\mathbf{y})^{[b_{i\ell;t}]_{+}}\right.\\\
\qquad\left.\displaystyle{}+\prod_{i=1}^{n}y_{i}^{[-c^{B;t_{0}}_{i\ell;t}]_{+}}\prod_{i=1}^{n}F^{B;t_{0}}_{i;t}(\mathbf{y})^{[-b_{i\ell;t}]_{+}}\right)\quad&\text{if}\
j=\ell,\\\ F^{B;t_{0}}_{j,t}(\mathbf{y})&\text{if}\ j\neq\ell,\end{cases}$
(2.12)

where $c^{B;t_{0}}_{i\ell;t}$ is the $(i,\ell)$ entry of $C^{B;t_{0}}_{t}$ and
$b_{i\ell;t}$ is the $(i,\ell)$ entry of $B_{t}$. The polynomials
$F^{B;t_{0}}_{j,t}(\mathbf{y})$ are called the _$F$ -polynomials_ at $t$.

The fact that seemingly rational functions $F^{B;t_{0}}_{j;t}(\mathbf{y})$ are
polynomials follows from the Laurent phenomenon of cluster variables [6,
Proposition 3.6]. We also remark that for any $j\in\\{1,\dots,n\\}$ and
$t\in\mathbb{T}_{n}$, $F^{B;t_{0}}_{j;t}(\mathbf{y})$ is an element of
$\mathbb{Q}_{\text{sf}}(y_{1},\dots,y_{n})$ because of (2.11) and (2.12).

In this paper, we refer to the recurrence relations (“mutations”) (2.8), (2.9)
and (2.12) as the _final-seed mutations_ in contrast to the initial-seed
mutations appearing later. Abusing of notation, we denote
$C_{t^{\prime}}^{B;{t_{0}}}=\mu_{\ell}\big{(}C_{t}^{B;{t_{0}}}\big{)}$,
$G_{t^{\prime}}^{B;{t_{0}}}=\mu_{\ell}\big{(}G_{t}^{B;{t_{0}}}\big{)}$, and
$F_{j:t^{\prime}}^{B;{t_{0}}}(\mathbf{y})=\mu_{\ell}\big{(}F_{j;t}^{B;{t_{0}}}(\mathbf{y})\big{)}$.

The $C$-matrices, the $G$-matrices, and the $F$-polynomials are important
because they factorize cluster variables and coefficients by the following
formulas:

######  Proposition 2.4 (separation formulas [6, Proposition 3.13, Corollary
6.3]).

Let $\\{\Sigma_{t}\\}_{t\in\mathbb{T}_{n}}$ be a cluster pattern with
coefficient in $\mathbb{P}$ with the initial seed (2.7). Then, for any
$t\in\mathbb{T}_{n}$ and $j\in\\{1,\dots,n\\}$, we have

$\displaystyle
x_{j;t}=\left(\prod_{k=1}^{n}x_{k}^{g^{B;t_{0}}_{kj;t}}\right)\frac{F_{j;t}^{B;t_{0}}|_{\mathcal{F}}(\hat{y}_{1},\dots,\hat{y}_{n})}{F_{j;t}^{B;t_{0}}|_{\mathbb{P}}(y_{1},\dots,y_{n})},$
(2.13) $\displaystyle
y_{j;t}=\prod_{k=1}^{n}y_{k}^{c^{B;t_{0}}_{kj;t}}\prod_{k=1}^{n}\big{(}F_{k;t}^{B;t_{0}}|_{\mathbb{P}}(y_{1},\dots,y_{n})\big{)}^{b_{kj;t}},$
(2.14)

where

$\displaystyle\hat{y}_{i}=y_{i}\mathop{\prod}\limits_{j=1}^{n}x_{j}^{b_{ji}},$

and $g_{ij;t}^{B;t_{0}}$ and $c_{ij;t}^{B;t_{0}}$ are the $(i,j)$ entry of
$G_{t}^{B;t_{0}}$ and $C_{t}^{B;t_{0}}$, respectively. Also, the rational
function $F_{j;t}^{B;t_{0}}|_{\mathcal{F}}(\hat{y}_{1},\dots,\hat{y}_{n})$ is
the element of $\mathcal{F}$ obtained by substituting $\hat{y}_{i}$ for
$y_{i}$ in $F_{j;t}^{B;t_{0}}(y_{1},\dots,y_{n})$.

The following fact is well-known:

######  Proposition 2.5.

  1. $(1)$

[6, equation (5.5)] For any $j\in\\{1,\dots,n\\}$ and $t\in\mathbb{T}_{n}$, we
have

$\displaystyle F_{j;t}^{B;t_{0}}|_{\text{\rm
Trop}(y_{1},\dots,y_{n})}(y_{1},\dots,y_{n})=1.$ (2.15)

  2. $(2)$

[6, equation (2.13)] Let $\\{\Sigma_{t}\\}_{t\in\mathbb{T}_{n}}$ be a cluster
pattern which has principal coefficients at $t_{0}$. Then, for any
$j\in\\{1,\dots,n\\}$ and $t\in\mathbb{T}_{n}$, we have

$\displaystyle y_{j;t}=\prod_{k=1}^{n}y_{k}^{c^{B;t_{0}}_{kj;t}}.$ (2.16)

###### Proof.

Because of (2.11) and (2.12), by using the tropicalization

$\displaystyle\pi\colon\ \mathbb{Q}_{\text{sf}}(y_{1},\dots,y_{n})$
$\displaystyle\longrightarrow\text{Trop}(y_{1},\dots,y_{n}),$ $\displaystyle
y_{i}$ $\displaystyle\longmapsto y_{i},$

we have (2.15). Moreover, setting $\mathbb{P}=\text{Trop}(y_{1},\dots,y_{n})$
in (2.14), we obtain (2.16) by (2.15). ∎

Next, we introduce another family of matrices, which are the “degree matrices”
of the $F$-polynomials.

######  Definition 2.6.

Let $B$ be any initial exchange matrix at $t_{0}$. For $i\in\\{1,\dots,n\\}$
and $t\in\mathbb{T}_{n}$, let $f^{B;t_{0}}_{1i;t},\dots,f^{B;t_{0}}_{ni;t}$ be
the maximal degrees of $y_{1},\dots,y_{n}$ in the $i$th $F$-polynomial
$F^{B;t_{0}}_{i;t}(y_{1},\dots,y_{n})$, respectively. We call the nonnegative
integer vectors
$\mathbf{f}^{B;t_{0}}_{i;t}=\left[\begin{smallmatrix}f_{1i;t}^{B;t_{0}}\\\
\vdots\\\ f_{ni;t}^{B;t_{0}}\end{smallmatrix}\right]$ the _$\mathbf{f}$
-vectors_ at $t$. We also call the nonnegative integer $n\times n$ matrix
$F_{t}^{B;t_{0}}$ with columns
$\mathbf{f}_{1;t}^{B;t_{0}},\dots,\mathbf{f}_{n;t}^{B;t_{0}}$ the _$F$
-matrix_ at $t$.

To avoid confusion the notation of $F$-matrices and $F$-polynomials, when we
write $F$-polynomials, we always write it with arguments.

We have the following description of the $F$-matrices: Consider a semifield
homomorphism

$\displaystyle\pi\colon\ \mathbb{Q}_{\text{sf}}(y_{1},\dots,y_{n})$
$\displaystyle\longrightarrow\text{Trop}\big{(}y_{1}^{-1},\dots,y_{n}^{-1}\big{)},$
$\displaystyle y_{i}$ $\displaystyle\longmapsto y_{i}.$

Then, we have

$\displaystyle\pi\big{(}F^{B;t_{0}}_{\ell;t}(y_{1},\dots,y_{n})\big{)}=F^{B;t_{0}}_{\ell;t}|_{\text{Trop}\left(y_{1}^{-1},\dots,y_{n}^{-1}\right)}(y_{1},\dots,y_{n})$
$\displaystyle\hphantom{\pi\big{(}F^{B;t_{0}}_{\ell;t}(y_{1},\dots,y_{n})\big{)}}{}=\prod_{i=1}^{n}\big{(}y_{i}^{-1}\big{)}^{-f_{i\ell;t}^{B;t_{0}}}=\prod_{i=1}^{n}y_{i}^{f_{i\ell;t}^{B;t_{0}}}.$
(2.17)

The $F$-matrices are uniquely determined by the following recurrence
relations:

######  Proposition 2.7.

Let $B$ be any initial exchange matrix at $t_{0}$. Then, the $F$-matrices have
the following recurrence: The initial condition is

$\displaystyle F^{B;t_{0}}_{t_{0}}={O_{n}},$ (2.18)

where $O_{n}$ is the zero matrix. For any edge
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$ in
$\mathbb{T}_{n}$, we have the recurrence relation,

$\displaystyle
F^{B;t_{0}}_{t^{\prime}}=F^{B;t_{0}}_{t}J_{\ell}+\max\big{(}\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+F^{B;t_{0}}_{t}[B_{t}]^{\bullet\ell}_{+},\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+F^{B;t_{0}}_{t}[-B_{t}]^{\bullet\ell}_{+}\big{)}.$
(2.19)

###### Proof.

Clearly, (2.18) is obtained by (2.11). Moreover, applying (2.17) to (2.12), we
have (2.19). ∎

We call the recurrence relation (2.19) the _final-seed mutations_ for the
$F$-matrices. As with the $C$-, $G$-matrices and the $F$-polynomials, we
denote $F^{B;t_{0}}_{t^{\prime}}=\mu_{\ell}\big{(}F^{B;t_{0}}_{t}\big{)}$.

Under the exchange of the initial exchange matrices $B$ and $-B$ at $t_{0}$,
we have the simple relations between the $C$-, $G$\- and $F$-matrices.

######  Theorem 2.8.

We have the following relations:

$\displaystyle C^{-B;t_{0}}_{t}=C^{B;t_{0}}_{t}+F^{B;t_{0}}_{t}B_{t},$ (2.20)
$\displaystyle G_{t}^{-B;t_{0}}=G_{t}^{B;t_{0}}+BF_{t}^{B;t_{0}},$ (2.21)
$\displaystyle F_{t}^{-B;t_{0}}=F_{t}^{B;t_{0}}.$ (2.22)

###### Proof.

Let
$\big{\\{}\Sigma_{t}^{B}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\big{\\}}_{t\in\mathbb{T}_{n}}$
be a cluster pattern with coefficients in any semifield $\mathbb{P}$ with the
initial seed $(\mathbf{x},\mathbf{y},B)$. Also, let
$\big{\\{}\Sigma_{t}^{-B}=(\mathbf{x}^{\prime}_{t},\mathbf{y}^{\prime}_{t},B^{\prime}_{t})\big{\\}}_{t\in\mathbb{T}_{n}}$
be a cluster pattern with coefficients in $\mathbb{P}$ with the initial seed
$\big{(}\mathbf{x},\mathbf{y}^{-1},-B\big{)}$. Then, by the definition of the
mutation (2.2), (2.3) and (2.4), we have [6, Proof of Proposition 5.3]

$\displaystyle\mathbf{x}^{\prime}_{t}=\mathbf{x}_{t},\qquad\mathbf{y}^{\prime}_{t}=\mathbf{y}_{t}^{-1},\qquad
B^{\prime}_{t}=-B_{t}.$ (2.23)

We also note that for the initial seed
$\Sigma^{-B}_{t_{0}}=(\mathbf{x}^{\prime},\mathbf{y}^{\prime},B^{\prime})=\big{(}\mathbf{x},\mathbf{y}^{-1},-B\big{)}$,
we have

$\displaystyle\hat{y^{\prime}_{i}}:=y_{i}^{\prime}\prod_{j=1}^{n}{x^{\prime}_{j}}^{b_{ji}^{\prime}}=y_{i}^{-1}\prod_{j=1}^{n}{x_{j}}^{-b_{ji}}=\hat{y}_{i}^{-1}.$

Now we set $\mathbb{P}=\text{Trop}\big{(}y_{1}^{-1},\dots,y_{n}^{-1}\big{)}$
and apply the separation formulas (2.13) and (2.14) to (2.23). Then, we obtain

$\displaystyle\left(\prod_{k=1}^{n}x_{k}^{g^{-B;t_{0}}_{kj;t}}\right)F_{j;t}^{-B;t_{0}}|_{\mathcal{F}}\big{(}\hat{y}^{-1}_{1},\dots,\hat{y}^{-1}_{n}\big{)}$
$\displaystyle\qquad{}=\left(\prod_{k=1}^{n}x_{k}^{g^{B;t_{0}}_{kj;t}}\right)\frac{F_{j;t}^{B;t_{0}}|_{\mathcal{F}}(\hat{y}_{1},\dots,\hat{y}_{n})}{F_{j;t}^{B;t_{0}}|_{\text{Trop}\left(y^{-1}_{1},\dots,y^{-1}_{n}\right)}(y_{1},\dots,y_{n})},$
(2.24)
$\displaystyle\left(\prod_{k=1}^{n}\big{(}y^{-1}_{k}\big{)}^{c^{-B;t_{0}}_{kj;t}}\right)^{-1}=\prod_{k=1}^{n}y_{k}^{c^{B;t_{0}}_{kj;t}}\prod_{k=1}^{n}\big{(}F_{k;t}^{B;t_{0}}|_{\text{Trop}\left(y_{1}^{-1},\dots,y_{n}^{-1}\right)}(y_{1},\dots,y_{n})\big{)}^{b_{kj;t}},$
(2.25)

by (2.15) and (2.16). Applying (2.17) to (2.25), we obtain

$\displaystyle\mathbf{c}_{j;t}^{-B;t_{0}}=\mathbf{c}_{j;t}^{B;t_{0}}+\sum_{i=1}^{n}b_{ij;t}\mathbf{f}^{B;t_{0}}_{i;t},$

thus we have (2.20). To show (2.21) from (2.24), let us set the
$\mathbb{Z}^{n}$-gradings in $\mathbb{Z}[x_{1}^{\pm 1},\dots,x_{n}^{\pm
1},\allowbreak y_{1},\dots,y_{n}]$ as follows [6, equations (6.1) and (6.2)]:

$\displaystyle\deg(x_{i})=\mathbf{e}_{i},\qquad\deg(y_{i})=-{\mathbf{b}}_{i},$

where $\mathbf{e}_{i}$ is the $i$th column vector of $I_{n}$ and
${\mathbf{b}}_{i}$ is the $i$th column vector of $B$. Then, we have

$\displaystyle\deg(\hat{y}_{i})=0.$

Hence comparing the $\mathbb{Z}^{n}$-gradings of both sides of (2.24), we
obtain

$\displaystyle\mathbf{g}_{j;t}^{-B;t_{0}}=\mathbf{g}_{j;t}^{B;t_{0}}+\sum_{i=1}^{n}f^{B;t_{0}}_{ij;t}\mathbf{b}_{i}.$

Therefore, we have (2.21). Let us prove (2.22). Substituting $x_{i}=1$
$(i=1,\dots,n)$ for (2.24), we have [6, equation (5.6)]

$\displaystyle
F_{j;t}^{-B;t_{0}}\big{(}y^{-1}_{1},\dots,y^{-1}_{n}\big{)}=\frac{F_{j;t}^{B;t_{0}}(y_{1},\dots,y_{n})}{F_{j;t}^{B;t_{0}}|_{\text{Trop}\left(y^{-1}_{1},\dots,y^{-1}_{n}\right)}(y_{1},\dots,y_{n})}.$
(2.26)

Under the exchange of $B$ and $-B$ in (2.26), we also have

$\displaystyle
F_{j;t}^{B;t_{0}}(y_{1},\dots,y_{n})=\frac{F_{j;t}^{-B;t_{0}}\big{(}y^{-1}_{1},\dots,y^{-1}_{n}\big{)}}{F_{j;t}^{-B;t_{0}}|_{\text{Trop}(y_{1},\dots,y_{n})}\big{(}y^{-1}_{1},\dots,y^{-1}_{n}\big{)}}.$
(2.27)

Hence combining (2.26) with (2.27), we obtain

$\displaystyle
F_{j;t}^{-B;t_{0}}|_{\text{Trop}(y_{1},\dots,y_{n})}\big{(}y_{1}^{-1},\dots,y_{n}^{-1}\big{)}F_{j;t}^{B;t_{0}}|_{\text{Trop}\left(y_{1}^{-1},\dots,y_{n}^{-1}\right)}(y_{1},\dots,y_{n})=1.$
(2.28)

Comparing exponents of $y_{i}$ of both sides of (2.28), we have

$\displaystyle-\mathbf{f}_{j;t}^{-B;t_{0}}+\mathbf{f}_{j;t}^{B;t_{0}}=\mathbf{0},$

thus we obtain (2.22). ∎

Thanks to the relation (2.20), we have the following alternating expression of
the final-seed mutations of the $F$-matrices:

######  Proposition 2.9.

Let $\varepsilon\in\\{\pm 1\\}$. For any edge
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$ in
$\mathbb{T}_{n}$, the matrices $F^{B;t_{0}}_{t}$ and
$F^{B;t_{0}}_{t^{\prime}}$ are related by

$\displaystyle
F^{B;t_{0}}_{t^{\prime}}=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[-\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}+\big{[}{-}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+\big{[}\varepsilon
C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}.$ (2.29)

###### Proof.

Firstly, we prove the case of $\varepsilon=1$ in (2.29). By (2.1), (2.19) and
(2.20), we have

$\displaystyle F^{B;t_{0}}_{t^{\prime}}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[-B_{t}]^{\bullet\ell}_{+}\big{)}+\max\big{(}\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+F^{B;t_{0}}_{t}B_{t}^{\bullet\ell},\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}\big{)}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[-B_{t}]^{\bullet\ell}_{+}\big{)}+\max\big{(}\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+\big{(}C^{-B;t_{0}}_{t}\big{)}^{\bullet\ell}-\big{(}C^{B;t_{0}}_{t}\big{)}^{\bullet\ell},\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}\big{)}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[-B_{t}]^{\bullet\ell}_{+}\big{)}+\max\big{(}\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+\big{(}C^{-B;t_{0}}_{t}\big{)}^{\bullet\ell},\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}\big{)}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[-B_{t}]^{\bullet\ell}_{+}\big{)}+\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+\big{[}C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}$

as desired. Secondly, we prove the case of $\varepsilon=-1$ in (2.29). In the
same way as $\varepsilon=1$, we have

$\displaystyle F^{B;t_{0}}_{t^{\prime}}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[B_{t}]^{\bullet\ell}_{+}\big{)}+\max\big{(}\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+},\big{[}{-}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}-F^{B;t_{0}}_{t}B_{t}^{\bullet\ell}\big{)}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[B_{t}]^{\bullet\ell}_{+}\big{)}+\max\big{(}\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+},\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}-\big{(}C^{-B;t_{0}}_{t}\big{)}^{\bullet\ell}\big{)}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+[B_{t}]^{\bullet\ell}_{+}\big{)}+\big{[}C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+\big{[}{-}C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}$

as desired. ∎

###  2.3 Final-seed mutations with sign-coherence of $\bm{C}$-matrices

In this subsection, we reduce the final-seed mutation formulas by applying the
sign-coherence of the $C$-matrices.

######  Definition 2.10.

Let $A$ be an (not necessarily square) integer matrix. We say that $A$ is
_column sign-coherent_ (resp.  _row sign-coherent_) if for any column (resp.
row) of $A$, its entries are either all nonnegative, or all nonpositive, and
not all zero.

When $A$ is column sign-coherent (resp. row sign-coherent), we can define its
$\ell$th _column sign_ $\varepsilon_{\bullet\ell}(A)$ (resp.  _row sign_
$\varepsilon_{\ell\bullet}(A)$) as the sign of nonzero entries of the $\ell$th
column (resp. row) of $A$. We have the following fundamental and nontrivial
result:

######  Theorem 2.11 ([7, Corollary 5.5]).

For any initial exchange matrix $B$, every $C$-matrix $C_{t}^{B;t_{0}}$
$(t\in\mathbb{T}_{n})$ is column sign-coherent.

The column signs of the $C$-matrix $C_{t}^{B;t_{0}}$ are called the _tropical
signs_ due to (2.16). Using them, the following reduced expression of the
final-seed mutations of the $C$\- and $G$-matrices are obtained:

######  Proposition 2.12 ([8, Proposition 1.3]).

For any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$ in
$\mathbb{T}_{n}$, we have

$\displaystyle
C^{B;t_{0}}_{t^{\prime}}=C^{B;t_{0}}_{t}\big{(}J_{\ell}+\big{[}\varepsilon_{\bullet\ell}\big{(}C^{B;t_{0}}_{t}\big{)}B_{t}\big{]}^{\ell\bullet}_{+}\big{)},$
(2.30) $\displaystyle
G^{B;t_{0}}_{t^{\prime}}=G^{B;t_{0}}_{t}\big{(}J_{\ell}+\big{[}{-}\varepsilon_{\bullet\ell}(C^{B;t_{0}}_{t})B_{t}\big{]}^{\bullet\ell}_{+}\big{)}.$
(2.31)

They are obtained from (2.8) and (2.9) by setting
$\varepsilon=\varepsilon_{\bullet\ell}\big{(}C_{t}^{B;t_{0}}\big{)}$ and
$\varepsilon=-\varepsilon_{\bullet\ell}\big{(}C_{t}^{B;t_{0}}\big{)}$,
respectively.

The following fact is shown by [6]:

######  Proposition 2.13 ([6, Proposition 5.6]).

For any initial exchange matrix $B$, the following are equivalent:

  1. $(i)$

The sign-coherence of the $C$-matrices holds.

  2. $(ii)$

Every polynomial $F_{\ell;t}^{B;t_{0}}(\mathbf{y})$ has constant term $1$.

  3. $(iii)$

Every polynomial $F_{\ell;t}^{B;t_{0}}(\mathbf{y})$ has a unique monomial of
maximal degree. Furthermore, this monomial has coefficient $1$, and it is
divisible by all the other occurring monomials.

In proposition 2.13, the equivalence of (ii) and (iii) is proved by [6,
Proposition 5.3] (see [6, Conjectures 5.4 and 5.5]).

######  Remark 2.14.

In the definition of the (column) sign-coherence in [6], the nonzero vector
property of column vectors are not assumed. However, this property can be
easily recovered, since $\det C_{t}^{B;t_{0}}=\pm 1$ due to (2.30).

By Theorem 2.11 and Proposition 2.13, we have the following description of the
$\mathbf{f}$-vectors:

######  Corollary 2.15.

The $\mathbf{f}$-vector $\mathbf{f}_{i;t}^{B;t_{0}}$ is the exponent vector of
the unique monomial with maximal degree of $F_{i;t}^{B;t_{0}}(\mathbf{y})$. In
other words, the unique monomial with maximal degree of
$F_{i;t}^{B;t_{0}}(\mathbf{y})$ is given by $y_{1}^{f_{1i;t}^{B;t_{0}}}\cdots
y_{n}^{f_{ni;t}^{B;t_{0}}}$.

Now, let us give the reduced expression of the final-seed mutations of the
$F$-matrices by using the tropical signs.

######  Proposition 2.16.

For any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$ in
$\mathbb{T}_{n}$, we have

$\displaystyle F^{B;t_{0}}_{t^{\prime}}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+\big{[}\varepsilon_{\bullet\ell}\big{(}C^{-B;t_{0}}_{t}\big{)}B_{t}\big{]}^{\bullet\ell}_{+}\big{)}+\big{[}\varepsilon_{\bullet\ell}\big{(}C^{-B;t_{0}}_{t}\big{)}C^{B;t_{0}}_{t}\big{]}^{\bullet{\ell}}_{+}$
$\displaystyle=F^{B;t_{0}}_{t}\big{(}J_{\ell}+\big{[}{-}\varepsilon_{\bullet\ell}\big{(}C^{B;t_{0}}_{t}\big{)}B_{t}\big{]}^{\bullet\ell}_{+}\big{)}+\big{[}\varepsilon_{\bullet\ell}\big{(}C^{B;t_{0}}_{t}\big{)}C^{-B;t_{0}}_{t}\big{]}^{\bullet{\ell}}_{+}.$
(2.32)

###### Proof.

Substituting
$\varepsilon=-\varepsilon_{\bullet\ell}\big{(}C^{-B;t_{0}}_{t}\big{)}$ or
$\varepsilon=\varepsilon_{\bullet\ell}\big{(}C^{B;t_{0}}_{t}\big{)}$ for
(2.29), we obtain (2.32). ∎

##  3 Initial-seed mutations

###  3.1 Initial-seed mutations of functions $\bm{\mathcal{Y}}$ and
$\bm{\mathcal{X}}$

We introduce the concept of the _initial-seed mutations_ which appears in [6,
8, 9] in the following way. Let $\mathbb{Q}_{\text{sf}}(\mathbf{y})$ be the
universal semifield with formal variables $\mathbf{y}=(y_{1},\dots,y_{n})$ in
Section 2.1. Let $\\{\Sigma_{t}\\}_{t\in\mathbb{T}_{n}}$ be the cluster
pattern with coefficients in $\mathbb{Q}_{\text{sf}}(\mathbf{y})$ where the
initial coefficients $\mathbf{y}_{t_{0}}$ are taken as the above formal
variables $\mathbf{y}$. Then, recursively applying the mutations (2.3) or
(2.5) from the initial coefficients, $y_{i;t}$ are written as a rational
function of $\mathbf{y}$:

$\displaystyle
y_{i;t}=\mathcal{Y}_{i;t}^{B;t_{0}}(\mathbf{y})\in\mathbb{Q}_{\text{sf}}(\mathbf{y}).$

Similarly, recursively applying the mutations (2.4), $x_{i;t}\in\mathcal{F}$
are written as a rational function of the initial cluster variables
$\mathbf{x}_{t_{0}}=\mathbf{x}$ with coefficients in
$\mathbb{Q}(\mathbb{Q}_{\text{sf}}(\mathbf{y}))$:

$\displaystyle
x_{i;t}=\mathcal{X}_{i;t}^{B;t_{0}}(\mathbf{x})\in\mathbb{Q}(\mathbb{Q}_{\text{sf}}(\mathbf{y}))(\mathbf{x}).$

Then, for any cluster pattern $\\{\Sigma_{t}\\}_{t\in\mathbb{T}_{n}}$ with
coefficients in $\mathbb{P}$, we recover $x_{i;t}$ and $y_{i;t}$ by the
specialisation
$\pi\colon\mathbb{Q}_{\text{sf}}(\mathbf{y})\rightarrow\mathbb{P}$ with
$y_{i}$ setting to be the initial coefficients of $\Sigma_{t_{0}}$. Let
$t_{1}\in\mathbb{T}_{n}$ be the vertex with
$\textstyle{t_{0}}$$\textstyle{t_{1}}$$\scriptstyle{k}$and let
$B_{1}=\mu_{k}(B)$. Then, the rational functions
$\mathcal{Y}_{i;t}^{B;t_{0}}(\mathbf{y})$ and
$\mathcal{Y}_{i;t}^{B_{1};t_{1}}(\mathbf{y})$ are related by

$\displaystyle\mathcal{Y}_{i;t}^{B_{1};t_{1}}(\mathbf{y})=\rho_{k}\big{(}\mathcal{Y}_{i;t}^{B;t_{0}}(\mathbf{y})\big{)},$

where $\rho_{k}$ is the semifield automorphism of
$\mathbb{Q}_{\text{sf}}(\mathbf{y})$ defined by

$\displaystyle\rho_{k}\colon\ \mathbb{Q}_{\text{sf}}(\mathbf{y})$
$\displaystyle\rightarrow\mathbb{Q}_{\text{sf}}(\mathbf{y}),$ $\displaystyle
y_{j}$ $\displaystyle\mapsto\begin{cases}y_{k}^{-1}&\text{if $j=k$,}\\\
y_{j}y_{k}^{[b_{kj}]_{+}}(y_{k}\oplus
1)^{-b_{kj}}&\text{otherwise.}\end{cases}$

Similarly, the rational functions $\mathcal{X}_{i;t}^{B;t_{0}}(\mathbf{x})$
and $\mathcal{X}_{i;t}^{B_{1};t_{1}}(\mathbf{x})$ are related by

$\displaystyle\mathcal{X}_{i;t}^{B_{1};t_{1}}(\mathbf{x})=\rho_{k}\big{(}\mathcal{X}_{i;t}^{B;t_{0}}(\mathbf{x})\big{)},$

where $\rho_{k}$ is the field automorphism of
$\mathbb{Q}(\mathbb{Q}_{\text{sf}}(\mathbf{y}))(\mathbf{x})$ defined by

$\displaystyle\rho_{k}\colon\
\mathbb{Q}(\mathbb{Q}_{\text{sf}}(\mathbf{y}))(\mathbf{x})$
$\displaystyle\rightarrow\mathbb{Q}(\mathbb{Q}_{\text{sf}}(\mathbf{y}))(\mathbf{x}),$
$\displaystyle y_{j}$ $\displaystyle\mapsto\begin{cases}y_{k}^{-1}&\text{if
$j=k$},\\\ y_{j}y_{k}^{[b_{kj}]_{+}}(y_{k}\oplus
1)^{-b_{kj}}&\text{otherwise},\end{cases}$ $\displaystyle x_{j}$
$\displaystyle\mapsto\begin{cases}\dfrac{y_{k}\mathop{\prod}\limits_{i=1}^{n}x_{i}^{[b_{ik}]_{+}}+\mathop{\prod}\limits_{i=1}^{n}x_{i}^{[-b_{ik}]_{+}}}{(y_{k}\oplus
1)x_{k}}&\text{if $j=k$},\\\ x_{j}&\text{otherwise}.\end{cases}$

We call them the _initial-seed mutations of the functions $\mathcal{Y}$ and
$\mathcal{X}$_.

###  3.2 Initial-seed mutations without sign-coherence of $\bm{C}$-matrices

We use the notations in Section 3.1 continuously. By Proposition 2.4, we have

$\displaystyle\mathcal{X}_{j;t}^{B;t_{0}}(\mathbf{x})=\left(\prod_{k=1}^{n}x_{k}^{g^{B;t_{0}}_{kj;t}}\right)\frac{F_{j;t}^{B;t_{0}}(\hat{y}_{1},\dots,\hat{y}_{n})}{F_{j;t}^{B;t_{0}}(y_{1},\dots,y_{n})},$
$\displaystyle\mathcal{Y}_{j;t}^{B;t_{0}}(\mathbf{y})=\prod_{k=1}^{n}y_{k}^{c^{B;t_{0}}_{kj;t}}\prod_{k=1}^{n}\big{(}F_{k;t}^{B;t_{0}}(y_{1},\dots,y_{n})\big{)}^{b_{kj;t}}.$

As with the final-seed mutation, we will define _the initial-seed mutations in
direction $k$ of the $C$-matrices_ (resp.  _the $G$-matrices, the
$F$-polynomials, the $F$-matrices_) as transformations from $C_{t}^{B;t_{0}}$
to $C_{t}^{B_{1};t_{1}}$ (resp. from $G_{t}^{B;t_{0}}$ to
$G_{t}^{B_{1};t_{1}}$, from $F_{j;t}^{B;t_{0}}$ to $F_{j;t}^{B_{1};t_{1}}$,
from $F_{t}^{B;t_{0}}$ to $F_{t}^{B_{1};t_{1}}$). We will deduce the initial-
seed mutations of the $C$-, $G$-matrices, the $F$-polynomials and the
$F$-matrices. In order to describe these initial-seed mutations, we introduce
the _$H$ -matrices_ according to [6].

######  Definition 3.1 ([6, equation (6.16)]).

Let $B$ be any initial exchange matrix at $t_{0}$. Then, for any $t$, the
$(i,j)$ entry of $H^{B;t_{0}}_{t}=\big{(}h_{ij;t}^{B;t_{0}}\big{)}$ is given
by

$\displaystyle
u^{h_{ij;t}^{B;t_{0}}}=F^{B;t_{0}}_{j;t}|_{\text{Trop}(u)}\big{(}u^{[-b_{i1}]_{+}},\dots,u^{-1},\dots,u^{[-b_{in}]_{+}}\big{)}\qquad\big{(}\text{$u^{-1}$
in the $i$th position}\big{)}.$

The matrix $H^{B;t_{0}}_{t}$ is called the _$H$ -matrix_ at $t$.

The following fact holds [6, Proof of Proposition 6.8].

######  Lemma 3.2.

We have the following equality:

$\displaystyle{y^{\prime}_{k}}^{h_{kj;t}^{B;t_{0}}}=F^{B;t_{0}}_{j;t}|_{{\rm
Trop}(y^{\prime}_{1},\dots,y^{\prime}_{n})}(y_{1},\dots,y_{n}),$ (3.1)

where $(y^{\prime}_{1},\dots,y^{\prime}_{n})$ are the coefficients at $t_{1}$
connected with $t_{0}$ by an edge labeled $k$ in $\mathbb{T}_{n}.$

###### Proof.

Consider the cluster pattern with coefficients in
$\text{Trop}(y^{\prime}_{1},\dots,y^{\prime}_{n})$. Let
$\mathbf{y}=(y_{1},\dots,y_{n})$ be the coefficients at $t_{0}$. Then,
$\mathbf{y}$ and $\mathbf{y}^{\prime}$ have the following relation:

$\displaystyle y_{i}=\begin{cases}{y^{\prime}_{k}}^{-1}&\text{if}\ i=k,\\\
{y^{\prime}_{i}}{y^{\prime}_{k}}^{[-b_{ki}]_{+}}&\text{if}\ i\neq
k.\end{cases}$ (3.2)

Therefore, for any $j\in\\{1,\dots,n\\}$, we have

$\displaystyle
F^{B;t_{0}}_{j;t}|_{\text{Trop}(y^{\prime}_{1},\dots,y^{\prime}_{n})}(y_{1},\dots,y_{n})$
$\displaystyle\;\>=F^{B;t_{0}}_{j;t}|_{\text{Trop}(y^{\prime}_{1},\dots,y^{\prime}_{n})}\big{(}y^{\prime}_{1}{y^{\prime}_{k}}^{[-b_{k1}]_{+}},\dots,{y^{\prime}_{k}}^{-1},\dots,y^{\prime}_{n}{y^{\prime}_{k}}^{[-b_{kn}]_{+}}\big{)}$
$\displaystyle\overset{\eqref{eq:F=1}}{=}F^{B;t_{0}}_{j;t}|_{\text{Trop}(y^{\prime}_{k})}\big{(}{y^{\prime}_{k}}^{[-b_{k1}]_{+}},\dots,{y^{\prime}_{k}}^{-1},\dots,{y^{\prime}_{k}}^{[-b_{kn}]_{+}}\big{)}$
$\displaystyle\;\>={y^{\prime}_{k}}^{h_{kj;t}^{B;t_{0}}}.$ ∎

The initial-seed mutations of the $C$\- and $G$-matrices are given as follows,
where the latter was given in [6, Proposition 6.8]:

######  Proposition 3.3.

Let $\textstyle{t_{0}}$$\textstyle{t_{1}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$, $\mu_{k}(B)=B_{1}$ and $\varepsilon\in\\{\pm 1\\}$. Then,
for any $t$, we have

$\displaystyle C^{B_{1};t_{1}}_{t}=\big{(}J_{k}+[-\varepsilon
B]^{k\bullet}_{+}\big{)}C^{B;t_{0}}_{t}+H_{t}(\varepsilon)^{k\bullet}B_{t},$
(3.3) $\displaystyle G^{B_{1};t_{1}}_{t}=\big{(}J_{k}+[\varepsilon B]^{\bullet
k}_{+}\big{)}G^{B;t_{0}}_{t}-BH_{t}(\varepsilon)^{k\bullet},$ (3.4)

where $H_{t}(+)=H^{B;t_{0}}_{t}$, $H_{t}(-)=H^{B_{1};t_{1}}_{t}$.

###### Proof.

We denote $C^{B;t_{0}}_{t}=(c_{ij;t})$,
$C^{B_{1};t_{1}}_{t}=(c^{\prime}_{ij;t})$, $H^{B;t_{0}}_{t}=(h_{ij;t})$, and
$H^{B_{1};t_{1}}_{t}=(h^{\prime}_{ij;t})$. The equation (3.4) is just [6,
Proposition 6.8], rewritten in matrix form.

Let us show (3.3). Consider the same cluster pattern as in the proof of Lemma
3.2. Then, applying (2.14) and (2.16) to any coefficient $y_{j;t}$, we have

$\displaystyle\prod_{i=1}^{n}{y^{\prime}_{i}}^{c^{\prime}_{ij;t}}=\prod_{i=1}^{n}{y_{i}}^{c_{ij;t}}\prod_{i=1}^{n}F^{B;t_{0}}_{i;t}|_{\text{Trop}(y^{\prime}_{1},\dots,y^{\prime}_{n})}(y_{1},\dots,y_{n})^{b_{ij;t}}.$
(3.5)

Substituting (3.2) for (3.5) and using (3.1), we have

$\displaystyle\prod_{i=1}^{n}{y^{\prime}_{i}}^{c^{\prime}_{ij;t}}=\left(\prod_{i\neq
k}{y^{\prime}_{i}}^{c_{ij;t}}{y^{\prime}_{k}}^{[-b_{ki}]_{+}c_{ij;t}}\right){y^{\prime}_{k}}^{-c_{kj;t}}\prod_{i=1}^{n}{y^{\prime}_{k}}^{h_{ki;t}b_{ij;t}}.$
(3.6)

Comparing exponents of $y^{\prime}_{i}$ of the both sides of (3.6), we obtain

$\displaystyle c^{\prime}_{ij;t}=\begin{cases}\displaystyle-
c_{kj;t}+\mathop{\sum}\limits_{\ell=1}^{n}[-b_{k\ell}]_{+}c_{\ell
j;t}+\mathop{\sum}\limits_{\ell=1}^{n}h_{k\ell;t}b_{\ell j;t}&\text{if}\
i=k,\\\ c_{ij;t}&\text{if}\ i\neq k.\end{cases}$

Also, by interchanging $t_{0}$ and $t_{1}$, we get

$\displaystyle c^{\prime}_{ij;t}=\begin{cases}\displaystyle-
c_{kj;t}+\mathop{\sum}\limits_{\ell=1}^{n}[b_{k\ell}]_{+}c_{\ell
j;t}+\mathop{\sum}\limits_{\ell=1}^{n}h^{\prime}_{k\ell;t}b_{\ell
j;t}&\text{if}\ i=k,\\\ c_{ij;t}&\text{if}\ i\neq k.\end{cases}$

Thus we have (3.3). ∎

The initial-seed mutations of the $F$-polynomials were given in [6, Proof of
Proposition 6.8] as follows:

######  Proposition 3.4 ([6, equation (6.21)]).

Let $\textstyle{t_{0}}$$\textstyle{t_{1}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$ and $\mu_{k}(B)=B_{1}$. Then, for any $j\in\\{1,\dots,n\\}$
and $t\in\mathbb{T}_{n}$, the polynomials $F^{B;t_{0}}_{j;t}(\mathbf{y})$ and
$F^{B_{1};t_{1}}_{j;t}(\mathbf{y})$ are related by

$\displaystyle F^{B_{1};t_{1}}_{j;t}(y_{1},\dots,y_{n})$
$\displaystyle=(1+y_{k})^{g_{kj;t}^{B;t_{0}}}{y_{k}}^{-h_{kj;t}^{B;t_{0}}}$
(3.7) $\displaystyle\quad\times
F^{B;t_{0}}_{j;t}\big{(}y_{1}{y_{k}}^{[-b_{k1}]_{+}}(y_{k}+1)^{b_{k1}},\dots,{y_{k}}^{-1},\dots,y_{n}{y_{k}}^{[-b_{kn}]_{+}}(y_{k}+1)^{b_{kn}}\big{)},$

where ${y_{k}}^{-1}$ is in the $k$th position.

The initial-seed mutation of the $F$-matrices are deduced from Proposition 3.4
as follows:

######  Proposition 3.5.

Let $\textstyle{t_{0}}$$\textstyle{t_{1}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$, $\mu_{k}(B)=B_{1}$ and $\varepsilon\in\\{\pm 1\\}$. Then for
any $t$, the matrices $F^{B;t_{0}}_{t}$ and $F^{B_{1};t_{1}}_{t}$ are related
by

$\displaystyle F^{B_{1};t_{1}}_{t}$ $\displaystyle=\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+(\varepsilon
G^{B;t_{0}}_{t})^{k\bullet}-H^{-B;t_{0}}_{t}(\varepsilon)^{k\bullet}-H^{B;t_{0}}_{t}(\varepsilon)^{k\bullet}$
(3.8) $\displaystyle=\big{(}J_{k}+[-\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+(\varepsilon
G^{-B;t_{0}}_{t})^{k\bullet}-H^{-B;t_{0}}_{t}(\varepsilon)^{k\bullet}-H^{B;t_{0}}_{t}(\varepsilon)^{k\bullet}.$
(3.9)

###### Proof.

Let us show (3.8) in case of $\varepsilon=1$. Substituting (3.7) for
$y_{i}=y^{\prime}_{i}$ and evaluating (3.7) at
$\text{Trop}\big{(}{y^{\prime}_{1}}^{-1},\dots,{y^{\prime}_{n}}^{-1}\big{)}$,
we have

$\displaystyle
F^{B_{1};t_{1}}_{j;t}|_{\text{Trop}({y^{\prime}_{1}}^{-1},\dots,{y^{\prime}_{n}}^{-1})}(y^{\prime}_{1},\dots,y^{\prime}_{n})$
$\displaystyle\;\>=(1\oplus
y^{\prime}_{k})^{g^{B;t_{0}}_{kj}}{y^{\prime}_{k}}^{-h^{B;t_{0}}_{kj}}$
$\displaystyle\qquad\times
F^{B;t_{0}}_{j;t}|_{\text{Trop}\left({y^{\prime}_{1}}^{-1},\dots,{y^{\prime}_{n}}^{-1}\right)}\big{(}y^{\prime}_{1}{y^{\prime}_{k}}^{[-b_{k1}]_{+}}(y^{\prime}_{k}\oplus
1)^{b_{k1}},\dots,{y^{\prime}_{k}}^{-1},\dots,y^{\prime}_{n}{y^{\prime}_{k}}^{[-b_{kn}]_{+}}(y^{\prime}_{k}\oplus
1)^{b_{kn}}\big{)}$
$\displaystyle\,\overset{\eqref{eq:y-mutation2}}{=}(1\oplus
y^{\prime}_{k})^{g^{B;t_{0}}_{kj}}{y^{\prime}_{k}}^{-h^{B;t_{0}}_{kj}}$
$\displaystyle\qquad\times
F^{B;t_{0}}_{j;t}|_{\text{Trop}\left({y^{\prime}_{1}}^{-1},\dots,{y^{\prime}_{n}}^{-1}\right)}\big{(}y^{\prime}_{1}{y^{\prime}_{k}}^{[b_{k1}]_{+}}\big{(}{y^{\prime}_{k}}^{-1}\oplus
1\big{)}^{b_{k1}},\dots,$
$\displaystyle\qquad\qquad{y^{\prime}_{k}}^{-1},\dots,y^{\prime}_{n}{y^{\prime}_{k}}^{[b_{kn}]_{+}}({y^{\prime}_{k}}^{-1}\oplus
1)^{b_{kn}}\big{)}$
$\displaystyle\;\>={y^{\prime}_{k}}^{g^{B;t_{0}}_{kj}}{y^{\prime}_{k}}^{-h^{B;t_{0}}_{kj}}F^{B;t_{0}}_{j;t}|_{\text{Trop}\left({y^{\prime}_{1}}^{-1},\dots,{y^{\prime}_{n}}^{-1}\right)}\big{(}{y^{\prime}_{1}}{y^{\prime}_{k}}^{[b_{k1}]_{+}},\dots,{y^{\prime}_{k}}^{-1},\dots,y^{\prime}_{n}{y^{\prime}_{k}}^{[b_{kn}]_{+}}\big{)}$
$\displaystyle\overset{\eqref{Fpoly=Fpoly2}}{=}{y^{\prime}_{k}}^{g^{B;t_{0}}_{kj}}{y^{\prime}_{k}}^{-h^{B;t_{0}}_{kj}}\frac{F_{j;t}^{-B;t_{0}}|_{\text{Trop}\left({y^{\prime}_{1}}^{-1},\dots,{y^{\prime}_{n}}^{-1}\right)}\big{(}{y^{\prime}_{1}}^{-1}{y^{\prime}_{k}}^{-[b_{k1}]_{+}},\dots,y^{\prime}_{k},\dots,{y^{\prime}_{n}}^{-1}{y^{\prime}_{k}}^{-[b_{kn}]_{+}}\big{)}}{F_{j;t}^{-B;t_{0}}|_{\text{Trop}(y_{1},\dots,y_{n})}\big{(}y^{-1}_{1},\dots,y^{-1}_{n}\big{)}|_{y_{i}\mapsto{y^{\prime}_{i}}{y^{\prime}_{k}}^{[b_{ki}]_{+}},y_{k}\mapsto{y^{\prime}_{k}}^{-1}}}$
$\displaystyle\;\;={y^{\prime}_{k}}^{g^{B;t_{0}}_{kj}}{y^{\prime}_{k}}^{-h^{B;t_{0}}_{kj}}{y^{\prime}_{k}}^{-h^{-B;t_{0}}_{kj}}\prod_{i\neq
k}\big{(}{y^{\prime}_{i}}^{f_{ij}}{y^{\prime}_{k}}^{[b_{ki}]_{+}f_{ij}}\big{)}{y^{\prime}_{k}}^{-f_{kj}}.$

Comparing the exponent of both sides, we have

$\displaystyle f^{B_{1};t_{1}}_{ij;t}=\begin{cases}\displaystyle
g^{B;t_{0}}_{kj;t}-h^{B;t_{0}}_{kj;t}-h^{-B;t_{0}}_{kj;t}+\mathop{\sum}\limits_{i=1}^{n}[b_{ki}]_{+}f^{B;t_{0}}_{ij;t}-f^{B;t_{0}}_{kj;t}\quad&\text{if}\
i=k,\\\ f^{B;t_{0}}_{ij;t}\quad&\text{if}\ i\neq k.\end{cases}$

Hence we obtain the desired equality (3.8). Also replacing $B$ with $-B$ in
(3.8) and applying (2.22) to it, we get (3.9). ∎

###  3.3 Initial-seed mutations with sign-coherence of $\bm{C}$-matrices

In this subsection, we reduce the initial-seed mutation formulas by applying
the sign-coherence of the $C$-matrices. Let us introduce a duality between the
$C$-matrices and the $G$-matrices, which is a result in [8], and give the
reduced form of the initial-seed mutations of the $C$\- and $G$-matrices.

Under the sign-coherence of the $C$-matrices (Theorem 2.11), we have the
following result:

######  Proposition 3.6.

  1. $(1)$

For any exchange matrix $B$ and $t_{0},t\in\mathbb{T}_{n}$, we have

$\displaystyle\big{(}G^{B:t_{0}}_{t}\big{)}^{T}=C^{B^{T}_{t};t}_{t_{0}}.$
(3.10)

  2. $(2)$

Let $\textstyle{t_{0}}$$\textstyle{t_{1}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$, $\mu_{k}(B)=B_{1}$ and $\varepsilon\in\\{\pm 1\\}$. Then, we
have

$\displaystyle C^{B_{1};t_{1}}_{t}$ $\displaystyle=\big{(}J_{k}+[-\varepsilon
B]^{k\bullet}_{+}\big{)}C^{B;t_{0}}_{t}-\big{[}{-}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}B_{t},$ (3.11) $\displaystyle
G^{B_{1};t_{1}}_{t}$ $\displaystyle=\big{(}J_{k}+[\varepsilon B]^{\bullet
k}_{+}\big{)}G^{B;t_{0}}_{t}+B\big{[}{-}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}.$ (3.12)

  3. $(3)$

We have the reduced forms of the initial-seed mutations as follows:

$\displaystyle C^{B_{1};t_{1}}_{t}$
$\displaystyle=\big{(}J_{k}+\big{[}{-}\varepsilon_{k\bullet}\big{(}G^{B;t_{0}}_{t}\big{)}B\big{]}^{k\bullet}_{+}\big{)}C^{B;t_{0}}_{t},$
(3.13) $\displaystyle G^{B_{1};t_{1}}_{t}$
$\displaystyle=\big{(}J_{k}+\big{[}\varepsilon_{k\bullet}\big{(}G^{B;t_{0}}_{t}\big{)}B\big{]}^{\bullet
k}_{+}\big{)}G^{B;t_{0}}_{t}.$ (3.14)

###### Proof.

Equalities (3.10) and (3.12) are the results in [8, equations (1.13) and
(4.1)]. Also, (3.13) is obtained by combining (3.10) with [8, equations (1.16)
and (2.7)]. Using (3.10), we have (3.11) by (2.9). We note that the
$G$-matrices have the row sign-coherence by (3.10). By substituting
$\varepsilon=\varepsilon_{k\bullet}\big{(}G^{B;t_{0}}_{t}\big{)}$ for (3.12),
we obtain (3.14). ∎

Through the duality (3.10), we can find out the dual equalities between the
unreduced form of the final-seed and initial-seed mutations, (2.9) and (3.11),
(2.8) and (3.12), respectively. Similarly, the reduced form of the final-seed
and initial-seed mutations (2.30) and (3.14), (2.31) and (3.13) are dual
equalities, respectively.

Using Proposition 3.6, we prove the conjecture [6, Conjecture 6.10], which is
the relation between the $H$-matrices and the $G$-matrices as follows:

######  Theorem 3.7 ([6, Conjecture 6.10]).

For any $t\in\mathbb{T}_{n}$, we have the following relation:

$\displaystyle H^{B;t_{0}}_{t}=-\big{[}{-}G^{B;t_{0}}_{t}\big{]}_{+}.$ (3.15)

###### Proof.

We assume the sign-coherence of the $C$-matrices. Then, comparing (3.4) with
(3.12) and setting $\varepsilon=1$, we get

$\displaystyle
B\big{[}{-}G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}=-B\big{(}H^{B;t_{0}}_{t}\big{)}^{k\bullet}.$
(3.16)

Since $k$ is arbitrary, we have

$\displaystyle B\big{[}{-}G^{B;t_{0}}_{t}\big{]}_{+}=-BH^{B;t_{0}}_{t}.$
(3.17)

If $B$ have no zero column vector, then choosing $i$ which satisfies
$b_{ik}\neq 0$, we have $[-g_{kj;t}]_{+}=-h_{kj;t}$ and thus we have (3.15) as
desired. We prove the case that $B$ have $m(\neq 0)$ zero column vectors.
Permuting labels of $n$-regular tree $\mathbb{T}_{n}$, we can assume

$\displaystyle B=\begin{bmatrix}B^{\prime}&O\\\ O&O\end{bmatrix},$

where $B^{\prime}$ is $(n-m)\times(n-m)$ matrix without zero column vector.
Under this assumption, for
$\textstyle{t_{0}}$$\textstyle{\cdots}$$\textstyle{t}$$\scriptstyle{i_{1}}$$\scriptstyle{\
\ i_{s}}$ in $\mathbb{T}_{n}$, we have

$\displaystyle\mu_{i_{s}}\cdots\mu_{i_{2}}\mu_{i_{1}}\big{(}G^{B;t_{0}}_{t_{0}}\big{)}|_{n-m}=\begin{bmatrix}\mu_{i_{\ell^{\prime}}}\cdots\mu_{i^{\prime}_{2}}\mu_{i^{\prime}_{1}}\big{(}G_{t_{0}}^{B^{\prime};t_{0}}\big{)}\\\
O\end{bmatrix},$
$\displaystyle\mu_{i_{s}}\cdots\mu_{i_{2}}\mu_{i_{1}}\big{(}H^{B;t_{0}}_{t_{0}}\big{)}|_{n-m}=\begin{bmatrix}\mu_{i^{\prime}_{\ell}}\cdots\mu_{i^{\prime}_{2}}\mu_{i^{\prime}_{1}}\big{(}H_{t_{0}}^{B^{\prime};t_{0}}\big{)}\\\
O\end{bmatrix},$

where $(i^{\prime}_{1},\dots,i^{\prime}_{\ell})$ is a sequence which is
obtained by removing $n-m+1,\dots,n$ from $(i_{1},\dots,i_{s})$, and $|_{n-m}$
means taking the left $n\times(n-m)$ submatrix. Also about the other diagonal
entries of $B$, we have the similar equalities. Thus we have

$\displaystyle G_{t}^{B;t_{0}}=G_{t^{\prime}}^{B^{\prime};t_{0}}\oplus
G_{t_{1}}^{(0);t_{0}}\oplus\cdots\oplus G_{t_{m}}^{(0);t_{0}},$ $\displaystyle
H_{t}^{B;t_{0}}=H_{t^{\prime}}^{B^{\prime};t_{0}}\oplus
H_{t_{1}}^{(0);t_{0}}\oplus\cdots\oplus H_{t_{m}}^{(0);t_{0}},$

where $t^{\prime}$ is a vertex of $\mathbb{T}_{m-n}$ which satisfies
$\Sigma_{t^{\prime}}=\mu_{i^{\prime}_{\ell}}\cdots\mu_{i^{\prime}_{1}}(\Sigma_{t_{0}})$
and

$\displaystyle t_{j}=\begin{cases}t_{0}&\text{if the number of $n-m+j$ in
$(i_{1},\dots,i_{s})$ is even},\\\ t^{\prime}_{0}&\text{if the number of
$n-m+j$ in $(i_{1},\dots,i_{s})$ is odd},\end{cases}$

in $\mathbb{T}_{1}$:
$\textstyle{t_{0}}$$\textstyle{t^{\prime}_{0}}$$\scriptstyle{j}$ for any
$j\in\\{1,\dots,m\\}$. Explicitly, we have

$\displaystyle
G^{B;t_{0}}_{t}=\begin{bmatrix}G^{B^{\prime};t_{0}}_{t^{\prime}}&&&O\\\
&(-1)^{N_{1}}&&\\\ &&\ddots&\\\ O&&&(-1)^{N_{m}}\end{bmatrix},$ $\displaystyle
H^{B;t_{0}}_{t}=\begin{bmatrix}H^{B^{\prime};t_{0}}_{t^{\prime}}&&&O\\\
&-[(-1)^{N_{1}+1}]_{+}&&\\\ &&\ddots&\\\
O&&&-[(-1)^{N_{m}+1}]_{+}\end{bmatrix},$

where $N_{j}$ is the number of $n-m+j$ in $(i_{1},\dots,i_{s})$. Since
$B^{\prime}$ has no zero column vectors and

$\displaystyle\begin{bmatrix}B^{\prime}\big{[}{-}G_{t^{\prime}}^{B^{\prime};t_{0}}\big{]}_{+}&O\\\
O&O\end{bmatrix}=\begin{bmatrix}-B^{\prime}H_{t^{\prime}}^{B^{\prime};t_{0}}&O\\\
O&O\end{bmatrix}$

holds by (3.17), we have
$\big{[}{-}G_{t^{\prime}}^{B^{\prime};t_{0}}\big{]}_{+}=-H_{t^{\prime}}^{B^{\prime};t_{0}}$.
By direct calculation, we also have
$\big{[}{-}G_{t_{j}}^{(0);t_{0}}\big{]}_{+}\allowbreak=-H_{t_{j}}^{(0);t_{0}}$
for all $j$. Therefore, we have (3.15) as desired. ∎

We point out the following equivalence.

######  Proposition 3.8.

The following statements are equivalent:

  1. $(i)$

The $C$-matrices have the column sign-coherence.

  2. $(ii)$

The $G$-matrices have the row sign-coherence, and equality (3.15) holds.

###### Proof.

We proved (i) $\Rightarrow$ (ii) by (3.10) and Theorem 3.7. We give the proof
of (ii) $\Rightarrow$ (i). To prove it, let us show (3.10). We note that we
can not use (3.10) directly because we do not assume the sign-coherence of the
$C$-matrices now. By assumption, we have
$H_{t}(\varepsilon)^{k\bullet}=-\big{[}{-}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}$. Substituting it for (3.3) and (3.4),
we have (3.11) and (3.12). Let us show (3.10) by the induction on the distance
between $t$ and $t_{0}$. If
$\big{(}G^{B:t_{0}}_{t}\big{)}^{T}=C^{B^{T}_{t};t}_{t_{0}}$ holds for some
$t\in\mathbb{T}_{n}$, then for $t^{\prime}\in\mathbb{T}_{n}$ such that
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$, using (3.11) and
(2.9), we have

$\displaystyle C^{B^{T}_{t^{\prime}};t^{\prime}}_{t_{0}}$
$\displaystyle=\big{(}J_{\ell}+\big{[}{-}\varepsilon
B_{t}^{T}\big{]}^{\ell\bullet}_{+}\big{)}C^{B_{t}^{T};t}_{t_{0}}-\big{[}{-}\varepsilon
G^{B_{t}^{T};t}_{t_{0}}\big{]}^{\ell\bullet}_{+}B^{T}$
$\displaystyle=\big{(}G^{B;t_{0}}_{t}\big{(}J_{\ell}+[-\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}-B\big{[}{-}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}\big{)}^{T}=\big{(}G^{B;t_{0}}_{t^{\prime}}\big{)}^{T}.$

Thus by the row sign-coherence of the $G$-matrices, we get the sign-coherence
of the $C$-matrices immediately. ∎

By Theorem 2.11 and Proposition 3.8, the (row) sign-coherence the $G$-matrices
and Theorem 3.7 hold. Furthermore, applying the sign-coherence of the
$C$-matrices is equivalent to applying the sign-coherence of the $G$-matrices
and (3.15). Using it, let us give a reduced expression of the initial-seed
mutations of $F$-matrices.

######  Proposition 3.9.

  1. $(1)$

Let $\textstyle{t_{0}}$$\textstyle{t_{1}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$, $\mu_{k}(B)=B_{1}$ and $\varepsilon\in\\{\pm 1\\}$. Then, we
have

$\displaystyle F^{B_{1};t_{1}}_{t}=\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+\big{[}{-}\varepsilon
G^{-B;t_{0}}_{t}\big{]}^{k\bullet}_{+}+\big{[}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}.$ (3.18)

  2. $(2)$

We have a reduced form of the initial-seed mutations as follows:

$\displaystyle F^{B_{1};t_{1}}_{t}$
$\displaystyle=\big{(}J_{k}+\big{[}\varepsilon_{k}\big{(}G^{-B;t_{0}}_{t}\big{)}B\big{]}^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+\big{[}\varepsilon_{k}\big{(}G^{-B;t_{0}}_{t}\big{)}G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}$
$\displaystyle=\big{(}J_{k}+\big{[}{-}\varepsilon_{k}\big{(}G^{B;t_{0}}_{t}\big{)}B\big{]}^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+\big{[}\varepsilon_{k}\big{(}G^{B;t_{0}}_{t}\big{)}G^{-B;t_{0}}_{t}\big{]}^{k\bullet}_{+}.$
(3.19)

###### Proof.

(1) Thanks to Theorem 3.7, we can substitute
$H^{B;t_{0}}_{t}\\!(\varepsilon)^{k\bullet}\\!=\\!{-}\big{[}{-}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}\\!$ and
$H^{-B;t_{0}}_{t}\\!(\varepsilon)^{k\bullet}\\!\allowbreak=-\big{[}{-}\varepsilon
G^{-B;t_{0}}_{t}\big{]}^{k\bullet}_{+}$ for (3.8). Then, we have

$\displaystyle F^{B_{1};t_{1}}_{t}$ $\displaystyle=\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+\big{(}\varepsilon
G^{B;t_{0}}_{t}\big{)}^{k\bullet}+\big{[}{-}\varepsilon
G^{-B;t_{0}}_{t}\big{]}^{k\bullet}_{+}+\big{[}{-}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}$
$\displaystyle=\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}+\big{[}{-}\varepsilon
G^{-B;t_{0}}_{t}\big{]}^{k\bullet}_{+}+\big{[}\varepsilon
G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+},$

as desired.

(2) Substituting $\varepsilon=\varepsilon_{k}\big{(}G^{-B;t_{0}}_{t}\big{)}$
or $\varepsilon=-\varepsilon_{k}\big{(}G^{B;t_{0}}_{t}\big{)}$ for (3.18), we
obtain (3.19). ∎

Like the duality between the final-seed and initial-seed mutations of the
$C$\- and $G$-matrices, (2.29) and (3.18), (2.32) and (3.19) are dual
equalities, respectively. We show the self-duality of the $F$-matrices, which
is analogous to the duality (3.10) between the $C$\- and $G$-matrices. This is
the main theorem in this paper.

######  Theorem 3.10.

For any exchange matrix $B$ and $t_{0},t\in\mathbb{T}_{n}$, we have

$\displaystyle\big{(}F_{t}^{B;t_{0}}\big{)}^{T}=F_{t_{0}}^{B_{t}^{T};t}.$
(3.20)

###### Proof.

We prove (3.20) by the induction on the distance between $t$ and $t_{0}$ in
$\mathbb{T}_{n}$. When $t=t_{0}$, we have
$\big{(}F_{t}^{B;t_{0}}\big{)}^{T}=O=F_{t_{0}}^{B_{t}^{T};t}$ as desired. We
show that if (3.20) holds for some $t\in\mathbb{T}_{n}$, then it also holds
for $t^{\prime}\in\mathbb{T}_{n}$ such that
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$. By the inductive
assumption, (2.32), Proposition 3.9 and (3.10), we have

$\displaystyle\big{(}F_{t^{\prime}}^{B;t_{0}}\big{)}^{T}$
$\displaystyle=\big{(}J_{\ell}+[\varepsilon_{\ell}\big{(}G^{-B^{T}_{t};t}_{t_{0}}\big{)}B_{t}^{T}]^{\ell\bullet}_{+}\big{)}\big{(}F^{B;t_{0}}_{t}\big{)}^{T}+\big{[}\varepsilon_{\ell}\big{(}G^{-B_{t}^{T};t}_{t_{0}}\big{)}G^{B_{t}^{T};t}_{t_{0}}\big{]}^{\ell\bullet}_{+}$
$\displaystyle=\big{(}J_{\ell}+[\varepsilon_{\ell}\big{(}G^{-B_{t}^{T};t}_{t_{0}}\big{)}B_{t}^{T}]^{\ell\bullet}_{+}\big{)}F^{B^{T}_{t};t}_{t_{0}}+\big{[}\varepsilon_{\ell}\big{(}G^{-B_{t}^{T};t}_{t_{0}}\big{)}G^{B_{t}^{T};t}_{t_{0}}\big{]}^{\ell\bullet}_{+}=F_{t_{0}}^{B_{t^{\prime}}^{T};t^{\prime}}$

as desired. ∎

###  3.4 Examples

We introduce an example for the final-seed and initial-seed mutations in the
case of $A_{2}$. Let $n=2$, and consider a tree $\mathbb{T}_{2}$ whose edges
are labeled as follows:

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We set $B=\left[\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\right]$ as the
initial exchange matrix at $t_{0}$. Then, the coefficients, the cluster
variables, the $C$-, $G$\- and $F$-matrices are given by Tables 1 and 2 [6,
Example 2.10].

$t$ | $\mathcal{Y}^{B;t_{0}}_{t}$ |  $\mathcal{X}^{B;t_{0}}_{t}$ 3pt2pt
---|---|---
$0$ | $y_{1}$ | $y_{2}$ | $x_{1}$ | $x_{2}$
$1$ | $y_{1}(y_{2}\oplus 1)$ | $\dfrac{1}{y_{2}}$ | $x_{1}$ | $\dfrac{x_{1}y_{2}+1}{(y_{2}\oplus 1)x_{2}}$
$2$ | $\dfrac{1}{y_{1}(y_{2}\oplus 1)}$ | $\dfrac{y_{1}y_{2}\oplus y_{1}\oplus 1}{y_{2}}$ | $\dfrac{x_{1}y_{1}y_{2}+y_{1}+x_{2}}{(y_{1}y_{2}\oplus y_{1}\oplus 1)x_{1}x_{2}}$ | $\dfrac{x_{1}y_{2}+1}{(y_{2}\oplus 1)x_{2}}$
$3$ | $\dfrac{y_{1}\oplus 1}{y_{1}y_{2}}$ | $\dfrac{y_{2}}{y_{1}y_{2}\oplus y_{1}\oplus 1}$ | $\dfrac{x_{1}y_{1}y_{2}+y_{1}+x_{2}}{(y_{1}y_{2}\oplus y_{1}\oplus 1)x_{1}x_{2}}$ | $\dfrac{y_{1}+x_{2}}{x_{1}(y_{1}\oplus 1)}$
$4$ | $\dfrac{y_{1}y_{2}}{y_{1}\oplus 1}$ | $\dfrac{1}{y_{1}}$ | $x_{2}$ | $\dfrac{y_{1}+x_{2}}{x_{1}(y_{1}\oplus 1)}$
$5$ | $y_{2}$ | $y_{1}$ | $x_{2}$ | $x_{1}$
Table 1: Coefficients and cluster variables in type $A_{2}$. $t$ | $C^{B;t_{0}}_{t}$ | $G^{B;t_{0}}_{t}$ |  $F^{B;t_{0}}_{t}$ 2pt
---|---|---|---
$0$ | $\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}$
$1$ | $\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}$
$2$ | $\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 1&1\end{bmatrix}$
3 | $\begin{bmatrix}-1&0\\\ -1&1\end{bmatrix}$ | $\begin{bmatrix}-1&-1\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}1&1\\\ 1&0\end{bmatrix}$
$4$ | $\begin{bmatrix}1&-1\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&-1\\\ 1&1\end{bmatrix}$ | $\begin{bmatrix}0&1\\\ 0&0\end{bmatrix}$
$5$ | $\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}$
Table 2: $C$-, $G$\- and $F$-matrices in type $A_{2}$.

We show the expressions of the coefficients and the cluster variables at
$t_{0}$ in Type $A_{2}$ in Table 3, and its counterpart the $C$-, $G$\- and
$F$-matrices in Table 4.

$t$ | $\mathcal{Y}^{B^{T}_{t};t}_{t_{0}}$ |  $\mathcal{X}^{B^{T}_{t};t}_{t_{0}}$ 5pt2pt
---|---|---
$0$ | $y_{1}$ | $y_{2}$ | $x_{1}$ | $x_{2}$
$1$ | $y_{1}(y_{2}\oplus 1)$ | $\dfrac{1}{y_{2}}$ | $x_{1}$ | $\dfrac{y_{2}x_{1}+1}{(y_{2}\oplus 1)x_{2}}$
$2$ | $\dfrac{y_{1}y_{2}\oplus y_{2}\oplus 1}{y_{1}}$ | $\dfrac{1}{y_{2}(y_{1}\oplus 1)}$ | $\dfrac{y_{1}x_{2}+1}{(y_{1}\oplus 1)x_{1}}$ | $\dfrac{y_{1}y_{2}x_{2}+y_{2}+x_{1}}{(y_{1}y_{2}\oplus y_{2}\oplus 1)x_{1}x_{2}}$
$3$ | $\dfrac{y_{1}\oplus 1}{y_{1}y_{2}}$ | $\dfrac{y_{2}}{y_{1}y_{2}\oplus y_{1}\oplus 1}$ | $\dfrac{y_{1}y_{2}x_{1}+y_{1}+x_{2}}{(y_{1}y_{2}\oplus y_{1}\oplus 1)x_{1}x_{2}}$ | $\dfrac{y_{1}+x_{2}}{(y_{1}\oplus 1)x_{1}}$
$4$ | $\dfrac{1}{y_{2}}$ | $\dfrac{y_{1}y_{2}}{y_{2}\oplus 1}$ | $\dfrac{y_{2}+x_{1}}{(y_{2}\oplus 1)x_{2}}$ | $x_{1}$
$5$ | $y_{2}$ | $y_{1}$ | $x_{2}$ | $x_{1}$
Table 3: Expressions of coefficients and cluster variables at $t_{0}$ in type $A_{2}$. $t$ | $C^{B^{T}_{t};t}_{t_{0}}$ | $G^{B^{T}_{t};t}_{t_{0}}$ |  $F^{B^{T}_{t};t}_{t_{0}}$ 2pt
---|---|---|---
$0$ | $\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}$
$1$ | $\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&1\end{bmatrix}$
$2$ | $\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}-1&0\\\ 0&-1\end{bmatrix}$ | $\begin{bmatrix}1&1\\\ 0&1\end{bmatrix}$
$3$ | $\begin{bmatrix}-1&0\\\ -1&1\end{bmatrix}$ | $\begin{bmatrix}-1&-1\\\ 0&1\end{bmatrix}$ | $\begin{bmatrix}1&1\\\ 1&0\end{bmatrix}$
$4$ | $\begin{bmatrix}0&1\\\ -1&1\end{bmatrix}$ | $\begin{bmatrix}1&1\\\ -1&0\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}$
$5$ | $\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}$ | $\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}$
Table 4: $C$-, $G$\- and $F$-matrices in type $A_{2}$ (moving the initial
vertex).

Comparing Table 2 with Table 4, we can see the duality of the $C$-, $G$\- and
$F$-matrices in (3.10) and (3.20).

##  4 Properties on principal extension of exchange matrices

In this section, we present some properties of the principal extension of the
exchange matrices. These properties yield alternative derivations of some
equalities in the previous sections and are interesting for their own right.

###  4.1 Principal extension

For an exchange matrix $B_{t}$,
$\tilde{B_{t}}=\left[\begin{smallmatrix}B_{t}\\\
C_{t}^{B;t_{0}}\end{smallmatrix}\right]$ $(t\in\mathbb{T}_{n})$ is the
extended exchange matrix [5, Section 5]. We fix the initial vertex
$t_{0}\in\mathbb{T}_{2n}$. Let us regard $\mathbb{T}_{n}$ as the subtree of
$\mathbb{T}_{2n}$ induced by vertices which are reachable from $t_{0}$ via
edges labeled by $1,\dots,n$. Let us consider the following “full extension of
$B$”. We set

$\displaystyle\overline{B}:=\begin{bmatrix}B&-I_{n}\\\ I_{n}&O\end{bmatrix}.$
(4.1)

Note that $\overline{B}$ is regular. Then, we obtain a family of $2n\times 2n$
skew-symmetrizable matrices $\\{\overline{B}_{t}\\}_{t\in\mathbb{T}_{2n}}$
such that $\overline{B}=\overline{B}_{t_{0}}$
$(t_{0}\in\mathbb{T}_{n}\subset\mathbb{T}_{2n})$ and they are related by the
mutation (2.2). Then, the left half of $\overline{B}_{t}$ is $\tilde{B}_{t}$
for $t\in\mathbb{T}_{n}$, and also $\overline{B}_{t}$ is regular. We call
$\\{\overline{B}_{t}\\}_{t\in\mathbb{T}_{2n}}$ the _principal extension_ of
$\\{B_{t}\\}_{t\in\mathbb{T}_{n}}$. The following proposition gives the
explicit expression of $\overline{B}_{t}$ $(t\in\mathbb{T}_{n})$:

######  Proposition 4.1.

Let
$\\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\\}_{t\in\mathbb{T}_{n}}$
be a cluster pattern with the initial vertex $t_{0}$. For any mutation series
$\mu_{i_{m}}\mu_{i_{m-1}}\cdots\mu_{i_{1}}\
(i_{1},\dots,i_{m}\in\\{1,\dots,n\\})$ of $\mathbb{T}_{n}$, we set
$t_{1},\dots,t_{{m-1}},t_{m},t$ as
$\textstyle{t_{0}}$$\textstyle{t_{1}}$$\textstyle{\cdots}$$\textstyle{t_{m-1}}$$\textstyle{t_{m}=t}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{m-1}}$$\scriptstyle{i_{m}}$
in $\mathbb{T}_{n}$. Then, the following equality holds:

$\displaystyle\overline{B}_{t}=\begin{bmatrix}B_{t}&-D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\\\
C_{t}^{B;t_{0}}&O\end{bmatrix},$ (4.2)

where $D$ is a skew-symmetrizer of $B$.

###### Proof.

We start with the proof of the following equality:

$\displaystyle\overline{B}_{t}=\begin{bmatrix}B_{t}&-D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\\\
C_{t}^{B;t_{0}}&\mathop{\sum}\limits_{s=0}^{m-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{[}{-}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{i_{s+1}\bullet}-\big{[}{-}C_{t_{s}}^{B;t_{0}}\big{]}_{+}^{\bullet
i_{s+1}}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{)}\end{bmatrix}.$
(4.3)

We prove (4.3) by the induction on the distance between $t$ and $t_{0}$ in
$\mathbb{T}_{n}$. The base case $m=0$ is immediate as
$\overline{B_{t_{0}}}=\overline{B}$. It remains to show that if (4.3) holds
for some $t\in\mathbb{T}_{n}$, then it also holds for
$t^{\prime}\in\mathbb{T}_{n}$ such
that$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$. The left
$2n\times n$ submatrix of $\overline{B}_{t}$ coincides with the extended
matrix $\tilde{B}_{t}$ in [5, Section 5]. Therefore, it suffices to prove the
right $2n\times n$ submatrix of $\overline{B}_{t}$. For any $1\leq i\leq n$
and $n+1\leq j\leq 2n$, we have

$\displaystyle b_{ij;t^{\prime}}$
$\displaystyle=\begin{cases}-b_{ij;t}&\text{if $i=\ell$,}\\\
b_{ij;t}+b_{i\ell;t}[b_{\ell j;t}]_{+}+[-b_{i\ell;t}]_{+}b_{\ell
j;t}&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}d_{i}^{-1}c_{j-n,i;t}d_{j}&\text{if $i=\ell$,}\\\
-d_{i}^{-1}c_{j-n,i;t}d_{j}+b_{i\ell;t}\big{[}{-}d_{\ell}^{-1}c_{j-n,\ell;t}d_{j}\big{]}_{+}-[-b_{i\ell;t}]_{+}d_{\ell}^{-1}c_{j-n,\ell;t}d_{j}&\text{otherwise}\end{cases}$
$\displaystyle=\begin{cases}-d_{i}^{-1}c_{j-n,i;t^{\prime}}d_{j}&\text{if
$i=\ell$,}\\\ -d_{i}^{-1}(c_{j-n,i;t}+b_{\ell
i;t}[-c_{j-n,\ell;t}]_{+}+[b_{\ell
i;t}]_{+}c_{j-n,\ell;t})d_{j}&\text{otherwise}\end{cases}$
$\displaystyle=-d_{i}^{-1}c_{j-n,i;t^{\prime}}d_{j}.$

By using the equality

$\displaystyle\big{(}C_{t}^{B;t_{0}}\big{[}{-}D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{\ell\bullet}-\big{[}{-}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\big{)}_{i-n,j-n}$
$\displaystyle\qquad{}=c_{i-n,\ell;t}\big{[}{-}d_{\ell}^{-1}c_{j-n,\ell;t}d_{j}\big{]}_{+}-[-c_{i-n,\ell;t}]_{+}d_{\ell}^{-1}c_{j-n,\ell;t}d_{j}$

and the inductive assumption, for any $n+1\leq i,j\leq 2n$, we have

$\displaystyle b_{ij;t^{\prime}}$ $\displaystyle=b_{ij;t}+b_{i\ell}[b_{\ell
j;t}]_{+}+[-b_{i\ell;t}]_{+}b_{\ell j;t}$
$\displaystyle=\sum_{s=0}^{{m-1}}\big{(}c_{i-n,{i_{s+1}};t_{s}}\big{[}{-}d_{i_{s+1}}^{-1}c_{j-n,{i_{s+1}};t_{s}}d_{j}\big{]}_{+}-[-c_{i-n,{i_{s+1}};t_{s}}]_{+}d_{i_{s+1}}^{-1}c_{j-n,{i_{s+1}};t_{s}}d_{j}\big{)}$
$\displaystyle\quad{}+c_{i-n,\ell;t}\big{[}{-}d_{\ell}^{-1}c_{j-n,\ell;t}d_{j}\big{]}_{+}-[-c_{i-n,\ell;t}]_{+}d_{\ell}^{-1}c_{j-n,\ell;t}d_{j}$
$\displaystyle=\sum_{s=0}^{m}\big{(}c_{i-n,{i_{s+1}};t_{s}}\big{[}{-}d_{i_{s+1}}^{-1}c_{j-n,{i_{s+1}};t_{s}}d_{j}\big{]}_{+}-[-c_{i-n,{i_{s+1}};t_{s}}]_{+}d_{i_{s+1}}^{-1}c_{j-n,{i_{s+1}};t_{s}}d_{j}\big{)}$
$\displaystyle=\sum_{s=0}^{m}\big{(}C_{t_{s}}^{B;t_{0}}\big{[}{-}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{{i_{s+1}}\bullet}-\big{[}{-}C_{t_{s}}^{B;t_{0}}\big{]}_{+}^{\bullet{i_{s+1}}}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{)}_{i-n,j-n},$
(4.4)

where $t_{m}=t$ and $\ i_{m+1}=\ell$. Thus we have (4.3). Moreover, applying
the sign-coherence of the $C$-matrices, the third equality of (4.4) is always
zero. Therefore, we have (4.2). ∎

Let us study the $C$-, $G$\- and $F$-matrices for the initial matrix
$\overline{B}$ in (4.1), especially for the subtree $\mathbb{T}_{n}$ in
$\mathbb{T}_{2n}$.

######  Theorem 4.2.

Let
$\\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\\}_{t\in\mathbb{T}_{n}}$
be a cluster pattern with the initial vertex $t_{0}$. For any mutation series
$\mu_{i_{m}}\mu_{i_{m-1}}\cdots\mu_{i_{1}}\
(i_{1},\dots,i_{m}\in\\{1,\dots,n\\})$ of $\mathbb{T}_{n}$, we set
$t_{1},\dots,t_{{m-1}},t_{m},t$ as
$\textstyle{t_{0}}$$\textstyle{t_{1}}$$\textstyle{\cdots}$$\textstyle{t_{m-1}}$$\textstyle{t_{m}=t}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{m-1}}$$\scriptstyle{i_{m}}$
in $\mathbb{T}_{n}$. Then, the following equalities hold:

$\displaystyle C_{t}^{\overline{B};t_{0}}=\begin{bmatrix}C_{t}^{B;t_{0}}&O\\\
O&I_{n}\end{bmatrix},$ (4.5) $\displaystyle
G_{t}^{\overline{B};t_{0}}=\begin{bmatrix}G_{t}^{B;t_{0}}&O\\\
O&I_{n}\end{bmatrix},$ (4.6) $\displaystyle
F_{\ell;t}^{\overline{B};t_{0}}(\mathbf{y})=\begin{cases}F_{\ell;t}^{B;t_{0}}(\mathbf{y})&\text{if
}\ell\in\\{1,\dots,n\\},\\\ 1&\text{if }\ell\in\\{n+1,\dots,2n\\}.\end{cases}$
(4.7)

###### Proof.

Firstly, we start the proof of the following equality:

$\displaystyle
C_{t}^{\overline{B};t_{0}}=\begin{bmatrix}C_{t}^{B;t_{0}}&\mathop{\sum}\limits_{s=0}^{m-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{[}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{i_{s+1}\bullet}-\big{[}C_{t_{s}}^{B;t_{0}}\big{]}_{+}^{\bullet
i_{s+1}}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{)}\\\
O&I_{n}\end{bmatrix}.$ (4.8)

We prove (4.8) by the induction on the distance between $t$ and $t_{0}$ in
$\mathbb{T}_{n}$. The base case $m=0$ is immediate as
$C_{t_{0}}^{\overline{B};t_{0}}=I_{2n}$. It remains to show that if (4.8)
holds for some $t\in\mathbb{T}_{n}$, then it also holds for
$t^{\prime}\in\mathbb{T}_{n}$ such
that$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$. By (2.8), we
have

$\displaystyle C_{t^{\prime}}^{\overline{B};t_{0}}$
$\displaystyle=C_{t}^{\overline{B};t_{0}}\big{(}J_{\ell}+[-\overline{B}_{t}]_{+}^{\ell\bullet}\big{)}+\big{[}C^{\overline{B};t_{0}}_{t}\big{]}_{+}^{\bullet\ell}\overline{B}_{t}$
$\displaystyle=\begin{bmatrix}C_{t}^{B;t_{0}}&X\\\
O&I_{n}\end{bmatrix}\left(\begin{bmatrix}J_{\ell}&O\\\
O&I_{n}\end{bmatrix}+\begin{bmatrix}[-B_{t}]_{+}^{\ell\bullet}&[D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D]_{+}^{\ell\bullet}\\\
O&O\end{bmatrix}\right)$
$\displaystyle\quad{}+\begin{bmatrix}\big{[}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}&O\\\
O&O\end{bmatrix}\begin{bmatrix}B_{t}&-D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\\\
C_{t}^{B;t_{0}}&Y\end{bmatrix}$
$\displaystyle=\begin{bmatrix}C_{t}^{B;t_{0}}\big{(}J_{\ell}+[-{B}_{t}]_{+}^{\ell\bullet}\big{)}+\big{[}C^{B;t_{0}}_{t}\big{]}_{+}^{\bullet\ell}{B}_{t}&Z\\\
O&I_{n}\end{bmatrix}=\begin{bmatrix}C_{t^{\prime}}^{B;t_{0}}&Z\\\
O&I_{n}\end{bmatrix},$

where

$\displaystyle
X=\mathop{\sum}\limits_{s=0}^{m-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{[}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{i_{s+1}\bullet}-\big{[}C_{t_{s}}^{B;t_{0}}\big{]}_{+}^{\bullet
i_{s+1}}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{)},$ $\displaystyle
Y=\mathop{\sum}\limits_{s=0}^{m-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{[}{-}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{i_{s+1}\bullet}-\big{[}{-}C_{t_{s}}^{B;t_{0}}\big{]}_{+}^{\bullet
i_{s+1}}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{)},$ $\displaystyle
Z=X+C_{t}^{B;t_{0}}\big{[}D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{\ell\bullet}-\big{[}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D.$

Calculating $Z$, we have

$\displaystyle
Z=\sum_{s=0}^{m}\big{(}C_{t_{s}}^{B;t_{0}}\big{[}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{i_{s+1}\bullet}-\big{[}C_{t_{s}}^{B;t_{0}}\big{]}_{+}^{\bullet
i_{s+1}}D^{-1}\big{(}C_{t_{s}}^{B;t_{0}}\big{)}^{T}D\big{)},$

where $t_{m}=t$ and $i_{m+1}=\ell$. Thus we have (4.8). Furthermore, since the
$(i,j)$ entry of

$\displaystyle
C_{t}^{B;t_{0}}\big{[}D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D\big{]}_{+}^{\ell\bullet}-\big{[}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}D^{-1}\big{(}C_{t}^{B;t_{0}}\big{)}^{T}D$

is

$\displaystyle
d^{-1}_{\ell}(c_{i\ell;t}[c_{j\ell;t}]_{+}-[c_{i\ell;t}]c_{j\ell;t})d_{j},$

we have $Z=O$ by applying the sign-coherence of the $C$-matrices. Hence we
have (4.5). Secondly, we prove (4.6) by the same way as (4.8). The base case
$m=0$ is immediate as $G_{t_{0}}^{\overline{B};t_{0}}=I_{2n}$. By the
inductive assumption, we have

$\displaystyle G_{t^{\prime}}^{\overline{B};t_{0}}$
$\displaystyle=G_{t}^{\overline{B};t_{0}}\big{(}J_{\ell}+[\overline{B}_{t}]_{+}^{\bullet\ell}\big{)}-\overline{B}\big{[}C_{t}^{\overline{B};t_{0}}\big{]}_{+}^{\bullet\ell}$
$\displaystyle=\begin{bmatrix}G_{t}^{B;t_{0}}&O\\\
O&I_{n}\end{bmatrix}\left(\begin{bmatrix}J_{\ell}&O\\\
O&I_{n}\end{bmatrix}+\begin{bmatrix}[{B}_{t}]_{+}^{\bullet\ell}&O\\\
\big{[}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}&O\end{bmatrix}\right)-\begin{bmatrix}B&-I_{n}\\\
I_{n}&O\end{bmatrix}\begin{bmatrix}\big{[}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}&O\\\
O&O\end{bmatrix}$
$\displaystyle=\begin{bmatrix}G_{t}^{{B};t_{0}}\big{(}J_{\ell}+[{B}_{t}]_{+}^{\bullet\ell}\big{)}-B\big{[}C_{t}^{B;t_{0}}\big{]}_{+}^{\bullet\ell}&O\\\
O&I_{n}\end{bmatrix}=\begin{bmatrix}G_{t^{\prime}}^{B;t_{0}}&O\\\
O&I_{n}\end{bmatrix}$

as desired. Finally, we prove (4.7) by the same way as (4.8) or (4.6). The
base case $m=0$ is immediate as
$F_{i;t_{0}}^{\overline{B};t_{0}}(\mathbf{y})=1$ for all $i$. Let
$C_{t}^{\overline{B};t_{0}}=(\overline{c}_{ij;t})$ and
$\overline{B}_{t}=(\overline{b}_{ij;t})$, and we abbreviate
$F_{i;t}^{B;t_{0}}(\mathbf{y})=F_{i;t}$ and
$F_{i;t}^{\overline{B};t_{0}}(\mathbf{y})=\overline{F}_{i;t}$. By the
inductive assumption and Proposition 4.1 and (4.8), we have

$\displaystyle\overline{F}_{i;t^{\prime}}$
$\displaystyle=\overline{F}_{i;t}=F_{i;t}=F_{i;t^{\prime}}\qquad\text{if
$i\neq\ell$,}$ $\displaystyle\overline{F}_{\ell;t^{\prime}}$
$\displaystyle=\frac{\prod\limits_{j=1}^{2n}y_{j}^{[\overline{c}_{j\ell;t}]_{+}}\prod\limits_{i=1}^{2n}\overline{F}_{i;t}^{[\overline{b}_{i\ell;t}]_{+}}+\prod\limits_{j=1}^{2n}y_{j}^{[-\overline{c}_{j\ell;t}]_{+}}\prod\limits_{i=1}^{2n}\overline{F}_{i;t}^{[-\overline{b}_{i\ell;t}]_{+}}}{\overline{F}_{\ell;t}}$
$\displaystyle=\frac{\prod\limits_{j=1}^{n}y_{j}^{[c_{j\ell;t}]_{+}}\prod\limits_{i=1}^{n}F_{i;t}^{[b_{i\ell;t}]_{+}}+\prod\limits_{j=1}^{n}y_{j}^{[c_{j\ell;t}]_{+}}\prod\limits_{i=1}^{n}F_{i;t}^{[-b_{i\ell;t}]_{+}}}{F_{\ell;t}}={F}_{\ell;t^{\prime}}$

as desired. ∎

######  Remark 4.3.

The equality (4.5) is the specialisation of [2, equation (2)]. Indeed, setting
$m=n$ and substituting $B$, $I_{n}$, $-I_{n}$ and $O$ for $B_{1}$, $B_{2}$,
$B_{3}$ and $B_{4}$ in $\tilde{B}=\left(\begin{smallmatrix}B_{1}&B_{2}\\\
B_{3}&B_{4}\\\ I_{n}&0\\\ 0&I_{m}\end{smallmatrix}\right)$ appearing in [2,
Proof of Theorem 4.5], we have (4.5) by [2, equation (2)].

######  Remark 4.4.

We did not use the sign-coherence of the $C$-matrices when we show (4.6) and
(4.7). Furthermore, (4.7) always holds when $\overline{B}$ is a $2n\times 2n$
matrix whose the upper left $n\times n$ submatrix is $B$.

By Theorem 4.2 and Remark 4.4, we have the following corollary:

######  Corollary 4.5.

Let
$\\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\\}_{t\in\mathbb{T}_{n}}$
be a cluster pattern with the initial vertex $t_{0}$. Then, for any
$t\in\mathbb{T}_{n}$, the following equalities hold:

$\displaystyle F_{t}^{\overline{B};t_{0}}=\begin{bmatrix}F_{t}^{B;t_{0}}&O\\\
O&O\end{bmatrix},$ (4.9) $\displaystyle
H^{\overline{B};t_{0}}_{t}=\begin{bmatrix}H^{B;t_{0}}_{t}&O\\\
O&O\end{bmatrix}.$ (4.10)

###### Proof.

We have (4.9) immediately by (4.7). Except for the lower left $n\times n$
submatrix, (4.10) is obtained from the definition of the $H$-matrices and
(4.7). For $i\in\\{n+1,\dots,2n\\}$ and $j\in\\{1,\dots,n\\}$, we have

$\displaystyle u^{h_{ij;t}}$
$\displaystyle=F^{\overline{B};t_{0}}_{j;t}|_{\text{Trop}(u)}\big{(}u^{[-\overline{b_{i1}}]_{+}},\dots,u^{-1},\dots,u^{[-\overline{b_{i,2n}}]_{+}}\big{)}\qquad\big{(}\text{$u^{-1}$
in the $i$th position}\big{)}$
$\displaystyle=F^{B;t_{0}}_{j;t}|_{\text{Trop}(u)}\big{(}u^{0},\dots,u^{0}\big{)}=1.$

Hence we obtain $h_{ij;t}=0$. ∎

###  4.2 Alternative derivations of Propositions 2.9, 3.5, Theorems 3.7 and
3.10

Using the principal extension, we give alternative derivations of Propositions
2.9, 3.5, Theorems 3.7 and 3.10. We think that these derivations are important
in point of the following. That is, we cancel out exchange matrix $B_{t}$ in
equations to derivate the desired equation in a process of proving. By the
principal extension, we can do it even if $B_{t}$ is not invertible in various
cases. We believe that we also apply this technique to any other proofs of
theorems in cluster algebra theory.

###### Alternative derivation of Proposition 2.9.

For any edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{\ell}$ in
$\mathbb{T}_{n}$, by using (2.9) and (2.21), we obtain two expressions of
$G^{-B;t_{0}}_{t^{\prime}}$ by the following diagram:

$\textstyle{G^{B;t_{0}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eqref{eq:G=G+BF}}$final-
seed mutation
(2.9)$\textstyle{G^{-B;t_{0}}_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$final-
seed mutation
(2.9)$\textstyle{G^{B;t_{0}}_{t^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eqref{eq:G=G+BF}}$$\textstyle{G^{-B;t_{0}}_{t^{\prime}}.}$

Applying (2.21) to $G^{B;t_{0}}_{t}$ before applying the final-seed mutation,
we have

$\displaystyle
G^{-B;t_{0}}_{t^{\prime}}=G_{t^{\prime}}^{B;t_{0}}+BF_{t^{\prime}}^{B;t_{0}}=G^{B;t_{0}}_{t}\big{(}J_{\ell}+[\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}-B\big{[}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+BF_{t^{\prime}}^{B;t_{0}}.$

On the other hand, applying the final-seed mutation to $G^{B;t_{0}}_{t}$
before applying (2.21), we have

$\displaystyle G^{-B;t_{0}}_{t^{\prime}}$
$\displaystyle=G^{-B;t_{0}}_{t}\big{(}J_{\ell}+[\varepsilon(-B_{t})]^{\bullet\ell}_{+}\big{)}-(-B)\big{[}\varepsilon
C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}$
$\displaystyle=\big{(}G_{t}^{B;t_{0}}+BF_{t}^{B;t_{0}}\big{)}\big{(}J_{\ell}+[-\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}+B\big{[}\varepsilon
C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}.$

Comparing these two expressions, we obtain

$\displaystyle BF^{B;t_{0}}_{t^{\prime}}=G^{B;t_{0}}_{t}(-\varepsilon
B_{t})^{\bullet\ell}+BF^{B;t_{0}}_{t}\big{(}J_{\ell}+[-\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}+B\big{[}\varepsilon
C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+B\big{[}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}.$ (4.11)

Hence if $B$ is invertible, we have

$\displaystyle F^{B;t_{0}}_{t^{\prime}}=B^{-1}G^{B;t_{0}}_{t}(-\varepsilon
B_{t})^{\bullet\ell}+F^{B;t_{0}}_{t}\big{(}J_{\ell}+[-\varepsilon
B_{t}]^{\bullet\ell}_{+}\big{)}+\big{[}\varepsilon
C^{-B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}+\big{[}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}.$ (4.12)

Since $B^{-1}G^{B;t_{0}}_{t}=C^{B;t_{0}}_{t}B_{t}^{-1}$, the first term of the
right hand side of (4.12) is $\big{(}{-}\varepsilon
C^{B;t_{0}}_{t}\big{)}^{\bullet\ell}$. Moreover, we note that we have

$\displaystyle\big{(}{-}\varepsilon
C^{B;t_{0}}_{t}\big{)}^{\bullet\ell}+\big{[}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}=\big{[}{-}\varepsilon
C^{B;t_{0}}_{t}\big{]}^{\bullet\ell}_{+}.$

Thus we have (2.29) as desired. When $B$ is not invertible, substituting
$\overline{B}$ for $B$ in (4.11), we have the equality which is substituted
$\overline{B}$ for $B$ in (4.12). Thanks to (4.8) and (4.9), comparing the
upper left $n\times n$ submatrix of it, we have (2.29) as desired. ∎

###### Alternative derivation of Proposition 3.5.

We prove (3.8). By using (3.3) and (2.20), we obtain two expressions of
$C^{-B_{1};t_{1}}_{t}$ by the following diagram:

Applying (2.20) to $C_{t}^{B;t_{0}}$ before applying the initial-seed
mutation, we have

$\displaystyle C^{-B_{1};t_{1}}_{t}$
$\displaystyle=\big{(}J_{k}+[-\varepsilon(-B)]^{k\bullet}_{+}\big{)}C^{-B;t_{0}}_{t}+H^{-B;t_{0}}_{t}(\varepsilon)^{k\bullet}(-B_{t})$
$\displaystyle=\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}\big{(}C^{B;t_{0}}_{t}+F^{B;t_{0}}_{t}B_{t}\big{)}-H^{-B;t_{0}}_{t}(\varepsilon)^{k\bullet}B_{t}.$

On the other hand, applying the initial-seed mutation to $C_{t}^{B;t_{0}}$
before applying (2.20), we have

$\displaystyle
C^{-B_{1};t_{1}}_{t}=C^{B_{1};t_{1}}_{t}+F^{B_{1};t_{1}}_{t}B_{t}=\big{(}J_{k}+[-\varepsilon
B]^{k\bullet}_{+}\big{)}C^{B;t_{0}}_{t}+H^{B;t_{0}}_{t}(\varepsilon)^{k\bullet}B_{t}+F^{B_{1};t_{1}}_{t}B_{t}.$

Comparing these two expressions, we have

$\displaystyle F^{B_{1};t_{1}}_{t}B_{t}=(\varepsilon
B)^{k\bullet}C^{B;t_{0}}_{t}+\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}B_{t}-H^{-B;t_{0}}_{t}(\varepsilon)^{k\bullet}B_{t}-H^{B;t_{0}}_{t}(\varepsilon)^{k\bullet}B_{t}.$

When $B$ is invertible, all $B_{t}$ also is. Therefore, we have

$\displaystyle F^{B_{1};t_{1}}_{t}=(\varepsilon
B)^{k\bullet}C^{B;t_{0}}_{t}B^{-1}_{t}+\big{(}J_{k}+[\varepsilon
B]^{k\bullet}_{+}\big{)}F^{B;t_{0}}_{t}-H^{-B;t_{0}}_{t}(\varepsilon)^{k\bullet}-H^{B;t_{0}}_{t}(\varepsilon)^{k\bullet}.$
(4.13)

By (2.10), $BC^{B;t_{0}}_{t}=G^{B;t_{0}}_{t}B_{t}$ for any $t$, the first term
of right hand side of (4.13) is $\big{(}\varepsilon
G^{B;t_{0}}_{t}\big{)}^{k\bullet}$. Hence we have desired equality. On the
other hand, when $B$ is not invertible, we consider replacing $B$ with
$\overline{B}$ in (4.13). Thanks to (4.6) and Corollary 4.5, we have (3.8). ∎

###### Alternative derivation of Theorem 3.7.

We get (3.16) by the same way as a previous proof. If $B$ is invertible, then
we have
$-\big{[}{-}G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}=\big{(}H^{B;t_{0}}_{t}\big{)}^{k\bullet}$.
Applying this equality to all $k=1,\dots,n$, we have
$H^{B;t_{0}}_{t}=-\big{[}{-}G^{B;t_{0}}_{t}\big{]}$. If $B$ is not invertible,
replacing $B$ with $\overline{B}$ in (3.16), we have
$-\big{[}{-}G^{\overline{B};t_{0}}_{t}\big{]}^{k\bullet}_{+}=\big{(}H^{\overline{B};t_{0}}_{t}\big{)}^{k\bullet}$.
Comparing the upper left $n\times n$ submatrices of both sides of it, we
obtain
$-\big{[}{-}G^{B;t_{0}}_{t}\big{]}^{k\bullet}_{+}=\big{(}H^{B;t_{0}}_{t}\big{)}^{k\bullet}$.
∎

###### Alternative derivation of Theorem 3.10.

By the equalities which substitute $B_{t}^{T}$, $t_{0}$ and $t$ for $B$,
$t_{0}$ and $t$ respectively in (2.20), (2.21) and (3.10), we have

$\displaystyle B\big{(}F_{t}^{B;t_{0}}\big{)}^{T}=BF_{t_{0}}^{B_{t}^{T};t}.$
(4.14)

When $B$ is invertible, we get (3.20) immediately. We prove the case that $B$
is not invertible. Replacing $B$ with $\overline{B}$ in (4.14), we have

$\displaystyle\big{(}F_{t}^{\overline{B};t_{0}}\big{)}^{T}=F_{t_{0}}^{\overline{B}_{t}^{T};t}$

because $\overline{B}$ is invertible. Since we have

$\displaystyle\big{(}F_{t}^{\overline{B};t_{0}}\big{)}^{T}=\begin{bmatrix}\big{(}F_{t}^{B;t_{0}}\big{)}^{T}&O\\\
O&O\end{bmatrix}$

and

$\displaystyle
F_{t_{0}}^{\overline{B}_{t}^{T};t}=F_{t_{0}}^{\overline{B^{T}}_{t};t}=\begin{bmatrix}F_{t_{0}}^{B_{t}^{T};t}&O\\\
O&O\end{bmatrix}$

by (4.7) and Remark 4.4, we have (3.20). ∎

### Acknowledgments

The authors are grateful to Professor Tomoki Nakanishi for useful comments and
advice. We also thanks Toshiya Yurikusa for insightful comments about the
principal extension. We also would like express our gratitude to the referees.

## References

  * [1] Buan A.B., Marsh R., Reineke M., Reiten I., Todorov G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618, arXiv:math.RT/0402054.
  * [2] Cao P., Li F., Uniformly column sign-coherence and the existence of maximal green sequences, arXiv:1712.00973.
  * [3] Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865–930, arXiv:math.AG/0311245.
  * [4] Fomin S., Shapiro M., Thurston D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), 83–146, arXiv:math.RA/0608367.
  * [5] Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497–529, arXiv:math.RT/0104151.
  * [6] Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112–164, arXiv:math.RA/0602259.
  * [7] Gross M., Hacking P., Keel S., Kontsevich M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497–608, arXiv:1411.1394.
  * [8] Nakanishi T., Zelevinsky A., On tropical dualities in cluster algebras, in Algebraic Groups and Quantum Groups, Contemp. Math., Vol. 565, Amer. Math. Soc., Providence, RI, 2012, 217–226, arXiv:1101.3736.
  * [9] Reading N., Stella S., Initial-seed recursions and dualities for $d$-vectors, Pacific J. Math. 293 (2018), 179–206, arXiv:1409.4723.