# The valuation pairing on an upper cluster algebra

Peigen Cao  Université Paris Cité, UFR de Mathématiques, CNRS, Institut de
Mathématiques de Jussieu–Paris Rive Gauche, IMJ-PRG, Bâtiment Sophie Germain,
75205 Paris Cedex 13, France
Present address: Einstein Institute of Mathematics, Edmond J. Safra Campus,
The Hebrew University of Jerusalem, Jerusalem 91904, Israel<EMAIL_ADDRESS>,  Bernhard Keller  Université Paris Cité
UFR de Mathématiques
CNRS
Institut de Mathématiques de Jussieu–Paris Rive Gauche, IMJ-PRG
Bâtiment Sophie Germain
75205 Paris Cedex 13
France<EMAIL_ADDRESS>https://webusers.imj-prg.fr/
bernhard.keller/ and  Fan Qin  School of Mathematical Sciences
Shanghai Jiao Tong University
Shanghai 200240
China<EMAIL_ADDRESS>
###### Abstract.

It is known that many (upper) cluster algebras are not unique factorization
domains. We exhibit the local factorization properties with respect to any
given seed $t$: any non-zero element in a full rank upper cluster algebra can
be uniquely written as the product of a cluster monomial in $t$ and another
element not divisible by the cluster variables in $t$. Our approach is based
on introducing the valuation pairing on an upper cluster algebra: it counts
the maximal multiplicity of a cluster variable among the factorizations of any
given element.

We apply the valuation pairing to obtain many results concerning factoriality,
$d$-vectors, $F$-polynomials and the combinatorics of cluster Poisson
variables. In particular, we obtain that full rank and primitive upper cluster
algebras are factorial; an explanation of $d$-vectors using valuation pairing;
a cluster monomial in non-initial cluster variables is determined by its
$F$-polynomial; the $F$-polynomials of non-initial cluster variables are
irreducible; and the cluster Poisson variables parametrize the exchange pairs
of the corresponding upper cluster algebra.

###### Key words and phrases:

Cluster algebra, valuation pairing, $d$-vectors, $F$-polynomials, cluster
Poisson variables, unique factorization domains

###### 2010 Mathematics Subject Classification:

13F60; 05E40

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 Main results
    3. 1.3 Contents
    4. 1.4 Convention and assumption
  2. 2 Preliminaries
    1. 2.1 Basics on cluster algebras and cluster Poisson algebras
    2. 2.2 $d$-vectors, $f$-vectors, $g$-vectors, $c$-vectors and $F$-polynomials
    3. 2.3 Compatibility degrees on the set of cluster variables
    4. 2.4 Unique factorization domains
  3. 3 The valuation pairing on an upper cluster algebra
  4. 4 Application to factoriality of upper cluster algebras
    1. 4.1 Characterizations for the factoriality of upper cluster algebras
    2. 4.2 Full rank, primitive upper cluster algebras are factorial
    3. 4.3 Examples of non-factorial upper cluster algebras
    4. 4.4 Ray fish theorem
  5. 5 Applications to $d$-vectors
    1. 5.1 $d$-compatibility degree and $d$-vectors via the valuation pairing
    2. 5.2 Local factorizations of cluster monomials
  6. 6 Application to $F$-polynomials
    1. 6.1 From $F$-polynomials to monomials in non-initial cluster variables
    2. 6.2 $F$-polynomials of non-initial cluster variables are irreducible
  7. 7 Application to combinatorics of cluster Poisson variables
    1. 7.1 Parametrization of $\mathscr{A}$-exchange pairs by cluster Poisson variables
    2. 7.2 Exchange graphs of cluster Poisson algebras

##  1\. Introduction

###  1.1. Background

Around the year 2000, Fomin and Zelevinsky introduced [FZ02] cluster algebras
with the aim of developing a combinatorial approach to the theory of canonical
bases in quantum groups and the closely related theory of total positivity in
algebraic groups. Since then, cluster algebras have been linked to numerous
other subjects and their study has flourished, cf. for example the surveys
[Lec10, Fom10, Kel12, KD20].

A cluster algebra $\mathcal{A}$ is a subalgebra of an ambient field
$\mathcal{F}$ generated by certain combinatorially defined generators (called
cluster variables), which are grouped into overlapping sets (called clusters)
of constant cardinality $n$. Different clusters are obtained from each other
by a sequence of mutations. One remarkable feature of cluster algebras is the
Laurent phenomenon [FZ02], that is, for any given cluster
$A_{t_{0}}=(A_{1;t_{0}},\ldots,A_{n;t_{0}})$, each cluster variable $A_{k;t}$
can be written as

$A_{k;t}=\frac{P(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{d_{1}}\cdots
A_{n;t_{0}}^{d_{n}}},$

where $P$ is a polynomial in variables from $A_{t_{0}}$ such that
$A_{i;t_{0}}$ does not divide $P$ for any $i$. The vector

${\bf d}^{t_{0}}(A_{k;t})=(d_{1},\ldots,d_{n})^{\rm T}$

is called the $d$-vector of $A_{k;t}$ with respect to $A_{t_{0}}$ and the
polynomial $P$ is called the numerator polynomial of $A_{k;t}$ with respect to
$A_{t_{0}}$.

For each cluster algebra $\mathcal{A}$, Fock–Goncharov defined [FG09] a pair
of varieties: the cluster $K_{2}$-variety and the cluster Poisson variety (we
follow the terminology of the appendix to [SW20]). The upper cluster algebra
$\mathcal{U}$ is defined to be the algebra of global functions on the cluster
$K_{2}$-variety, and the cluster Poisson algebra $\mathcal{X}$ is defined to
be the algebra of the global functions on the cluster Poisson variety. The
clusters of $\mathcal{U}$ correspond to local toric charts and the cluster
variables correspond to local coordinates (which happen to be global
functions) of the cluster $K_{2}$-variety. The cluster Poisson variety admits
a canonical atlas of dual toric charts whose coordinates are the cluster
Poisson variables (but they are not global) of $\mathcal{X}$.

###  1.2. Main results

An upper cluster algebra is constructed from seeds. Each seed contains a
cluster and a skew-symmetrizable matrix $B$ called its exchange matrix, see
Section 2. Moreover, when the upper cluster algebra has geometric type, we can
associate to $t$ the extended exchange matrix $\tilde{B}$, see Definition
2.10. The upper cluster algebra is said to be full rank or primitive if
$\tilde{B}$ is. Such properties are independent of the choice of the seed.

#### Valuation pairings, factoriality and the ray fish theorem

It is known from [GLS13] that many (upper) cluster algebras are not unique
factorization domains. In order to study the local factorization properties of
upper cluster algebras, we introduce the valuation pairing (see Definition
3.1) on any upper cluster algebra $\mathcal{U}$. To each pair $(A_{k;t},M)$
consisting of a cluster variable $A_{k;t}$ and an element $M$ in
$\mathcal{U}$, it associates the largest integer $s$ (possibly infinity) such
that $M/A_{k;t}^{s}$ still belongs to $\mathcal{U}$. We write
$(A_{k;t}||M)_{v}=s$. Using the valuation pairing we prove that any full rank
upper cluster algebra has the following local unique factorization property:
For each seed $t$ of $\mathcal{U}$, any non-zero element $M$ can be uniquely
factorized as $M=N\cdot L$, where $N$ is a cluster monomial in $t$ and $L$ is
an element in $\mathcal{U}$ not divisible by any cluster variable in $t$. We
will give many applications to $d$-vectors, $F$-polynomials, factoriality of
upper cluster algebras and combinatorics of cluster Poisson variables.

As an application to factoriality of upper cluster algebras, we prove that a
full rank upper cluster algebra $\mathcal{U}$ with initial seed $t_{0}$ is
factorial if and only if each exchange binomial of $t_{0}$ is irreducible in
the corresponding polynomial ring (see Theorem 4.4). In particular, full rank,
primitive upper cluster algebras are factorial (see Theorem 4.8 (i)). These
include principal coefficient upper cluster algebras as a special case. For
full rank, primitive upper cluster algebras, we also prove that the numerator
polynomials of non-initial cluster variables are irreducible (see Theorem 4.8
(ii)).

The starfish theorem in [BFZ05], cf. also Theorem 2.16 of the present paper,
plays a very important role in this paper. It states that any full rank upper
cluster algebra $\mathcal{U}$ can be written as the intersection of $n+1$
Laurent polynomial rings. Here $n$ is the rank of $\mathcal{U}$. Thanks to the
results on factoriality of upper cluster algebras, we show that any full rank,
primitive upper cluster algebra can be written as the intersection of two
Laurent polynomial rings (see Theorem 4.18). We call this the ray fish
theorem.

#### Application to $d$-vectors

In [CL20], Li and the first author of the present paper proved that there
exists a well-defined function $(-||-)_{d}$ on the set of cluster variables,
which is called the $d$-compatibility degree. The values of the
$d$-compatibility degree $(-||-)_{d}$ are given by the components of the
$d$-vectors. One remarkable property of the $d$-compatibility degree is that
it uniquely determines how the set of cluster variables is grouped into
clusters. As an application to $d$-vectors, we show how to express the
$d$-compatibility degree and the $d$-vectors using the valuation pairing for
full rank upper cluster algebras (see Theorem 5.1). As an application, in the
full rank, primitive case, we prove that if $M$ is a monomial in non-initial
cluster variables, then $M$ and its $d$-vector are uniquely determined by the
numerator polynomial $P_{M}$ of $M$ (see Proposition 5.6).

#### Application to $F$-polynomials

Let $B$ be an $n\times n$ skew-symmetrizable integer matrix. The
$F$-polynomial $F_{k;t}^{B;t_{0}}$ associated with $(B,t_{0};k,t)$ may be
constructed by an explicit recursion or using the cluster algebra
$\mathcal{A}$ with principal coefficients [FZ07] associated with $(B,t_{0})$:
Let $A_{t_{0}}=(A_{1;t_{0}},\ldots,A_{n;t_{0}})$ be the initial cluster of
$\mathcal{A}$ and $A_{k;t}$ a cluster variable of $\mathcal{A}$. In this case,
the cluster variable $A_{k;t}$ can be written as a Laurent polynomial in
$\mathbb{Z}[Z_{1},\ldots,Z_{n}][A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$. Then the $F$-polynomial of $A_{k;t}$ is the specialization given by

$F_{k;t}^{B;t_{0}}(Z_{1},\ldots,Z_{n}):=A_{k;t}|_{A_{1;t_{0}}=\ldots=A_{n;t_{0}}=1}\in\mathbb{Z}[Z_{1},\ldots,Z_{n}].$

When $B$ and $t_{0}$ is clear, we just write $F_{k;t}^{B;t_{0}}$ for
$F_{k;t}$.

Let $M=\prod\limits_{i=1}^{s}M_{i}$ be a monomial in cluster variables, where
each $M_{i}$ is a cluster variable of $\mathcal{A}$. The $F$-polynomial of $M$
is the polynomial $F_{M}:=\prod\limits_{i=1}^{s}F_{M_{i}}$, where $F_{M_{i}}$
is the $F$-polynomial of the cluster variable $M_{i}$.

The $F$-polynomials are the non tropical ingredients of the canonical
expressions [NZ12] for both, the cluster variables and the cluster Poisson
variables. They are fundamental in the additive categorification of cluster
algebras (see for example [CC06, DWZ10] and the surveys [BM06, GLS08, Kel10,
Kel12, Pla18]) and in their link to Donaldson–Thomas theory (see for example
[KS08, Nag13, Bri17]). It is known that $F$-polynomials enjoy many nice
properties, for example, they have positive coefficients and constant term
$1$. We refer the readers to [LS15, DWZ10, GHKK18] for these results and to
[GY20, Gyo21, FG19a, Fei19a, Fei19b, LP22] for some recent work on
$F$-polynomials.

As an application to $F$-polynomials, we prove that if $M$ is a monomial in
non-initial cluster variables, then $M$ is uniquely determined by its
$F$-polynomial $F_{M}$ (see Theorem 6.1). We also prove that the
$F$-polynomials of non-initial cluster variables are irreducible (see Theorem
6.2).

#### Application to cluster Poisson variables

As an application to combinatorics of cluster Poisson variables, we give
several equivalent characterizations of when two cluster Poisson variables are
equal (see Theorem 7.5). Recall that each mutation $t^{\prime}=\mu_{k}(t)$
gives an $\mathscr{A}$-exchange pair $(A_{k;t},A_{k;t^{\prime}})$ of the upper
cluster algebra $\mathcal{U}$ and an $\mathscr{X}$-exchange pair
$(X_{k;t},X_{k;t^{\prime}})$ of the cluster Poisson algebra $\mathcal{X}$.

As the first application of Theorem 7.5, we prove that the cluster Poisson
variables of a cluster Poisson algebra $\mathcal{X}$ parametrize the
$\mathscr{A}$-exchange pairs of the upper cluster algebra $\mathcal{U}$ of the
same type as $\mathcal{X}$ (see Theorem 7.6). This extends the corresponding
result by Sherman-Bennett [SB19] from the finite type case to full generality.

As the second application of Theorem 7.5, we prove that the
$\mathscr{X}$-seeds of $\mathcal{X}$ whose Poisson clusters contain particular
cluster Poisson variables form a connected subgraph of the exchange graph of
$\mathcal{X}$ (see Theorem 7.10). This is analogous to the result on the
exchange graphs of cluster algebras given in [CL20], cf. also Theorem 2.17 of
this paper.

###  1.3. Contents

This paper is organized as follows: In Section 2, some basic definitions,
notations and known results are introduced. In Section 3, we introduce the
valuation pairing $(-||-)_{v}$ on any upper cluster algebra and prove the
local unique factorization property for full rank upper cluster algebras (see
Theorem 3.7). In Sections 4, 5, 6, 7, we give the applications to $d$-vectors,
$F$-polynomials, factoriality of upper cluster algebras and combinatorics of
cluster Poisson variables. To be more precise:

In Section 4.1, we prove that a full rank upper cluster algebra $\mathcal{U}$
with initial seed $t_{0}$ is factorial if and only if each exchange binomial
of $t_{0}$ is irreducible in the corresponding polynomial ring (see Theorem
4.4). In particular, we show in Section 4.2 that full rank, primitive upper
cluster algebras are factorial (see Theorem 4.8 (i)). Moreover, for these
upper cluster algebras, we also show that the numerator polynomials of non-
initial cluster variables are irreducible (see Theorem 4.8 (ii)). In Section
4.3, we give some examples of non-factorial upper cluster algebras. In Section
4.4, we prove the ray fish theorem, which states that any full rank, primitive
upper cluster algebra $\mathcal{U}$ can be written as the intersection of two
Laurent polynomial rings (see Theorem 4.18).

In Section 5.1, we show how to express the $d$-compatibility degree and the
$d$-vectors using the valuation pairing for full rank upper cluster algebras
(see Theorem 5.1). In Section 5.2, we give a local factorization for cluster
monomials (see Theorem 5.4). As an application, in the full rank, primitive
case, we prove that if $M$ is a monomial in non-initial cluster variables,
then $M$ and its $d$-vector are uniquely determined by the numerator
polynomial $P_{M}$ of $M$ (see Proposition 5.6).

In Section 6.1, we prove that if $M$ is a monomial in non-initial cluster
variables, then $M$ is uniquely determined by its $F$-polynomial $F_{M}$ (see
Theorem 6.1). In Section 6.2, we prove that the $F$-polynomials of non-initial
cluster variables are irreducible (see Theorem 6.2).

In Section 7, we give several equivalent characterizations of when two cluster
Poisson variables are equal (see Theorem 7.5). As the first application, we
prove that the cluster Poisson variables of a cluster Poisson algebra
$\mathcal{X}$ parametrize the $\mathscr{A}$-exchange pairs of the upper
cluster algebra $\mathcal{U}$ of the same type as $\mathcal{X}$ (see Theorem
7.6). As the second application, we prove that the $\mathscr{X}$-seeds of
$\mathcal{X}$ whose Poisson clusters contain particular cluster Poisson
variables form a connected subgraph of the exchange graph of $\mathcal{X}$
(see Theorem 7.10).

The following diagram gives the logical dependence among the proofs of the
main theorems in this paper.

$\begin{array}[]{ccc}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
28.43057pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 58.40279pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmunique}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\;}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
150.79156pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.61107pt\raise-38.60011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
163.0821pt\raise-38.60011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
150.79156pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmind}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces\;}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
249.61508pt\raise-38.60011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
238.59698pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmup}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
267.02756pt\raise-38.60011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
199.42204pt\raise-38.60011pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmstrong}}$}}}}}}}{\hbox{\kern
52.43057pt\raise-48.54456pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmcufd}\;\;\ref{thmufd}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\;}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
28.43059pt\raise-48.54456pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
150.79156pt\raise-48.54456pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmffirr}\;\;}$}}}}}}}{\hbox{\kern
237.20813pt\raise-48.54456pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmfpoly}\;\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
199.42204pt\raise-87.14467pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
59.79164pt\raise-97.08911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmbij}\;}$}}}}}}}{\hbox{\kern
150.79156pt\raise-97.08911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{mainthm}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\;\;}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
119.43051pt\raise-97.08911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
238.59698pt\raise-97.08911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
238.59698pt\raise-97.08911pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\rm
Theorem}\;\ref{thmgraph}}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{array}$

###  1.4. Convention and assumption

Throughout this article, $\mathbb{K}$ is a field of characteristic $0$ or
$\mathbb{K}=\mathbb{Z}$ and all upper cluster algebras are considered as
algebras over $\mathbb{KP}$, where $\mathbb{P}$ is some abelian multiplicative
group and $\mathbb{KP}$ the corresponding group ring.

We always assume that the exchange binomials of upper cluster algebras in this
article are not invertible in $\mathbb{KP}$. Note that when
$\mathbb{K}=\mathbb{Z}$, this condition is always satisfied. When $\mathbb{K}$
is a field and we have a trivial exchange relation

$A_{k}A_{k}^{\prime}=P_{k}\in(\mathbb{KP})^{\times}$

in an upper cluster algebra $\mathcal{U}$, we can always freeze the cluster
variable $A_{k}$ and consider the upper cluster algebra
$\mathcal{U}^{\dagger}$ with smaller rank over
$\mathbb{KP}^{\dagger}=\mathbb{KP}[A_{k}^{\pm}]$. Note that
$\mathcal{U}^{\dagger}$ and $\mathcal{U}$ are isomorphic as
$\mathbb{K}$-algebras.

### Acknowledgements

The authors are very grateful to Luc Pirio, whose conjectures have inspired a
good part of the results of this paper. P. Cao would like to thank Xiaofa
Chen, Yu Wang and Yilin Wu for their helpful discussions during his stay in
Paris. B. Keller is indebted to Lauren Williams and Melissa Sherman-Bennett
for their interest and for stimulating conversations.

P. Cao thanks the Fondation Sciences Mathématiques de Paris (FSMP) for funding
his postdoctoral stay (2019–2020) at the Université de Paris during which the
first two authors began work on this paper. He also thanks the ERC (Grant No.
669655) for funding his postdoctoral stay (2021–present) at The Hebrew
University of Jerusalem.

##  2\. Preliminaries

###  2.1. Basics on cluster algebras and cluster Poisson algebras

An $n\times n$ integer matrix $B$ is skew-symmetrizable if there is an integer
diagonal matrix $S$ with strictly positive diagonal entries such that $SB$ is
skew-symmetric. Such an $S$ is said to be a skew-symmetrizer of $B$.

######  Definition 2.1 (Matrix mutation).

Let $\tilde{B}=\begin{pmatrix}B_{n\times n}\\\ C_{m\times
n}\end{pmatrix}=(b_{ij})$ be an $(n+m)\times n$ integer matrix with $B$ skew-
symmetrizable. The mutation $\mu_{k}(\tilde{B})$ of $\tilde{B}$ at
$k\in\\{1,\ldots,n\\}$ is the new matrix
$\mu_{k}(\tilde{B})=\tilde{B}^{\prime}=\begin{pmatrix}B_{n\times
n}^{\prime}\\\ C_{m\times n}^{\prime}\end{pmatrix}=(b_{ij}^{\prime})$ given by

$\displaystyle b_{ij}^{\prime}=\begin{cases}-b_{ij}~{},&\text{if }i=k\text{ or
}j=k;\\\ b_{ij}+{\rm sgn}(b_{ik}){\rm
max}(b_{ik}b_{kj},0)~{},&\text{otherwise}.\end{cases}$

It is not hard to check that the submatrix $B^{\prime}$ of
$\tilde{B}^{\prime}$ is still skew-symmetrizable with the same skew-
symmetrizer as $B$ and $\mu_{k}$ is an involution.

######  Proposition 2.2.

[BFZ05, Lemma 3.2] Matrix mutations preserve the rank of $\tilde{B}$.

Recall that $(\mathbb{P},\oplus,\cdot)$ is a semifield if $(\mathbb{P},\cdot)$
is an abelian multiplicative group endowed with a binary operation of
auxiliary addition $\oplus$ which is commutative, associative and satisfies
that the multiplication distributes over the auxiliary addition.

The tropical semifield $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$ is the free
(multiplicative) abelian group generated by $Z_{1},\ldots,Z_{m}$ with
auxiliary addition $\oplus$ defined by

$\prod\limits_{i}Z_{i}^{a_{i}}\oplus\prod\limits_{i}Z_{i}^{b_{i}}=\prod\limits_{i}Z_{i}^{{\rm
min}(a_{i},b_{i})}.$

Let $\mathbb{Q}_{\rm sf}(Z_{1},\ldots,Z_{m})$ be the set of all non-zero
rational functions in $m$ independent variables $Z_{1},\ldots,Z_{m}$, which
can be written as subtraction-free rational expressions in
$Z_{1},\ldots,Z_{m}$. The set $\mathbb{Q}_{\rm sf}(Z_{1},\ldots,Z_{m})$ is a
semifield with respect to the usual operations of multiplication and addition.
It is called an universal semifield.

######  Definition 2.3 ($\mathscr{X}$-seed and cluster Poisson seed).

(i) A (labeled) $\mathscr{X}$-seed over a semifield $\mathbb{P}$ is a pair
$(B,X)$, where

  * •

$B=(b_{ij})$ is an $n\times n$ skew-symmetrizable integer matrix, called an
exchange matrix ;

  * •

$X=(X_{1},\ldots,X_{n})$ is an $n$-tuple of elements in $\mathbb{P}$. We call
$X$ the $\mathscr{X}$-cluster and $X_{1},\ldots,X_{n}$ the
$\mathscr{X}$-variables of $(B,X)$.

(ii) Let $(B,X)$ be an $\mathscr{X}$-seed over a semifield $\mathbb{P}$. If
$\mathbb{P}=\mathbb{Q}_{\rm sf}(X_{1},\ldots,X_{n})$, we call $(B,X)$ a
cluster Poisson seed, $X$ the Poisson cluster, and $X_{1},\ldots,X_{n}$ the
cluster Poisson variables of $(B,X)$.

Throughout this article, we assume that $\mathbb{K}$ is a field of
characteristic $0$ or $\mathbb{K}=\mathbb{Z}$ and we take the ambient field
$\mathcal{F}$ to be the field of rational functions in $n$ independent
variables with coefficients in $\mathbb{KP}$.

######  Definition 2.4 ($\mathscr{A}$-seed).

A (labeled) $\mathscr{A}$-seed over $\mathbb{P}$ is a triple $(B,X,A)$, where

  * •

$(B,X)$ forms an $\mathscr{X}$-seed over $\mathbb{P}$;

  * •

$A=(A_{1},\dots,A_{n})$ is an $n$-tuple such that $\\{A_{1},\dots,A_{n}\\}$ is
a free generating set of $\mathcal{F}$ over $\mathbb{KP}$. We call $A$ the
cluster and $A_{1},\dots,A_{n}$ the cluster variables of $(B,X,A)$;

######  Definition 2.5 ($\mathscr{X}$-seed mutation and $\mathscr{X}$-exchange
pair).

Let $(B,X)$ be an $\mathscr{X}$-seed over $\mathbb{P}$. Define the
$\mathscr{X}$-seed mutation of $(B,X)$ at $k\in\\{1,\ldots,n\\}$ as a new pair
$\mu_{k}(B,X)=(B^{\prime},X^{\prime})$, where $B^{\prime}=\mu_{k}(B)$ and
$X^{\prime}=(X_{1}^{\prime},\ldots,X_{n}^{\prime})$ is given by

$\displaystyle X_{i}^{\prime}=\begin{cases}X_{k}^{-1}~{},&\text{if }i=k;\\\
X_{i}X_{k}^{{\rm max}(b_{ki},0)}(1\bigoplus X_{k})^{-b_{ki}}~{},&\text{if
}i\neq k.\end{cases}$

In this case, $(X_{k},X_{k}^{\prime})$ is called an $\mathscr{X}$-exchange
pair.

######  Definition 2.6 ($\mathscr{A}$-seed mutation and $\mathscr{A}$-exchange
pair).

Let $(B,X,A)$ be an $\mathscr{A}$-seed over $\mathbb{P}$. Define the mutation
of $(B,X,A)$ at $k\in\\{1,\ldots,n\\}$ as a new triple
$\mu_{k}(B,X,A)=(B^{\prime},X^{\prime},A^{\prime})$, where
$(B^{\prime},X^{\prime})=\mu_{k}(B,X)$ and
$A^{\prime}=(A_{1}^{\prime},\ldots,A_{n}^{\prime})$ is given by

$A_{i}^{\prime}=\begin{cases}A_{i},&\text{if }i\neq k;\\\
A_{k}^{-1}(\frac{X_{k}}{1\oplus
X_{k}}\prod\limits_{b_{jk}>0}A_{j}^{b_{jk}}+\frac{1}{1\oplus
X_{k}}\prod\limits_{b_{jk}<0}A_{j}^{-b_{jk}}),&\text{if }i=k.\end{cases}$

In this case, $(A_{k},A_{k}^{\prime})$ is called an $\mathscr{A}$-exchange
pair. The binomial

$\frac{X_{k}}{1\oplus
X_{k}}\prod\limits_{b_{jk}>0}A_{j}^{b_{jk}}+\frac{1}{1\oplus
X_{k}}\prod\limits_{b_{jk}<0}A_{j}^{-b_{jk}}$

is called the $k$-th exchange binomial of $(B,X,A)$.

It is not hard to check that each mutation $\mu_{k}$ maps a seed
($\mathscr{X}$-seed or cluster Poisson seed or $\mathscr{A}$-seed) to a new
seed of the same type and that $\mu_{k}$ is an involution.

Let $\mathbb{T}_{n}$ be the $n$-regular tree. Let us label the edges of
$\mathbb{T}_{n}$ by $1,\dots,n$ such that the $n$ different edges adjacent to
the same vertex of $\mathbb{T}_{n}$ receive different labels.

######  Definition 2.7 (Seed pattern).

A seed pattern $\mathcal{S}$ over $\mathbb{P}$ is an assignment of a seed
$\Sigma_{t}$ ($\mathscr{X}$-seed or cluster Poisson seed or
$\mathscr{A}$-seed) to every vertex $t$ of the $n$-regular tree
$\mathbb{T}_{n}$ such that $\Sigma_{t^{\prime}}=\mu_{k}(\Sigma_{t})$ for any
edge $t^{~{}\underline{\quad k\quad}}~{}t^{\prime}$.

Note that the datum of an initial seed at a vertex $t_{0}$ determines a unique
seed pattern. Now we give some symbols which are used in the sequel. We always
write $B_{t}=(b_{ij}^{t})$, $X_{t}=(X_{1;t},\ldots,X_{n;t})$ and
$A_{t}=(A_{1;t},\ldots,A_{n;t})$. For simplicity, we will also use $t$ to
denote the seed at $t\in\mathbb{T}_{n}$.

Two (labeled) seeds are equivalent if they are the same up to relabeling.

######  Definition 2.8 (Exchange graph).

Let $\mathcal{S}$ be a seed pattern. The exchange graph ${\bf
EG}(\mathcal{S})$ of $\mathcal{S}$ is a graph whose vertices are in bijection
with the seeds (up to equivalence) of $\mathcal{S}$ and whose edges correspond
to the seed mutations.

######  Proposition 2.9.

[FZ07, Proposition 3.9] Let $\mathcal{S}$ be an $\mathscr{A}$-seed pattern.
With each $\mathscr{A}$-seed $t=(B_{t},X_{t},A_{t})$, we associate an
$\mathscr{X}$-seed $(B_{t},\hat{X}_{t})$, where
$\hat{X}_{t}=(\hat{X}_{1;t},\ldots,\hat{X}_{n;t})$ is given by
$\hat{X}_{j;t}:=X_{j;t}\prod\limits_{i=1}^{n}A_{i;t}^{b_{ij}^{t}}$. Then
$\hat{\mathcal{S}}=\\{(B_{t},\hat{X}_{t})\\}_{t\in\mathbb{T}_{n}}$ forms an
$\mathscr{X}$-seed pattern.

Now we recall some seed patterns ($\mathscr{X}$-seed patterns or
$\mathscr{A}$-seed patterns) over particular semifields.

  * •

A seed pattern $\mathcal{S}$ over $\mathbb{P}$ is said to be of geometric
type, if $\mathbb{P}$ is a tropical semifield, say $\mathbb{P}={\rm
Trop}(Z_{1},\ldots,Z_{m})$.

  * •

A seed pattern $\mathcal{S}$ over $\mathbb{P}$ is said to be with principal
coefficients at $t_{0}$, if $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{n})$ and
$X_{i;t_{0}}=Z_{i}$ for $i=1,\ldots,n$.

  * •

A seed pattern $\mathcal{S}$ over $\mathbb{P}$ is said to be with universal
coefficient semifield, if
$\mathbb{P}=\mathbb{Q}_{\rm{sf}}(X_{1;t_{0}},\ldots,X_{n;t_{0}})$ for some
seed $t_{0}$ of $\mathcal{S}$.

######  Definition 2.10 (Coefficient matrices and extended exchange matrices).

Let $\mathcal{S}$ be a seed pattern of geometric type with coefficient
semifield $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$. We know that each
$\mathscr{X}$-variable $X_{k;t}$ has the form

$X_{k;t}=Z_{1}^{c_{1k}^{t}}\cdots Z_{m}^{c_{mk}^{t}}.$

The matrix $C_{t}:=(c_{ij}^{t})_{m\times n}$ is called the coefficient matrix
at $t$ and the $(n+m)\times n$ matrix $\tilde{B}_{t}=\begin{pmatrix}B_{t}\\\
C_{t}\end{pmatrix}$ is called the extended exchange matrix at $t$.

######  Proposition 2.11.

[FZ02, Proposition 5.8] Let $\mathcal{S}$ be a seed pattern of geometric type.
Then for any edge $t^{~{}\underline{\quad k\quad}}~{}t^{\prime}$ in
$\mathbb{T}_{n}$, we have $\tilde{B}_{t^{\prime}}=\mu_{k}(\tilde{B}_{t})$,
where $\tilde{B}_{t}$ and $\tilde{B}_{t^{\prime}}$ are the extended exchange
matrices at $t$ and $t^{\prime}$.

######  Definition 2.12 (Cluster Poisson algebra).

Let $\mathcal{S}_{\rm uc}$ be an $\mathscr{X}$-seed pattern with universal
coefficient semifield. The cluster Poisson algebra
$\mathcal{X}=\mathcal{X}(\mathcal{S}_{\rm uc})$ associated with
$\mathcal{S}_{\rm uc}$ is the intersection

$\mathcal{X}=\bigcap\limits_{t\in\mathbb{T}_{n}}{\mathcal{L}}(t),$

where ${\mathcal{L}}(t)=\mathbb{K}[X_{1;t}^{\pm 1},\ldots,X_{n;t}^{\pm 1}]$.

######  Definition 2.13 (Cluster algebra and upper cluster algebra).

Let $\mathcal{S}$ be an $\mathscr{A}$-seed pattern over a semifield
$\mathbb{P}$.

(i) The cluster algebra $\mathcal{A}=\mathcal{A}(\mathcal{S})$ associated with
$\mathcal{S}$ is the $\mathbb{KP}$-subalgebra of $\mathcal{F}$ generated by
the cluster variables of $\mathcal{S}$, namely,
$\mathcal{A}=\mathbb{KP}[A_{1;t},\ldots,A_{n;t}|t\in\mathbb{T}_{n}]$.

(ii) The upper cluster algebra $\mathcal{U}=\mathcal{U}(\mathcal{S})$
associated with $\mathcal{S}$ is the intersection

$\mathcal{U}=\bigcap\limits_{t\in\mathbb{T}_{n}}{\mathcal{L}}(t),$

where ${\mathcal{L}}(t)=\mathbb{KP}[A_{1;t}^{\pm 1},\ldots,A_{n;t}^{\pm 1}]$.

Since cluster Poisson algebras and (upper) cluster algebras are defined from
seed patterns, we can talk about the exchange graphs of these algebras. We can
also talk about the (upper) cluster algebras of geometric type, with principal
coefficients and with universal coefficient semifield.

######  Theorem 2.14 (Laurent phenomenon and positivity).

Let $\mathcal{A}$ be a cluster algebra with coefficient semifield $\mathbb{P}$
and initial seed $t_{0}$. The following statements hold.

  * (i)

[FZ02] Each cluster variable $A_{k;t}$ of $\mathcal{A}$ can be written as a
Laurent polynomial in $\mathbb{ZP}[A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$.

  * (ii)

[GHKK18] The coefficients in above Laurent polynomial are positive.

  * (iii)

[FZ03] If $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$, then each cluster
variable $A_{k;t}$ of $\mathcal{A}$ can be written as a Laurent polynomial in
$\mathbb{Z}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$.

A geometric upper cluster algebra with initial seed $t_{0}$ is called a full
rank upper cluster algebra if its initial extended exchange matrix
$\tilde{B}_{t_{0}}$ has full rank.

######  Remark 2.15.

For a full rank upper cluster algebra $\mathcal{U}$, we know that every
extended exchange matrix $\tilde{B}_{t}$ of $\mathcal{U}$ has full rank, since
mutations preserve the rank of the extended exchange matrices (see Proposition
2.2).

######  Theorem 2.16 (Starfish theorem).

Let $\mathcal{U}$ be a full rank upper cluster algebra and $t_{0}$ a seed of
$\mathcal{U}$. Then we have

$\mathcal{U}=\bigcap\limits_{i=0}^{n}{\mathcal{L}}(t_{i}),$

where $t_{j}=\mu_{j}(t_{0})$ for $j=1,\ldots,n$ and
${\mathcal{L}}(t_{i})=\mathbb{KP}[A_{1;t_{i}}^{\pm 1},\ldots,A_{n;t_{i}}^{\pm
1}]$ for $i=0,1,\ldots,n$.

###### Proof.

For the case $\mathbb{K}=\mathbb{Z}$, the proof can be found in [BFZ05] and
the proof there also works for the case where $\mathbb{K}$ is a field of
characteristic $0$. ∎

######  Theorem 2.17.

[CL20] Let $\mathcal{A}$ be a cluster algebra with coefficient semifield
$\mathbb{P}$. Then the seeds of $\mathcal{A}$ whose clusters contain
particular cluster variables form a connected subgraph of the exchange graph
of $\mathcal{A}$.

The following result is a direct corollary.

######  Corollary 2.18.

[CL20] Let $\mathcal{A}$ be a cluster algebra with coefficient semifield
$\mathbb{P}$ and $t_{1},t_{2}$ two seeds of $\mathcal{A}$. If $t_{1}$ and
$t_{2}$ have at least $n-1$ common cluster variables, then in the exchange
graph of $\mathcal{A}$ either $t_{1}$ and $t_{2}$ represent the same vertex or
there is an edge between $t_{1}$ and $t_{2}$.

Recall that a cluster algebra is acyclic if it has an acyclic seed: a seed
with an exchange matrix $B=(b_{ij})$, such that there is no sequence of
indices $i_{1},\ldots,i_{\ell+1}\in\\{1,\ldots,n\\}$ with
$b_{i_{j},i_{j+1}}>0$ for all $j$ and $i_{\ell+1}=i_{1}$.

######  Proposition 2.19.

[BFZ05, Mul14] Let $\mathcal{A}$ be an acyclic cluster algebra and
$\mathcal{U}$ the corresponding upper cluster algebra. Then
$\mathcal{A}=\mathcal{U}$.

###  2.2. $d$-vectors, $f$-vectors, $g$-vectors, $c$-vectors and
$F$-polynomials

Let $M$ be any non-zero element in
${\mathcal{L}}(t_{0})=\mathbb{KP}[A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$. Then $M$ has the form

$\displaystyle
M=\frac{P_{M}(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{d_{1}}\cdots
A_{n;t_{0}}^{d_{n}}},$

where $P_{M}\in\mathbb{KP}[A_{1;t_{0}},\ldots,A_{n;t_{0}}]$ with
$A_{j;t_{0}}\nmid P_{M}$ for $j=1,\ldots,n$. The vector

${\bf d}^{t_{0}}(M)=(d_{1},\ldots,d_{n})^{\rm T}$

is called the $d$-vector of $M$ with respect to
$A_{t_{0}}=(A_{1;t_{0}},\ldots,A_{n;t_{0}})$ and the polynomial $P_{M}$ is
called the numerator polynomial of $M$ with respect to $A_{t_{0}}$.

######  Remark 2.20.

Notice that for any $0\neq M,N\in\mathcal{L}(t_{0})$, we have

${\bf d}^{t_{0}}(M\cdot N)={\bf d}^{t_{0}}(M)+{\bf d}^{t_{0}}(N).$

######  Definition 2.21 ($F$-polynomial).

Let $B$ be an $n\times n$ skew-symmetrizable integer matrix and $\mathcal{A}$
a principal coefficient cluster algebra at $t_{0}$ with $B_{t_{0}}=B$. Let
$A_{k;t}$ be a cluster variable of $\mathcal{A}$, which can be written as a
Laurent polynomial in

$\mathbb{Z}[Z_{1},\ldots,Z_{n}][A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}],$

by the Laurent phenomenon. The polynomial

$F_{k;t}^{B;t_{0}}(Z_{1},\ldots,Z_{n}):=A_{k;t}|_{A_{1;t_{0}}=\ldots=A_{n;t_{0}}=1}\in\mathbb{Z}[Z_{1},\ldots,Z_{n}],$

which only depends on $(B,t_{0};k,t)$, is called an $F$-polynomial of $B$.
Note that we always view $F_{k;t}^{B;t_{0}}$ as a polynomial in new variables
$Y_{1},\ldots,Y_{n}$ in the sequel.

######  Proposition 2.22.

[FZ07, Proposition 5.2] The $F$-polynomial
$F_{k;t}^{B;t_{0}}(Y_{1},\ldots,Y_{n})$ is not divisible by any $Y_{j}$.

Let $B$ be an $n\times n$ skew-symmetrizable matrix and $t_{0}$ a vertex of
$\mathbb{T}_{n}$. Now we define two families of integer matrices
$\\{D_{t}^{B;t_{0}}\\}_{t\in\mathbb{T}_{n}}$ and
$\\{F_{t}^{B;t_{0}}\\}_{t\in\mathbb{T}_{n}}$.

######  Definition 2.23 ($D$-matrices and $F$-matrices).

Let $B$ be an $n\times n$ skew-symmetrizable matrix and $t_{0}$ a vertex of
$\mathbb{T}_{n}$.

(i) [FZ07] The matrix $D_{t}^{B;t_{0}}=(d_{ij;t}^{B;t_{0}})$ is uniquely
determined by the initial condition $B_{t_{0}}=B$ and
$D_{t_{0}}^{B;t_{0}}=-I_{n}$, together with the following recurrence
relations:

$d_{ij;t^{\prime}}^{B;t_{0}}=\begin{cases}d_{ij;t}^{B;t_{0}},&\text{if }j\neq
k;\\\ -d_{ik;t}^{B;t_{0}}+{\rm
max}\\{\sum\limits_{b_{lk}^{t}>0}d_{il;t}^{B;t_{0}}b_{lk}^{t},\sum\limits_{b_{lk}^{t}<0}-d_{il;t}^{B;t_{0}}b_{lk}^{t}\\},&\text{if
}j=k,\end{cases}$

for any edge $t^{~{}\underline{\quad k\quad}}~{}t^{\prime}$ in
$\mathbb{T}_{n}$. The matrices $\\{D_{t}^{B;t_{0}}\\}_{t\in\mathbb{T}_{n}}$
are called the $D$-matrices of $B$.

(ii) [FG19b] Let $F_{k;t}^{B;t_{0}}(Z_{1},\ldots,Z_{n})$ be the $F$-polynomial
given in Definition 2.21, and $f_{ik;t}^{B;t_{0}}$ be the maximal exponent of
$Z_{i}$ appearing in $F_{k;t}^{B;t_{0}}$. The vector

${\bf f}_{k;t}^{B;t_{0}}=(f_{1k;t}^{B;t_{0}},\ldots,f_{nk;t}^{B;t_{0}})^{\rm
T}$

is called an $f$-vector of $B$ and the matrix
$F_{t}^{B;t_{0}}=(f_{ij;t}^{B;t_{0}})_{n\times n}$ is called an $F$-matrix of
$B$.

######  Remark 2.24.

Let $\mathcal{A}$ be a cluster algebra with initial exchange matrix $B$ at
$t_{0}$. Then the $k$-th column vector ${\bf d}_{k;t}^{B;t_{0}}$ of
$D_{t}^{B;t_{0}}$ is exactly the $d$-vector ${\bf d}^{t_{0}}(A_{k;t})$ of
$A_{k;t}$ with respect to $A_{t_{0}}$, by [FZ07, (7.7)].

######  Proposition 2.25.

Let $\mathcal{U}$ be an upper cluster algebra and $M$ a non-zero element in
$\mathcal{U}$. Let $t_{0},t_{1}$ be two seeds of $\mathcal{U}$ with
$t_{1}=\mu_{k}(t_{0})$. Let ${\bf d}=(d_{1},\ldots,d_{n})^{\rm T}$ and ${\bf
d}^{\prime}=(d_{1}^{\prime},\ldots,d_{n}^{\prime})^{\rm T}$ be the $d$-vectors
of $M$ with respect to $t_{0}$ and $t_{1}$ respectively. Then
$d_{i}=d_{i}^{\prime}$ for any $i\neq k$.

###### Proof.

The proof is the same as that of [RS18, Proposition 2.5]. ∎

######  Theorem 2.26 (Canonical expressions).

Let $B$ be an $n\times n$ skew-symmetrizable matrix and

$\\{F_{1;t}^{B;t_{0}},\ldots,F_{n;t}^{B;t_{0}}\\}_{t\in\mathbb{T}_{n}}$

the $F$-polynomials given in Definition 2.21.

  * (i)

[FZ07] Let $\mathcal{A}$ be a cluster algebra with principal coefficients at
$t_{0}$ and with $B_{t_{0}}=B$. Then for any cluster variable $A_{k;t}$ of
$\mathcal{A}$, there exists a unique vector ${\bf
g}_{k;t}^{B;t_{0}}=(g_{1k;t}^{B;t_{0}},\ldots,g_{nk;t}^{B;t_{0}})^{\rm
T}\in\mathbb{Z}^{n}$ such that

$\displaystyle A_{k;t}=A_{t_{0}}^{{\bf g}_{k;t}^{B;t_{0}}}\cdot
F_{k;t}^{B;t_{0}}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}}),$

where $A_{t_{0}}^{{\bf
g}_{k;t}^{B;t_{0}}}=\prod\limits_{i=1}^{n}A_{i;t}^{g_{ik;t}^{B;t_{0}}}$,
$\hat{X}_{j;t_{0}}=X_{j;t_{0}}\prod\limits_{i=1}^{n}A_{i;t_{0}}^{b_{ij}^{t_{0}}}=Z_{j}\prod\limits_{i=1}^{n}A_{i;t_{0}}^{b_{ij}^{t_{0}}}$.

  * (ii)

[NZ12] Let $\mathcal{X}$ be the cluster Poisson algebra with $B_{t_{0}}=B$.
Then for any cluster Poisson variable $X_{k;t}$ of $\mathcal{X}$, there exists
a unique vector ${\bf
c}_{k;t}^{B;t_{0}}=(c_{1k;t}^{B;t_{0}},\ldots,c_{nk;t}^{B;t_{0}})^{\rm
T}\in\mathbb{Z}^{n}$ such that

$X_{k;t}=X_{t_{0}}^{{\bf
c}_{k;t}^{B;t_{0}}}\cdot\prod\limits_{i=1}^{n}(F_{i;t}^{B;t_{0}}(X_{1;t_{0}},\ldots,X_{n;t_{0}}))^{b_{ik}^{t}}.$

######  Definition 2.27 ($g$-vectors and $c$-vectors).

Keep the notations of Theorem 2.26.

(i) The vector ${\bf
g}_{k;t}^{B;t_{0}}=(g_{1k;t}^{B;t_{0}},\ldots,g_{nk;t}^{B;t_{0}})^{\rm
T}\in\mathbb{Z}^{n},$ which only depends on $(B,t_{0};k,t)$, is called a
$g$-vector of $B$. The matrix $G_{t}^{B;t_{0}}=({\bf
g}_{ij;t}^{B;t_{0}})_{n\times n}$ is called a $G$-matrix of $B$ at $t$.

(ii) The vector ${\bf
c}_{k;t}^{B;t_{0}}=(c_{1k;t}^{B;t_{0}},\ldots,c_{nk;t}^{B;t_{0}})^{\rm
T}\in\mathbb{Z}^{n},$ which only depends on $(B,t_{0};k,t)$, is called a
$c$-vector of $B$. The matrix $C_{t}^{B;t_{0}}=({\bf
c}_{ij;t}^{B;t_{0}})_{n\times n}$ is called a $C$-matrix of $B$ at $t$.

######  Theorem 2.28.

[NZ12, CL18] Let $B$ be an $n\times n$ skew-symmetrizable matrix and $S$ a
skew-symmetrizer of $B$. Then for any vertices $t_{0},t\in\mathbb{T}_{n}$, we
have

$SC_{t}^{B;t_{0}}S^{-1}(G_{t}^{B;t_{0}})^{\rm T}=I_{n}.$

###  2.3. Compatibility degrees on the set of cluster variables

In this subsection, we recall that the $d$-compatibility degree given in
[CL20] and $f$-compatibility degree given in [FG19a].

######  Definition 2.29 (Compatibility degrees).

Let $\mathcal{A}$ be a cluster algebra and $\mathbb{A}$ the set of cluster
variables of $\mathcal{A}$.

(i) [CL20] The $d$-compatibility degree
$(-||-)_{d}:\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{Z}_{\geq-1}$ is
defined by

$(A_{i;t_{0}}||A_{j;t})_{d}:=d_{ij;t}^{B_{t_{0}};t_{0}},$

where $d_{ij;t}^{B_{t_{0}};t_{0}}$ is the $(i,j)$-entry of the $D$-matrix
$D_{t}^{B_{t_{0}};t_{0}}$.

(ii) [FG19a] The $f$-compatibility degree
$(-||-)_{f}:\mathbb{A}\times\mathbb{A}\rightarrow\mathbb{N}$ is defined by

$(A_{i;t_{0}}||A_{j;t})_{f}:=f_{ij;t}^{B_{t_{0}};t_{0}},$

where $f_{ij;t}^{B_{t_{0}};t_{0}}$ is the $(i,j)$-entry of the $F$-matrix
$F_{t}^{B_{t_{0}};t_{0}}$.

Now we summarize some properties of the $d$-compatibility degree $(-||-)_{d}$
and the $f$-compatibility degree $(-||-)_{f}$. We mainly refer to [CL20] and
[FG19a].

######  Proposition 2.30.

[CL20, FG19a] The $d$-compatibility degree $(-||-)_{d}$ and the
$f$-compatibility degree $(-||-)_{f}$ have the following properties.

  * (1)

They are well-defined.

  * (2)

The following statements are equivalent.

    * (i)

There exists a cluster $A_{t^{\prime}}$ containing both $A_{i;t_{0}}$ and
$A_{j;t}$;

    * (ii)

$(A_{i;t_{0}}||A_{j;t})_{d}\leq 0$;

    * (iii)

$(A_{j;t}||A_{i;t_{0}})_{d}\leq 0$;

    * (iv)

$(A_{i;t_{0}}||A_{j;t})_{f}=0$;

    * (v)

$(A_{j;t}||A_{i;t_{0}})_{f}=0$.

  * (3)

The following statements are equivalent.

    * (i)

There exists no cluster $A_{t^{\prime}}$ containing both $A_{i;t_{0}}$ and
$A_{j;t}$;

    * (ii)

$(A_{i;t_{0}}||A_{j;t})_{d}>0$;

    * (iii)

$(A_{j;t}||A_{i;t_{0}})_{d}>0$;

    * (iv)

$(A_{i;t_{0}}||A_{j;t})_{f}>0$;

    * (v)

$(A_{j;t}||A_{i;t_{0}})_{f}>0$.

  * (4)

The following statements are equivalent.

    * (i)

$A_{j;t}=A_{i;t_{0}}$;

    * (ii)

$(A_{i;t_{0}}||A_{j;t})_{d}<0$;

    * (iii)

$(A_{j;t}||A_{i;t_{0}})_{d}<0$;

    * (iv)

$(A_{i;t_{0}}||A_{j;t})_{d}=-1$;

    * (v)

$(A_{j;t}||A_{i;t_{0}})_{d}=-1$.

  * (5)

The following statements are equivalent.

    * (i)

$A_{i;t_{0}}\neq A_{j;t}$ and there exists a cluster $A_{t^{\prime}}$
containing both $A_{i;t_{0}}$ and $A_{j;t}$;

    * (ii)

$(A_{i;t_{0}}||A_{j;t})_{d}=0$;

    * (iii)

$(A_{j;t}||A_{i;t_{0}})_{d}=0$;

######  Remark 2.31.

(i) Roughly speaking, the integer $(A_{i;t_{0}}||A_{j;t})_{d}$ is defined to
be the $i$-th component of the $d$-vector of $A_{j;t}$ with respect to
$A_{t_{0}}$. By Proposition 2.25 and Theorem 2.17, the $d$-compatibility
degree $(-||-)_{d}$ is actually a well-defined function on
$\mathbb{A}\times(\mathcal{U}\backslash\\{0\\})$, where $\mathcal{U}$ is the
corresponding upper cluster algebra.

(ii) By Remark 2.20, for any $0\neq M,N\in\mathcal{U}$, we have

$(A_{i;t_{0}}||M\cdot N)_{d}=(A_{i;t_{0}}||M)_{d}+(A_{i;t_{0}}||N)_{d}.$

Recall that a cluster monomial in seed $t$ is a monomial in cluster variables
in $t$.

######  Corollary 2.32.

Let $\mathcal{A}$ be a cluster algebra with coefficient semifield $\mathbb{P}$
and $A_{k;t_{0}}$ a cluster variable of $\mathcal{A}$. Let $t$ be a seed of
$\mathcal{A}$ and $M=A_{1;t}^{c_{1}}\cdots A_{n;t}^{c_{n}}$ a cluster monomial
in $t$. Denote $I=\\{i|c_{i}>0\\}$. The following statements hold.

  * (i)

If $(A_{k;t_{0}}||M)_{d}<0$, then there exists $j_{0}\in I$ such that
$A_{k;t_{0}}=A_{j_{0};t}$. Namely, $A_{k;t_{0}}$ appears in $M$.

  * (ii)

If $(A_{k;t_{0}}||M)_{d}\leq 0$, then $(A_{k;t_{0}}||A_{i;t})_{d}\leq 0$ for
any $i\in I$.

###### Proof.

(i) By $c_{1},\ldots,c_{n}\geq 0$ and $(A_{k;t_{0}}||M)_{d}<0$, we have

$(A_{k;t_{0}}||M)_{d}=\sum_{i=1}^{n}c_{i}(A_{k;t_{0}}||A_{i;t})_{d}=\sum_{i\in
I}c_{i}(A_{k;t_{0}}||A_{i;t})_{d}<0.$

So there must exist some $j_{0}\in I$ such that
$(A_{k;t_{0}}||A_{j_{0};t})_{d}<0$. Then by Proposition 2.30 (4), we get
$A_{k;t_{0}}=A_{j_{0};t}$.

(ii) Assume by contradiction there exists some $i_{0}\in I$ such that

$(A_{k;t_{0}}||A_{i_{0};t})_{d}>0.$

Then by $(A_{k;t_{0}}||M)_{d}=\sum\limits_{i\in
I}c_{i}(A_{k;t_{0}}||A_{i;t})_{d}\leq 0$, there must exist some $i_{1}\in I$
such that $(A_{k;t_{0}}||A_{i_{1};t})_{d}<0$. So we get
$A_{k;t_{0}}=A_{i_{1};t}$, by Proposition 2.30 (4). Thus $A_{i_{0};t}$ and
$A_{k;t_{0}}=A_{i_{1};t}$ are in the same cluster. Then by Proposition 2.30
(2), we get $(A_{k;t_{0}}||A_{i_{0};t})_{d}\leq 0$. This contradicts
$(A_{k;t_{0}}||A_{i_{0};t})_{d}>0$. So $(A_{k;t_{0}}||A_{i;t})_{d}\leq 0$ for
any $i\in I$. ∎

Let $\mathcal{A}$ be a cluster algebra and $\mathbb{A}$ the set of cluster
variables of $\mathcal{A}$. We say that two cluster variables $A_{i;t_{0}}$
and $A_{j;t}$ of $\mathcal{A}$ are compatible if there exists a cluster
$A_{t^{\prime}}$ of $\mathcal{A}$ containing both $A_{i;t_{0}}$ and $A_{j;t}$,
which is equivalent to $(A_{i;t_{0}}||A_{j;t})_{d}\leq 0$, by Proposition 2.30
(2). A subset $U$ of $\mathbb{A}$ is called a compatible set if any two
cluster variables in $U$ are compatible.

######  Theorem 2.33.

[CL20] Let $\mathcal{A}$ be a cluster algebra and $\mathbb{A}$ the set of
cluster variables of $\mathcal{A}$. Then

  * (i)

a subset $U$ of $\mathbb{A}$ is a compatible set if and only if it is a subset
of some cluster of $\mathcal{A}$;

  * (ii)

a subset $U$ of $\mathbb{A}$ is a maximal compatible set if and only it is a
cluster of $\mathcal{A}$.

###  2.4. Unique factorization domains

Recall that an integral domain is a non zero commutative ring without zero
divisors. From now on, we always assume that $R$ is an integral domain and we
denote by $R^{\times}$ the set of invertible elements in $R$.

Two non-zero elements $r_{1},r_{2}\in R$ are associate if there exists some
invertible element $s\in R^{\times}$ such that $r_{2}=sr_{1}$. A non-
invertible element $r$ in $R$ is irreducible, if any factorization
$r=r_{1}r_{2}$ with $r_{1},r_{2}\in R$ implies that either $r_{1}$ or $r_{2}$
belongs to $R^{\times}$. A non-invertible element $r$ in $R$ is prime, if
whenever $r\mid r_{1}r_{2}$ for some $r_{1},r_{2}\in R$, then $r\mid r_{1}$ or
$r\mid r_{2}$. Every prime element is irreducible, but the converse is not
true in general.

######  Definition 2.34.

An integral domain $R$ is factorial if the following hold:

  * (i)

Every non-zero non-invertible element $r\in R$ can be written as a product
$r=a_{1}\cdots a_{s}$ of irreducible elements $a_{i}\in R$.

  * (ii)

If $a_{1}\cdots a_{s}=b_{1}\cdots b_{t}$ with $a_{i},b_{j}\in R$ irreducible
for all $i$ and $j$, then $s=t$ and there is a bijection
$\sigma:\\{1,\ldots,s\\}\rightarrow\\{1,\ldots,s\\}$ such that $a_{i}$ and
$b_{\sigma(i)}$ are associate for all $1\leq i\leq s$.

Factorial domains are often also called unique factorization domains.

It is easy to see that in a factorial domain, all irreducible elements are
prime.

######  Theorem 2.35.

Let $\mathcal{A}$ be a geometric cluster algebra with coefficient semifield
$\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$ and $\mathcal{U}$ the
corresponding upper cluster algebra. Then we have

  * (i)

[GLS13, Theorem 1.3]

    * (a)

$\mathcal{A}^{\times}=\\{\lambda Z_{1}^{c_{1}}\cdots
Z_{m}^{c_{m}}|\lambda\in\mathbb{K}^{\times},\;c_{1},\ldots,c_{m}\in\mathbb{Z}\\}$.

    * (b)

any cluster variable is irreducible in $\mathcal{A}$.

  * (ii)

    * (a)

$\mathcal{U}^{\times}=\\{\lambda Z_{1}^{c_{1}}\cdots
Z_{m}^{c_{m}}|\lambda\in\mathbb{K}^{\times},\;c_{1},\ldots,c_{m}\in\mathbb{Z}\\}$.

    * (b)

any cluster variable is irreducible in $\mathcal{U}$.

###### Proof.

The proof of (i) can be found in [GLS13] and the proof of (ii) is the same as
that of (i). ∎

##  3\. The valuation pairing on an upper cluster algebra

In this section, we introduce the valuation pairing $(-||-)_{v}$ on an upper
cluster algebra and prove the local unique factorization property for full
rank upper cluster algebras.

######  Definition 3.1 (Valuation pairing).

Let $\mathcal{U}$ be an upper cluster algebra and $\mathbb{A}$ the set of
cluster variables of $\mathcal{U}$. The pairing

$\displaystyle(-||-)_{v}:\mathbb{A}\times\mathcal{U}$
$\displaystyle\rightarrow$ $\displaystyle\mathbb{N}\cup\\{\infty\\}$
$\displaystyle(A_{k;t},M)$ $\displaystyle\mapsto$ $\displaystyle{\rm
max}\\{s\in\mathbb{N}|M/A_{k;t}^{s}\in\mathcal{U}\\}$

is called the valuation pairing on $\mathcal{U}$.

It is easy to see that $(A_{k;t}||M)_{v}=0$ if and only if
$M/A_{k;t}\notin\mathcal{U}$.

######  Example 3.2.

Let $\mathcal{U}$ be an upper cluster algebra with a seed $t$ and $M$ a
cluster variable in $t$. Then

$(A_{k;t}||M)_{v}=\begin{cases}1,&\text{if}\;M=A_{k;t};\\\
0,&\text{if}\;M=A_{j;t}\;\text{and}\;j\neq k.\end{cases}$

Now we summarize some useful and easy facts on the valuation pairing in the
following proposition.

######  Proposition 3.3.

Let $\mathcal{U}$ be an upper cluster algebra and $t_{0}$ a seed of
$\mathcal{U}$. Let $M$ and $L$ be two elements in $\mathcal{U}$. The following
statements hold.

  * (i)

If $M/A_{k;t_{0}}^{m}\notin U$ for some $m\in\mathbb{Z}$, then we also have
$M/A_{k;t_{0}}^{m+1}\notin U$.

  * (ii)

If $M\neq 0$, then there exists $m>0$ such that

$M/A_{k;t_{0}}^{m}\notin\mathcal{U},$

In particular, $(A_{k;t_{0}}||M)_{v}<\infty$.

  * (iii)

$(A_{k;t_{0}}||M)_{v}=\infty$ if and only if $M=0$.

  * (iv)

$(A_{k;t_{0}}||M+L)_{v}\geq{\rm
min}\\{(A_{k;t_{0}}||M)_{v},(A_{k;t_{0}}||L)_{v}\\}$.

  * (v)

For any $s\geq 0$, we have $(A_{k;t_{0}}||A_{k;t_{0}}^{s}\cdot
M)_{v}=s+(A_{k;t_{0}}||M)_{v}$.

###### Proof.

(i) Assume by contradiction that $M/A_{k;t_{0}}^{m+1}\in U$, then

$M/A_{k;t_{0}}^{m}=A_{k;t_{0}}\cdot(M/A_{k;t_{0}}^{m+1})\in\mathcal{U}.$

This is a contradiction. So $M/A_{k;t_{0}}^{m+1}\notin U$.

(ii) Without loss of generality, we can assume that $k=n$. As an element in

${\mathcal{L}}(t_{0})=\mathbb{KP}[A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]=\mathbb{KP}[A_{1;t_{0}}^{\pm 1},\ldots,A_{n-1;t_{0}}^{\pm
1}][A_{n;t_{0}}^{\pm 1}],$

$M$ can be written as

$M=\sum\limits_{s\in\mathbb{Z}}A_{n;t_{0}}^{s}L_{s},$

where $L_{s}$ is a Laurent polynomial in $\mathbb{KP}[A_{1;t_{0}}^{\pm
1},\ldots,A_{n-1;t_{0}}^{\pm 1}]$, i.e., $L_{s}$ does not contain any
$A_{n;t}^{\pm 1}$. By $M\neq 0$, we know that there exists some $s_{0}$ such
that $L_{s_{0}}\neq 0$. Let $t_{1}=\mu_{n}(t_{0})$ and thus
$t_{0}=\mu_{n}(t_{1})$. Applying the exchange relation, we have

$A_{i;t_{0}}=A_{i;t_{1}}\;\;\text{and}\;\;\;A_{n;t_{0}}=A_{n;t_{1}}^{-1}P$

where $i=1,\ldots,n-1$ and $P\in\mathbb{KP}[A_{1;t_{1}}^{\pm
1},\ldots,A_{n-1;t_{1}}^{\pm 1}]$ is the $n$-th exchange binomial of the seed
$t_{1}$. Now each $L_{s}$ can be also viewed as a Laurent polynomial in
$\mathbb{KP}[A_{1;t_{1}}^{\pm 1},\ldots,A_{n-1;t_{1}}^{\pm 1}]$. By our
assumption in Subsection 1.4, we know that $P$ is not invertible in
$\mathbb{KP}[A_{1;t_{1}}^{\pm 1},\ldots,A_{n-1;t_{1}}^{\pm 1}]$. Then by the
fact that $\mathbb{KP}[A_{1;t_{1}}^{\pm 1},\ldots,A_{n-1;t_{1}}^{\pm 1}]$ is
factorial and $L_{s_{0}}$ belongs to $\mathbb{KP}[A_{1;t_{1}}^{\pm
1},\ldots,A_{n-1;t_{1}}^{\pm 1}]$, we know that there must exist some $m_{0}$
large enough such that

$L_{s_{0}}/P^{m_{0}}\notin\mathbb{KP}[A_{1;t_{1}}^{\pm
1},\ldots,A_{n-1;t_{1}}^{\pm 1}].$

Set $m=m_{0}+s_{0}$. Then we know that

$M/A_{n;t_{0}}^{m}=\sum\limits_{s\in\mathbb{Z}}A_{n;t_{0}}^{s-m}L_{s}=\sum\limits_{s\in\mathbb{Z}}A_{n;t_{1}}^{m-s}(L_{s}/P^{m-s})$

cannot be a Laurent polynomial in $\mathbb{KP}[A_{1;t_{1}}^{\pm
1},\ldots,A_{n;t_{1}}^{\pm 1}]$. So $M/A_{n;t_{0}}^{m}$ is not in
$\mathcal{U}$. Then by (i), we know that $(A_{k;t_{0}}||M)_{v}<m<\infty$.

(iii) follows from (ii) and the fact $(A_{k;t_{0}}||0)_{v}=\infty$.

(iv) and (v) follow from the definition of the valuation pairing. ∎

######  Definition 3.4 (Local factorization).

Let $\mathcal{U}$ be an upper cluster algebra and $t$ a seed of $\mathcal{U}$.
Let $M=N\cdot L$ be a factorization in $\mathcal{U}$. If $N$ is a cluster
monomial in $t$ and $L\in\mathcal{U}$ satisfies $(A_{k;t}||L)_{v}=0$, that is,
$L/A_{k;t}\notin\mathcal{U}$ for $k=1,\ldots,n$, we call $M=N\cdot L$ a local
factorization of $M$ with respect to $t$.

######  Proposition 3.5 (Existence of local factorization).

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and $t_{0}$ a seed of $\mathcal{U}$. Then any $0\neq
M\in\mathcal{U}$ admits a local factorization with respect to $t_{0}$.

###### Proof.

We first define a cluster monomial $N=A_{1;t_{0}}^{m_{1}}\cdots
A_{n;t_{0}}^{m_{n}}$ in $t_{0}$. Let $m_{1}:=(A_{1;t_{0}}||M)_{v}$ and
$m_{2},\ldots,m_{n}$ are defined by induction. If $m_{k}$ has been defined,
$m_{k+1}$ is defined as follows:

$m_{k+1}:=(A_{k+1;t_{0}}||\frac{M}{A_{1;t_{0}}^{m_{1}}\cdots
A_{k;t_{0}}^{m_{k}}})_{v}.$

Thus the non-negative integers $m_{1},\ldots,m_{n}$ are defined. So we get a
cluster monomial $N=A_{1;t_{0}}^{m_{1}}\cdots A_{n;t_{0}}^{m_{n}}$ in $t_{0}$
and we know that $L:=M/N\in\mathcal{U}$.

Now we show that $L=M/N$ satisfies $(A_{k;t_{0}}||L)_{v}=0$, that is,
$L/A_{k;t_{0}}\notin\mathcal{U}$ for $k=1,\ldots,n$. Assume by contradiction
that there exists some $k\in\\{1,\ldots,n\\}$ such that

$L/A_{k;t_{0}}=\frac{M}{A_{1;t_{0}}^{m_{1}}\cdots
A_{k-1;t_{0}}^{m_{k-1}}A_{k;t_{0}}^{m_{k}+1}A_{k+1;t_{0}}^{m_{k+1}}\cdots
A_{n;t_{0}}^{m_{n}}}\in\mathcal{U}.$

Thus we get

$\frac{M}{A_{1;t_{0}}^{m_{1}}\cdots
A_{k-1;t_{0}}^{m_{k-1}}A_{k;t_{0}}^{m_{k}+1}}=A_{k+1;t_{0}}^{m_{k+1}}\cdots
A_{n;t_{0}}^{m_{n}}(L/A_{k;t_{0}})\in\mathcal{U},$

which contradicts the choice of $m_{k}$. So $L/A_{k;t_{0}}\notin\mathcal{U}$
for $k=1,\ldots,n$. Thus $M=N\cdot L$ is a local factorization of $M$ with
respect to $t_{0}$. This completes the proof. ∎

######  Lemma 3.6 (Reduction Lemma).

Let $\mathcal{U}$ be a full rank upper cluster algebra and $t_{0}$ a seed of
$\mathcal{U}$. Let $L$ be a non-zero element in $\mathcal{U}$ and
$N=A_{1;t_{0}}^{m_{1}}\cdots A_{n;t_{0}}^{m_{n}}$ a cluster monomial in
$t_{0}$. Then the following statements hold.

  * (i)

$L/A_{k;t_{0}}^{s}\in\mathcal{U}$ if and only if
$L/A_{k;t_{0}}^{s}\in\mathcal{L}(t_{k})$, where $s\in\mathbb{N}$,
$t_{k}=\mu_{k}(t_{0})$ and

$\mathcal{L}(t_{k})=\mathbb{KP}[A_{1;t_{k}}^{\pm 1},\ldots,A_{n;t_{k}}^{\pm
1}].$

  * (ii)

$(A_{k;t_{0}}||L)_{v}={\rm
max}\\{s\in\mathbb{N}|L/A_{k;t_{0}}^{s}\in\mathcal{L}(t_{k})\\}$.

  * (iii)

$(A_{k;t_{0}}||N\cdot L)_{v}=m_{k}+(A_{k;t_{0}}||L)_{v}$.

###### Proof.

(i) By the starfish Theorem 2.16, we know that
$\mathcal{U}=\bigcap\limits_{i=0}^{n}\mathcal{L}(t_{i})$. Since $A_{k;t_{0}}$
is a cluster variable in $\mathcal{L}(t_{i})$ for any
$i\in\\{0,1,\ldots,n\\}\backslash\\{k\\}$, we know that
$L/A_{k;t_{0}}^{s}\in\mathcal{L}(t_{i})$ for any
$i\in\\{0,1,\ldots,n\\}\backslash\\{k\\}$. Thus we get that
$L/A_{k;t_{0}}^{s}\in\mathcal{U}$ if and only if
$L/A_{k;t_{0}}^{s}\in\mathcal{L}(t_{k})$.

(ii) follows from (i) and the definition of the valuation pairing.

(iii) Without loss of generality, we assume $k=1$. Since
$A_{2;t_{0}}^{m_{2}}\cdots A_{n;t_{0}}^{m_{n}}$ is invertible in
$\mathcal{L}(t_{1})$ and by (ii), we have

$(A_{1;t_{0}}||N\cdot L)_{v}=(A_{1;t_{0}}||\frac{N\cdot
L}{A_{2;t_{0}}^{m_{2}}\cdots
A_{n;t_{0}}^{m_{n}}})_{v}=(A_{1;t_{0}}||A_{1;t_{0}}^{m_{1}}\cdot L)_{v}.$

Then by Proposition 3.3 (v), we get

$(A_{1;t_{0}}||A_{1;t_{0}}^{m_{1}}\cdot L)_{v}=m_{1}+(A_{1;t_{0}}||L)_{v}.$

This completes the proof. ∎

######  Theorem 3.7 (Local unique factorization property).

Let $\mathcal{U}$ be a full rank upper cluster algebra and $t_{0}$ a seed of
$\mathcal{U}$. Then any $0\neq M\in\mathcal{U}$ admits a unique local
factorization with respect to $t_{0}$.

###### Proof.

The existence of local factorization of $M$ with respect to $t_{0}$ is known
from Proposition 3.5. Now we show the uniqueness. Let $M=N\cdot L$ be a local
factorization of $M$ with respect to $t_{0}$, where
$N=A_{1;t_{0}}^{m_{1}}\cdots A_{n;t_{0}}^{m_{n}}$ is a cluster monomial in
$t_{0}$ and $L\in\mathcal{U}$ satisfies $(A_{k;t_{0}}||L)_{v}=0$ for
$k=1,\ldots,n$. By the Reduction Lemma 3.6 (iii), we have

$(A_{k;t_{0}}||M)_{v}=(A_{k;t_{0}}||N\cdot
L)_{v}=m_{k}+(A_{k;t_{0}}||L)_{v}=m_{k}.$

So $m_{k}=(A_{k;t_{0}}||M)_{v}$ is uniquely determined by $M$. Namely, $N$ is
uniquely determined by $M$. This completes the proof. ∎

Now we give a counter-example to the local unique factorization property in
the case where $\tilde{B}_{t_{0}}$ is not full rank.

######  Example 3.8.

Let $\mathcal{U}$ be a geometric upper cluster algebra with initial extended
exchange matrix $\tilde{B}_{t_{0}}$ given by

$\tilde{B}_{t_{0}}=B_{t_{0}}=\begin{pmatrix}0&-1&0\\\ 1&0&-1\\\
0&1&0\end{pmatrix}.$

Clearly, $\tilde{B}_{t_{0}}$ is not full rank. Denote $t_{1}=\mu_{1}(t_{0})$
and $t_{3}=\mu_{3}(t_{0})$. Applying the exchange relations, we have the
following equality.

(3.1) $\displaystyle
A_{1;t_{0}}A_{1;t_{1}}=A_{2;t_{0}}+1=A_{3;t_{0}}A_{3;t_{3}}.$

It is easy to check that both $A_{1;t_{1}}$ and $A_{3;t_{3}}$ are not
divisible by any cluster variable in $t_{0}$ (alternatively, one can also
refer to Theorem 5.4 (ii)). So the equality (3.1) gives two different local
factorizations of $A_{2;t_{0}}+1$ with respect to the initial seed $t_{0}$.

##  4\. Application to factoriality of upper cluster algebras

In this section we give several equivalent characterizations for the
factoriality of upper cluster algebras. As an application, we show that full
rank, primitive upper cluster algebras are factorial.

###  4.1. Characterizations for the factoriality of upper cluster algebras

We first give an observation which shows why the valuation pairing can be used
to study the factoriality of upper cluster algebras.

######  Proposition 4.1 (Observation).

Let $\mathcal{U}$ be a geometric upper cluster algebra and $A_{k;t}$ a cluster
variable of $\mathcal{U}$. Then the following two statements are equivalent.

  * (i)

$A_{k;t}$ is prime in $\mathcal{U}$.

  * (ii)

For any non-zero elements $M$ and $L$ in $\mathcal{U}$, we have the following
equality:

(4.1) $\displaystyle(A_{k;t}||M\cdot
L)_{v}=(A_{k;t}||M)_{v}+(A_{k;t}||L)_{v}.$

###### Proof.

(i)$\Rightarrow$ (ii): This is clear from the definition of the valuation
pairing.

(ii)$\Rightarrow$ (i): For any $0\neq M,L\in\mathcal{U}$ with $A_{k;t}|M\cdot
L$ in $\mathcal{U}$, we claim that either $A_{k;t}|M$ in $\mathcal{U}$ or
$A_{k;t}|L$ in $\mathcal{U}$. Otherwise, we have that both $M$ and $L$ are not
divisible by $A_{k;t}$ in $\mathcal{U}$. Then we have

$(A_{k;t}||M)_{v}=0=(A_{k;t}||L)_{v}.$

Now by our assumption, we have

$(A_{k;t}||M\cdot L)_{v}=(A_{k;t}||M)_{v}+(A_{k;t}||L)_{v}=0.$

This contradicts $A_{k;t}|M\cdot L$ in $\mathcal{U}$. So either $A_{k;t}|M$ or
$A_{k;t}|L$ in $\mathcal{U}$.

By Theorem 2.35 (ii), we know that $A_{k;t}$ is not invertible in
$\mathcal{U}$. Hence, $A_{k;t}$ is prime in $\mathcal{U}$. ∎

It is natural to ask under which conditions the equality (4.1) always holds.
Thanks to Reduction Lemma 3.6, we can answer this question for full rank upper
cluster algebras.

######  Proposition 4.2.

Let $\mathcal{U}$ be a full rank upper cluster algebra and $t$ a seed of
$\mathcal{U}$. Put $t_{k}=\mu_{k}(t)$ and let $P_{k;t}$ be the $k$-th exchange
binomial of $t$. Then the following statements are equivalent.

  * (i)

$(A_{k;t}||M\cdot L)_{v}=(A_{k;t}||M)_{v}+(A_{k;t}||L)_{v}$ holds for any non-
zero elements $M$ and $L$ in $\mathcal{U}$.

  * (ii)

$A_{k;t}$ is prime in $\mathcal{L}(t_{k})=\mathbb{K}[Z_{1}^{\pm
1},\ldots,Z_{m}^{\pm 1}][A_{1;t_{k}}^{\pm 1},\ldots,A_{n;t_{k}}^{\pm 1}]$.

  * (iii)

$P_{k;t}$ is prime in $\mathcal{L}(t_{k})$.

  * (iv)

$P_{k;t}$ is prime in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}]$,
which is actually equal to
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{k}},\ldots,\hat{A}_{k;t_{k}},\ldots,A_{n;t_{k}}]$.

  * (v)

$P_{k;t}$ is irreducible in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}]$.

###### Proof.

(i)$\Longrightarrow$ (ii): By our assumption in Subsection 1.4, we know that
$A_{k;t}$ is not invertible in $\mathcal{L}(t_{k})$. Now assume by
contradiction that $A_{k;t}$ is not prime in $\mathcal{L}(t_{k})$. Then there
exists $M^{\prime},L^{\prime}\in\mathcal{L}(t_{k})$ such that both
$M^{\prime}$ and $L^{\prime}$ are not divisible by $A_{k;t}$ in
$\mathcal{L}(t_{k})$, but the product $M^{\prime}\cdot L^{\prime}$ is
divisible by $A_{k;t}$ in $\mathcal{L}(t_{k})$. Since $M^{\prime}$ and
$L^{\prime}$ are in

$\mathcal{L}(t_{k})=\mathbb{K}[Z_{1}^{\pm 1},\ldots,Z_{m}^{\pm
1}][A_{1;t_{k}}^{\pm 1},\ldots,A_{n;t_{k}}^{\pm 1}],$

there exists a cluster monomial $A_{t_{k}}^{\bf v}$ in $t_{k}$ such that
$M:=A_{t_{k}}^{\bf v}M^{\prime}$ and $L:=A_{t_{k}}^{\bf v}L^{\prime}$ belong
to $\mathcal{U}$.

Because $A_{t_{k}}^{\bf v}$ is invertible in $\mathcal{L}(t_{k})$, we know
that both $M$ and $L$ are not divisible by $A_{k;t}$ in $\mathcal{L}(t_{k})$,
but the product $M\cdot L$ is divisible by $A_{k;t}$ in $\mathcal{L}(t_{k})$.
By the Reduction Lemma 3.6 (ii), we have

$(A_{k;t}||M)_{v}=0=(A_{k;t}||L)_{v}\;\;\text{but}\;\;(A_{k;t}||M\cdot
L)_{v}>0.$

This contradicts that $(A_{k;t}||M\cdot
L)_{v}=(A_{k;t}||M)_{v}+(A_{k;t}||L)_{v}$ holds for any non-zero elements $M$
and $L$ in $\mathcal{U}$.

(ii)$\Longrightarrow$ (i): This follows from the Reduction Lemma 3.6 (ii).

(ii)$\Longleftrightarrow$ (iii): Applying the exchange relation, we have
$A_{k;t}A_{k;t_{k}}=P_{k;t}$. Since $A_{k;t_{k}}$ is invertible in
$\mathcal{L}(t_{k})$, we know that $A_{k;t}$ is prime in $\mathcal{L}(t_{k})$
if and only if $P_{k;t}$ is prime in $\mathcal{L}(t_{k})$.

(iii)$\Longleftrightarrow$ (iv): Notice that $\mathcal{L}(t_{k})$ is a
localization of
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{k}},\ldots,A_{n;t_{k}}]$ at
$Z_{1},\ldots,Z_{m},A_{1;t_{k}},\ldots,A_{n;t_{k}}$. Since
$P_{k;t}=P_{k;t_{k}}$ is a binomial in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{k}},\ldots,A_{n;t_{k}}]$

and it is not divisible by any $Z_{i}$ and $A_{j;t_{k}}$, we know that
$P_{k;t}$ is prime in $\mathcal{L}(t_{k})$ if and only if it is prime in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{k}},\ldots,A_{n;t_{k}}]$. Because
$P_{k;t}$ is actually a binomial in variables from

$\\{Z_{1},\ldots,Z_{m},A_{1;t_{k}},\ldots,\hat{A}_{k;t_{k}},\ldots,A_{n;t_{k}}\\}=\\{Z_{1},\ldots,Z_{m},A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}\\},$

we know that $P_{k;t}$ is prime in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{k}},\ldots,A_{n;t_{k}}]$ if and only
if it is prime in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}].$

Hence, $P_{k;t}$ is prime in $\mathcal{L}(t_{k})$ if and only if it is prime
in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}].$

(iv)$\Longleftrightarrow$ (v): This follows from the fact that the polynomial
ring

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}]$

is factorial. ∎

######  Lemma 4.3.

[GELS19, Corollary 1.23] Let $R$ be a domain and $S\subseteq
R\setminus\\{0\\}$ a multiplicative set generated by prime elements. Then $R$
is factorial if and only if the localization $S^{-1}R$ is.

######  Theorem 4.4.

Let $\mathcal{U}$ be a full rank upper cluster algebra with initial seed
$t_{0}$. Then the following statements are equivalent.

  * (i)

$\mathcal{U}$ is factorial.

  * (ii)

Any initial cluster variable $A_{k;t_{0}}$ is prime in $\mathcal{U}$.

  * (iii)

Any initial cluster variable $A_{k;t_{0}}$ satisfies

$(A_{k;t_{0}}||M\cdot L)_{v}=(A_{k;t_{0}}||M)_{v}+(A_{k;t_{0}}||L)_{v},$

for any non-zero elements $M$ and $L$ in $\mathcal{U}$.

  * (iv)

Any exchange binomial $P_{k;t_{0}}$ of $t_{0}$ is irreducible in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,\hat{A}_{k;t_{0}},\ldots,A_{n;t_{0}}].$

###### Proof.

(i)$\Longrightarrow$ (ii): By Theorem 2.35 (ii), we know that any cluster
variable is irreducible in $\mathcal{U}$. Because $\mathcal{U}$ is factorial,
we get that any irreducible element is prime in $\mathcal{U}$. In particular,
any initial cluster variable $A_{k;t_{0}}$ is prime in $\mathcal{U}$.

(ii)$\Longrightarrow$ (i): Let $S$ be the multiplicative set generated by
initial cluster variables. By the Laurent phenomenon, we have

$S^{-1}\mathcal{U}=\mathcal{L}(t_{0})=\mathbb{Z}[Z_{1}^{\pm
1},\ldots,Z_{n}^{\pm 1}][A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm 1}],$

which is factorial. Since the initial cluster variables are prime in
$\mathcal{U}$ and by Lemma 4.3, we get that $\mathcal{U}$ is factorial.

The equivalence of (ii), (iii) and (iv) follows from Proposition 4.2. ∎

######  Remark 4.5.

Garcia Elsener et al. in [GELS19, Theorem 5.1] prove the equivalence of (i)
and (iv) in Theorem 4.4 for geometric cluster algebra $\mathcal{A}$ (possibly
not full rank) with an acyclic initial seed. Note that in acyclic case,
$\mathcal{A}=\mathcal{U}$, by Proposition 2.19. Our method here is very
different with that in [GELS19].

###  4.2. Full rank, primitive upper cluster algebras are factorial

In this subsection, we show that full rank and primitive upper cluster
algebras are factorial.

Recall that a vector $(b_{1},\ldots,b_{n+m})^{\rm T}\in\mathbb{Z}^{n+m}$ is
primitive if the greatest common divisor $d\in\mathbb{N}$ of
$b_{1},\ldots,b_{n+m}$ is $1$. An extended exchange matrix $\tilde{B}_{t}$ is
primitive if each column vector of $\tilde{B}_{t}$ is primitive.

######  Definition 4.6 (Primitive upper cluster algebra).

Let $\mathcal{U}$ be a geometric upper cluster algebra with initial seed
$t_{0}$. We say that $\mathcal{U}$ is a primitive upper cluster algebra, if
the initial extended exchange matrix $\tilde{B}_{t_{0}}$ is primitive.

######  Proposition 4.7.

Let $\mathcal{U}$ be a primitive upper cluster algebra. Then for any seed $t$,
the following statements hold.

  * (i)

[LP16, Lemma 4.3] $\tilde{B}_{t}$ is primitive.

  * (ii)

[LP16, Lemma 4.1] The $k$-th exchange binomial of $t$ is irreducible in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t},\ldots,\hat{A}_{k;t},\ldots,A_{n;t}],$

where $k=1,\ldots,n$.

######  Theorem 4.8.

Let $\mathcal{U}$ be a full rank, primitive upper cluster algebra with initial
seed $t_{0}$, and let

$M=\frac{P_{M}(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{d_{1}}\cdots
A_{n;t_{0}}^{d_{n}}}$

be a non-initial cluster variable of $\mathcal{U}$, where ${\bf
d}:=(d_{1},\ldots,d_{n})^{\rm T}$ is the $d$-vector of $M$ with respect to
$t_{0}$. Then

  * (i)

$\mathcal{U}$ is factorial.

  * (ii)

the numerator polynomial $P_{M}$ of $M$ is irreducible in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}].$

###### Proof.

(i) By Proposition 4.7 (ii), we know that every exchange binomial of $t_{0}$
is irreducible in the corresponding polynomial ring. Then the result follows
from Theorem 4.4 (i)(iv).

(ii) We first show that $P_{M}$ is not invertible in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}]$. Since $M$ is
a non-initial cluster variable and by Proposition 2.30 (4), we have ${\bf
d}\in\mathbb{N}^{n}$ and thus $A_{t_{0}}^{\bf d}\in\mathcal{U}$. By
$M=P_{M}/A_{t_{0}}^{\bf d}$, we get $MA_{t_{0}}^{\bf d}=P_{M}$. By Theorem
2.35 (2), we know that both $M$ and $A_{t_{0}}^{\bf d}$ are not invertible in
$\mathcal{U}$. So $P_{M}=MA_{t_{0}}^{\bf d}$ is not invertible in
$\mathcal{U}$. Namely, we have

$P_{M}\notin\mathcal{U}^{\times}=\\{\lambda Z_{1}^{c_{1}}\cdots
Z_{m}^{c_{m}}|\lambda\in\mathbb{K}^{\times},\;c_{1},\ldots,c_{m}\in\mathbb{Z}\\}.$

In particular, $P_{M}\notin\mathbb{K}^{\times}$. So $P_{M}$ is not invertible
in $\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}]$.

We claim that $P_{M}$ is not divisible by any $Z_{i}$ and $A_{j;t_{0}}$ in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}].$

Since $P_{M}$ is a numerator polynomial, it is not divisible by any
$A_{j;t_{0}}$. There are two approaches to prove $Z_{i}\nmid P_{M}$. The first
one is that we prove the result by induction on the minimal length of a
sequence of mutations from the initial seed to the final seed. The second one
is that we just view $Z_{i}$ as an exchangeable cluster variable. Then by the
fact that $M$ and $Z_{i}$ are in the same cluster and are not equal, we get
$(Z_{i}||M)_{d}=0$, by Proposition 2.30 (5). This implies that $P_{M}$ is not
divisible by $Z_{i}$.

We claim that $P_{M}$ is not invertible in

$\mathcal{L}(t_{0})=\mathbb{K}[Z_{1}^{\pm 1},\ldots,Z_{m}^{\pm
1}][A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm 1}].$

This follows from the two facts that $P_{M}$ is not invertible in

$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}]$

and $P_{M}$ is not divisible by any $Z_{i}$ and $A_{j;t_{0}}$ in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}]$.

Now we show that $P_{M}$ is irreducible in $\mathcal{L}(t_{0})$. Assume that
$P_{M}=L_{1}L_{2}$ for some $L_{1},L_{2}\in\mathcal{L}(t_{0})$. We know that
there exist vectors ${\bf v}_{1},{\bf v}_{2}\in\mathbb{N}^{n}$ such that

$U_{1}:=A_{t_{0}}^{{\bf
v}_{1}}L_{1}\in\mathcal{U}\;\;\text{and}\;\;\;U_{2}:=A_{t_{0}}^{{\bf
v}_{2}}L_{2}\in\mathcal{U}.$

By $M=\frac{P_{M}}{A_{t_{0}}^{\bf d}}=\frac{L_{1}L_{2}}{A_{t_{0}}^{\bf
d}}=\frac{U_{1}U_{2}}{A_{t_{0}}^{{\bf d}+{\bf v}_{1}+{\bf v}_{2}}}$, we get

(4.2) $\displaystyle MA_{t_{0}}^{{\bf d}+{\bf v}_{1}+{\bf v}_{2}}=U_{1}U_{2},$

which is an equality in $\mathcal{U}$, since ${\bf d},{\bf v}_{1},{\bf
v}_{2}\in\mathbb{N}^{n}$. Because $\mathcal{U}$ is factorial and $M$ is
irreducible in $\mathcal{U}$, we get that either $U_{1}$ or $U_{2}$ is
divisible by $M$ in $\mathcal{U}$. Without loss of generality, we assume that
$U_{1}^{\prime}:=U_{1}/M\in\mathcal{U}$. Then by the equality (4.2), we get
$U_{1}^{\prime}U_{2}=A_{t_{0}}^{{\bf d}+{\bf v}_{1}+{\bf v}_{2}},$ which
implies that both $U_{1}^{\prime}$ and $U_{2}$ are invertible in
$\mathcal{L}(t_{0})$. So $L_{2}=U_{2}/A_{t_{0}}^{{\bf v}_{2}}$ is invertible
in $\mathcal{L}(t_{0})$. Hence, $P_{M}$ is irreducible in
$\mathcal{L}(t_{0})$.

Since $P_{M}$ is irreducible in $\mathcal{L}(t_{0})$ and by the fact that
$P_{M}$ is not divisible by any $Z_{i}$ and $A_{j;t_{0}}$ in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}],$ we get that
$P_{M}$ is irreducible in
$\mathbb{K}[Z_{1},\ldots,Z_{m}][A_{1;t_{0}},\ldots,A_{n;t_{0}}]$. This
completes the proof. ∎

######  Remark 4.9.

(i) Note that any principal coefficient upper cluster algebra is primitive and
has full rank. So principal coefficient upper cluster algebras are always
factorial.

(ii) A different proof of Theorem 4.8 (i) via the factoriality of Cox rings
can be found in [GHK15, Corollary 4.7].

(iii) Garcia Elsener et al. in [GELS19] prove that any principal coefficient
cluster algebra $\mathcal{A}$ with an acyclic initial exchange matrix is
factorial. Note that in the acyclic case, we have $\mathcal{A}=\mathcal{U}$,
by Proposition 2.19.

######  Corollary 4.10.

Let $\mathcal{U}$ be a full rank, primitive upper cluster algebra and
$A_{k;t_{0}}$ a cluster variable of $\mathcal{U}$. Then for any non-zero
elements $M$ and $L$ in $\mathcal{U}$, we have

$(A_{k;t_{0}}||M\cdot L)_{v}=(A_{k;t_{0}}||M)_{v}+(A_{k;t_{0}}||L)_{v}.$

###### Proof.

This follows from Theorem 4.4 (i)(iii) and Theorem 4.8 (i). ∎

###  4.3. Examples of non-factorial upper cluster algebras

######  Proposition 4.11.

[GLS13, Corollary 6.5] Let $\mathcal{A}$ be a geometric cluster algebra with
initial extended exchange matrix

$\tilde{B}_{t_{0}}=B_{t_{0}}=\begin{pmatrix}0&-c\\\ d&0\end{pmatrix},$

where $c\geq 1$ and $d\geq 3$ an odd number. Then $\mathcal{A}$ is not
factorial.

######  Remark 4.12.

Note that in Proposition 4.11, $\mathcal{A}$ is an acyclic cluster algebra. In
this case, $\mathcal{A}$ coincides with its upper cluster algebra
$\mathcal{U}$, by Proposition 2.19. So Proposition 4.11 provides many upper
cluster algebras which have full rank but are not factorial.

Let $\mathcal{U}$ be the geometric upper cluster algebra with initial extended
exchange matrix

$\tilde{B}_{t_{0}}=B_{t_{0}}=\begin{pmatrix}0&-1\\\ 3&0\end{pmatrix}.$

We know that $\mathcal{U}$ is a full rank cluster algebra. It is non-
factorial, by Remark 4.12. In the view point of Theorem 4.4, the non-
factoriality of $\mathcal{U}$ is due to the fact that the first exchange
binomial $A_{2;t_{0}}^{3}+1=(A_{2;t_{0}}+1)(A_{2;t_{0}}^{2}-A_{2;t_{0}}+1)$ of
$t_{0}$ is not irreducible.

The following is a concrete example where $\mathcal{U}$ is primitive but not
factorial.

######  Example 4.13.

Let $\mathcal{A}$ be a geometric cluster algebra with the following initial
extended exchange matrix

$\tilde{B}_{t_{0}}=B_{t_{0}}=\begin{pmatrix}0&-1&0\\\ 1&0&-1\\\
0&1&0\end{pmatrix}.$

It is easy to see that $\tilde{B}_{t_{0}}$ is primitive and acyclic but not
full rank. It is known from [GLS13, Proposition 6.1] that $\mathcal{A}$ is not
factorial. Since $\mathcal{A}$ is acyclic, $\mathcal{A}$ coincides with its
upper cluster algebra $\mathcal{U}$, by Proposition 2.19. So
$\mathcal{U}=\mathcal{A}$ is not factorial.

###  4.4. Ray fish theorem

Recall that the starfish theorem states that any full rank upper cluster
algebra $\mathcal{U}$ can be written as the intersection of $n+1$ Laurent
polynomial rings. In this subsection, we show that any full rank, primitive
upper cluster algebra can be written as the intersection of two Laurent
polynomial rings. We call this the ray fish theorem.

######  Proposition 4.14.

[GLS13, Corollary 4.2] Let $\mathcal{A}$ be a geometric cluster algebra with
coefficient semifield $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$ and
$\mathcal{U}$ its upper cluster algebra. Let $t,t_{0}$ be two seeds of
$\mathcal{A}$ with no common cluster variables. If $\mathcal{A}$ is factorial,
then

$\mathcal{A}=\mathcal{U}=\mathcal{L}(t)\cap\mathcal{L}(t_{0}),$

where $\mathcal{L}(t)=\mathbb{KP}[A_{1;t}^{\pm 1},\ldots,A_{n;t}^{\pm 1}]$ and
$\mathcal{L}(t_{0})=\mathbb{KP}[A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$.

######  Remark 4.15.

By Proposition 4.14, a factorial geometric cluster algebra must coincide with
its upper cluster algebra. This provides a reason why we focus on the
factoriality of upper cluster algebras rather than cluster algebras.

######  Corollary 4.16.

Let $\mathcal{U}$ be a full rank, primitive upper cluster algebra and
$\mathcal{A}$ the corresponding cluster algebra. Then
$\mathcal{A}=\mathcal{U}$ if and only if $\mathcal{A}$ is factorial.

###### Proof.

“$\Longrightarrow$”: Since $\mathcal{U}$ is full rank and primitive and by
Theorem 4.8, we know that $\mathcal{U}$ is factorial. Then the factoriality of
$\mathcal{A}$ follows from $\mathcal{A}=\mathcal{U}$.

“$\Longleftarrow$”: This follows from Proposition 4.14. ∎

######  Lemma 4.17.

Let $\mathcal{U}$ be a geometric upper cluster algebra with coefficient
semifield $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$ and
$A_{i;t_{0}},A_{j;t}$ two cluster variables of $\mathcal{U}$. Then
$A_{i;t_{0}}$ and $A_{j;t}$ are associate in $\mathcal{U}$ if and only if
$A_{i;t_{0}}=A_{j;t}$.

###### Proof.

Assume that $A_{i;t_{0}}$ and $A_{j;t}$ are associate in $\mathcal{U}$. Then

(4.3) $\displaystyle A_{j;t}=\lambda Z_{1}^{c_{1}}\cdots
Z_{m}^{c_{m}}A_{i;t_{0}}$

for some $\lambda\in\mathbb{K}^{\times}$ and
$c_{1},\ldots,c_{m}\in\mathbb{Z}$, by Theorem 2.35 (2). Notice that the
equality (4.3) can be viewed as the Laurent expansion of $A_{j;t}$ with
respect to $t_{0}$. It is easy to see that $(A_{i;t_{0}}||A_{j;t})_{d}=-1$.
Then by Proposition 2.30 (4), we get $A_{j;t}=A_{i;t_{0}}$.

Conversely, assume $A_{i;t_{0}}=A_{j;t}$. Clearly, we have that $A_{i;t_{0}}$
and $A_{j;t}$ are associate in $\mathcal{U}$. This completes the proof. ∎

Inspired by Proposition 4.14, we give the following result.

######  Theorem 4.18 (Ray fish theorem).

Let $\mathcal{U}$ be a full rank, primitive upper cluster algebra with
coefficient semifield $\mathbb{P}={\rm Trop}(Z_{1},\ldots,Z_{m})$ and
$t,t_{0}$ two seeds of $\mathcal{U}$ with no common cluster variables. Then
$\mathcal{U}=\mathcal{L}(t)\cap\mathcal{L}(t_{0})$, where
$\mathcal{L}(t)=\mathbb{KP}[A_{1;t}^{\pm 1},\ldots,A_{n;t}^{\pm 1}]$ and
$\mathcal{L}(t_{0})=\mathbb{KP}[A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$.

###### Proof.

The proof is the same as that of [GLS13, Corollary 4.2]. For convenience of
the reader, we repeat it here.

Notice that the inclusion
$\mathcal{U}\subseteq\mathcal{L}(t)\cap\mathcal{L}(t_{0})$ is clear. Now we
show the converse inclusion. We know that any
$M\in\mathcal{L}(t)\cap\mathcal{L}(t_{0})$ has the following form:

$M=\frac{P(A_{1;t},\ldots,A_{n;t})}{A_{1;t}^{v_{1}}\cdots
A_{n;t}^{v_{n}}}=\frac{Q(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{s_{1}}\cdots
A_{n;t_{0}}^{s_{n}}},$

where
$P\in\mathbb{KP}[A_{1;t},\ldots,A_{n;t}],\;Q\in\mathbb{KP}[A_{1;t_{0}},\ldots,A_{n;t_{0}}]\subseteq\mathcal{U}$
and $v_{i},s_{i}\geq 0$ for any $i=1,\ldots,n$. Thus we have
$A_{1;t_{0}}^{s_{1}}\cdots A_{n;t_{0}}^{s_{n}}P=QA_{1;t}^{v_{1}}\cdots
A_{n;t}^{v_{n}}.$

By Theorem 2.35 (2), we know that $A_{i;t_{0}}$ and $A_{j;t}$ are irreducible
in $\mathcal{U}$ for any $i$ and $j$. By Lemma 4.17 and the fact that $t$ and
$t_{0}$ have no common cluster variables, we know that $A_{i;t_{0}}$ and
$A_{j;t}$ are non-associate for all $1\leq i,j\leq n$. Since $\mathcal{U}$ is
full rank and primitive and by Theorem 4.8, we know that $\mathcal{U}$ is
factorial. So the equality $A_{1;t_{0}}^{s_{1}}\cdots
A_{n;t_{0}}^{s_{n}}P=QA_{1;t}^{v_{1}}\cdots A_{n;t}^{v_{n}}$ implies that $P$
is divisible by $A_{1;t}^{v_{1}}\cdots A_{n;t}^{v_{n}}$ in $\mathcal{U}$.
Namely, there exists $P^{\prime}\in\mathcal{U}$ such that
$P=A_{1;t}^{v_{1}}\cdots A_{n;t}^{v_{n}}P^{\prime}$. Thus

$M=\frac{P}{A_{1;t}^{v_{1}}\cdots A_{n;t}^{v_{n}}}=P^{\prime}\in\mathcal{U}.$

So we have $\mathcal{L}(t)\cap\mathcal{L}(t_{0})\subseteq\mathcal{U}$ and thus
$\mathcal{U}=\mathcal{L}(t)\cap\mathcal{L}(t_{0})$. ∎

##  5\. Applications to $d$-vectors

###  5.1. $d$-compatibility degree and $d$-vectors via the valuation pairing

In this subsection, we show how to express $d$-compatibility and $d$-vectors
using the valuation pairing for full rank upper cluster algebras.

######  Theorem 5.1.

Let $\mathcal{U}$ be a full rank upper cluster algebra with initial seed
$t_{0}$. Let

$M=\frac{P_{M}(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{d_{1}}\cdots
A_{n;t_{0}}^{d_{n}}}$

be a non-zero element in $\mathcal{U}$, where ${\bf
d}_{M}:=(d_{1},\ldots,d_{n})^{\rm T}$ is the $d$-vector of $M$ with respect to
$A_{t_{0}}$. Then the following statements hold.

  * (i)

For any $k=1,\ldots,n$, we have

$d_{k}=(A_{k;t_{0}}||M)_{d}=(A_{k;t_{0}}||P_{M})_{v}-(A_{k;t_{0}}||M)_{v}.$

  * (ii)

If $(A_{k;t_{0}}||M)_{v}=0$ for some $k$, that is,
$M/A_{k;t_{0}}\notin\mathcal{U}$, then

$d_{k}=(A_{k;t_{0}}||M)_{d}=(A_{k;t_{0}}||P_{M})_{v}\geq 0.$

  * (iii)

If $(A_{k;t_{0}}||M)_{v}=0$ for $k=1,\ldots,n$, then ${\bf d}_{M}$ is uniquely
determined by $P_{M}$.

###### Proof.

(i) We know that there exist two vectors

${\bf d}^{+}=(d_{1}^{+},\ldots,d_{n}^{+})^{\rm T},\;{\bf
d}^{-}=(d_{1}^{-},\ldots,d_{n}^{-})^{\rm T}\in\mathbb{N}^{n}$

such that

${\bf d}_{M}={\bf d}^{+}-{\bf d}^{-}.$

By $M=P_{M}/A_{t_{0}}^{{\bf d}_{M}}=P_{M}/A_{t_{0}}^{{\bf d}^{+}-{\bf
d}^{-}}$, we have $A_{t_{0}}^{{\bf d}^{+}}M=A_{t_{0}}^{{\bf d}^{-}}P_{M}$. By
the Reduction Lemma 3.6 (iii), we get

$(A_{k;t_{0}}||M)_{v}+d_{k}^{+}=(A_{k;t_{0}}||A_{t_{0}}^{{\bf
d}_{+}}M)_{v}=(A_{k;t_{0}}||A_{t_{0}}^{{\bf
d}_{-}}P_{M})_{v}=(A_{k;t_{0}}||P_{M})_{v}+d_{k}^{-}.$

So we have

$d_{k}=d_{k}^{+}-d_{k}^{-}=(A_{k;t_{0}}||P_{M})_{v}-(A_{k;t_{0}}||M)_{v}.$

(ii) and (iii) follow from (i). ∎

###  5.2. Local factorizations of cluster monomials

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and initial seed $t_{0}$. In this subsection, we prove that if
$M$ is a cluster monomial in non-initial cluster variables, then
$M/A_{k;t_{0}}\notin\mathcal{U}$, that is, $(A_{k;t_{0}}||M)_{v}=0$ for
$k=1,\ldots,n$. In particular, we can give a local factorization for any
cluster monomial.

######  Lemma 5.2.

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and $t$ a seed of $\mathcal{U}$. If
$N=A_{1;t}^{b_{1}}A_{2;t}^{b_{2}}\cdots A_{n;t}^{b_{n}}\in\mathcal{U}$ for
some $b_{1},b_{2},\ldots,b_{n}\in\mathbb{Z}$, then
$b_{1},b_{2},\ldots,b_{n}\geq 0$.

###### Proof.

Assume by contradiction that there exists some $b_{i_{0}}<0$. Without loss of
generality, we just assume $b_{1}<0$. Let $t^{\prime}=\mu_{1}(t)$ and thus
$t=\mu_{1}(t^{\prime})$. Applying the exchange relation, we know that

$A_{1;t}A_{1;t^{\prime}}=P\in\mathbb{KP}[A_{2;t^{\prime}},\ldots,A_{n;t^{\prime}}],$

where $P$ is the first exchange binomial of $t^{\prime}$. Then we know that

$\displaystyle N=A_{1;t}^{b_{1}}A_{2;t}^{b_{2}}\cdots
A_{n;t}^{b_{n}}=\frac{A_{1;t^{\prime}}^{-b_{1}}A_{2;t^{\prime}}^{b_{2}}\cdots
A_{n;t^{\prime}}^{b_{n}}}{P^{-b_{1}}},$

which is the expansion of $N$ with respect to $A_{t^{\prime}}$. By
$N\in\mathcal{U}$, we get that

$N=\frac{A_{1;t^{\prime}}^{-b_{1}}A_{2;t^{\prime}}^{b_{2}}\cdots
A_{n;t^{\prime}}^{b_{n}}}{P^{-b_{1}}}\in\mathcal{L}(t^{\prime})=\mathbb{KP}[A_{1;t^{\prime}}^{\pm
1},\ldots,A_{n;t^{\prime}}^{\pm 1}].$

Thus we have $1/P^{-b_{1}}\in\mathcal{L}(t^{\prime}).$ Because both
$P^{-b_{1}}$ and $1/P^{-b_{1}}$ are in $\mathcal{L}(t^{\prime})$, we must have
$P^{-b_{1}}=\lambda A_{1;t^{\prime}}^{c_{1}}\cdots A_{n;t^{\prime}}^{c_{n}}$
for some invertible element $\lambda$ in $\mathbb{KP}$ and
$c_{1},\ldots,c_{n}\in\mathbb{Z}$. Since $P$ is a polynomial in
$\mathbb{KP}[A_{2;t^{\prime}},\ldots,A_{n;t^{\prime}}]$ with
$A_{i;t^{\prime}}\nmid P$ for any $i$, we get that $c_{1}=\ldots=c_{n}=0$ and
$P^{-b_{1}}=\lambda$ is invertible in $\mathbb{KP}$. By our assumption in
Subsection 1.4, we know that $P^{-b_{1}}=\lambda$ is not invertible in
$\mathbb{KP}$. This concludes a contradiction. So we must have
$b_{1},\ldots,b_{n}\geq 0$. ∎

######  Proposition 5.3.

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and $t_{0},t$ two seeds of $\mathcal{U}$. Let
$M=A_{1;t_{0}}^{c_{1}}\cdots A_{n;t_{0}}^{c_{n}}$ be a cluster monomial in
$A_{t_{0}}$ and $N=A_{1;t}^{b_{1}}\cdots A_{n;t}^{b_{n}}$ a Laurent monomial
in $A_{t}$. Let

$I=\\{i|c_{i}>0\\}\;\;\text{and}\;\;\;J=\\{j|b_{j}>0\\}.$

If there exists an invertible element $\lambda$ in $\mathbb{KP}$ such that
$M=\lambda N$, then

  * (i)

$N$ is a cluster monomial in $A_{t}$, i.e., we have $b_{1},\ldots,b_{n}\geq
0$;

  * (ii)

there exists a bijection $\sigma:I\rightarrow J$ such that
$A_{i;t_{0}}=A_{\sigma(i);t}$ and $c_{i}=b_{\sigma(i)}$. In particular, we
have $M=N$.

###### Proof.

(i) By $M=\lambda N$, we know that $N=\lambda^{-1}M\in\mathcal{U}$. Then by
Lemma 5.2, we know that $N$ is a cluster monomial in $A_{t}$.

(ii) Let $A_{j;t}$ be a cluster variable appearing in $N$. Then by (i), we
have $b_{j}>0$ and $j\in J$. Because we have

$\displaystyle\sum\limits_{i\in
I}^{n}c_{i}(A_{j;t}||A_{i;t_{0}})_{d}=(A_{j;t}||M)_{d}=(A_{j;t}||\lambda
N)_{d}=(A_{j;t}||N)_{d}=-b_{j}<0,$

there must exist some $j^{\prime}\in I$ such that

$(A_{j;t}||A_{j^{\prime};t_{0}})_{d}<0.$

Then by Proposition 2.30 (4), we have $A_{j;t}=A_{j^{\prime};t_{0}}$. So we
have

$c_{j^{\prime}}=-(A_{j^{\prime};t_{0}}||M)_{d}=-(A_{j;t}||M)_{d}=-(A_{j;t}||N)_{d}=b_{j}.$

Similarly, we can show that any cluster variable $A_{i;t_{0}}$ appearing in
$M$ also appears in $N$, and the multiplicity of $A_{i;t_{0}}$ in $N$ is equal
to the multiplicity of $A_{i;t_{0}}$ in $M$. Then the result follows. ∎

######  Theorem 5.4 (Local factorization of cluster monomials).

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and initial seed $t_{0}$. Then the following statements hold.

  * (i)

If $L$ is a cluster monomial of $\mathcal{U}$ which does not contain the
initial cluster variable $A_{k;t_{0}}$ for some $k$, then
$L/A_{k;t_{0}}\notin\mathcal{U}$, that is, $(A_{k;t_{0}}||L)_{v}=0$.

  * (ii)

If $L$ is a cluster monomial in non-initial cluster variables, then
$L/A_{k;t_{0}}\notin\mathcal{U}$, that is, $(A_{k;t_{0}}||L)_{v}=0$ for
$k=1,\ldots,n$.

  * (iii)

Let $M=\prod\limits_{i=1}^{n}A_{i;t}^{b_{i}}$ be a cluster monomial in a seed
$t$ of $\mathcal{U}$ and $I=\\{i|A_{i;t}\in A_{t_{0}}\\}$, then
$M=\prod\limits_{i\in I}A_{i;t}^{b_{i}}\cdot\prod\limits_{j\notin
I}A_{j;t}^{b_{j}}$ is a local factorization of $M$ with respect to $t_{0}$.

###### Proof.

(i) Assume by contradiction that $L/A_{k;t_{0}}\in\mathcal{U}$. In the
following proof, we will deduce a contradiction.

Because the initial cluster variable $A_{k;t_{0}}$ does not appear in $L$,
there exists a seed $t$ such that $L$ is a cluster monomial in $A_{t}$ and
$A_{k;t_{0}}$ does not belong to $A_{t}$. Consider the Laurent expansion of
$A_{k;t_{0}}$ with respect to $A_{t}$, say

$A_{k;t_{0}}=\frac{P(A_{1;t},\ldots,A_{n;t})}{A_{1;t}^{d_{1}^{\prime}}\cdots
A_{n;t}^{d_{n}^{\prime}}},$

where $P\in\mathbb{KP}[A_{1;t},\ldots,A_{n;t}]$ with $A_{i;t}\nmid P$ for
$i=1,\ldots,n$.

Now we have that

$\displaystyle L/A_{k;t_{0}}=\frac{L\cdot(A_{1;t}^{d_{1}^{\prime}}\cdots
A_{n;t}^{d_{n}^{\prime}})}{P},$

which is the expansion of $L/A_{k;t_{0}}$ with respect to $A_{t}$. By
$L/A_{k;t_{0}}\in\mathcal{U}$, we have

$\frac{L\cdot(A_{1;t}^{d_{1}^{\prime}}\cdots
A_{n;t}^{d_{n}^{\prime}})}{P}\in{\mathcal{L}}(t):=\mathbb{KP}[A_{1;t}^{\pm
1},\ldots,A_{n;t}^{\pm 1}].$

Because $L\cdot(A_{1;t}^{d_{1}^{\prime}}\cdots A_{n;t}^{d_{n}^{\prime}})$ is a
monic Laurent monomial in ${\mathcal{L}}(t)$, we get $1/P\in{\mathcal{L}}(t)$.
Since both $P$ and $1/P$ are in ${\mathcal{L}}(t)$, we must have $P=\lambda
A_{1;t}^{c_{1}}\cdots A_{n;t}^{c_{n}}$ for some invertible element $\lambda$
in $\mathbb{KP}$ and $c_{1},\ldots,c_{n}\in\mathbb{Z}$. Because $P$ is a
polynomial in $\mathbb{KP}[A_{1;t},\ldots,A_{n;t}]$ with $A_{i;t}\nmid P$ for
any $i$, we get $c_{1}=\ldots=c_{n}=0$ and thus $P=\lambda$ is invertible in
$\mathbb{KP}$. Applying Proposition 5.3 to

$A_{k;t_{0}}=\frac{P}{A_{1;t}^{d_{1}^{\prime}}\cdots
A_{n;t}^{d_{n}^{\prime}}}=\lambda\cdot A_{1;t}^{-d_{1}^{\prime}}\cdots
A_{n;t}^{-d_{n}^{\prime}},$

we get $A_{k;t_{0}}\in A_{t}$, which contradicts $A_{k;t_{0}}\notin A_{t}$. So
$L/A_{k;t_{0}}\notin\mathcal{U}$.

(ii) follows from (i).

(iii) follows from (ii) and the definition of local factorization. ∎

######  Corollary 5.5.

Let $\mathcal{U}$ be a full rank upper cluster algebra with initial seed
$t_{0}$. Let

$M=\frac{P_{M}(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{d_{1}}\cdots
A_{n;t_{0}}^{d_{n}}}$

be a cluster monomial of $\mathcal{U}$, where ${\bf
d}_{M}:=(d_{1},\ldots,d_{n})^{\rm T}$ is the $d$-vector of $M$ with respect to
$A_{t_{0}}$. Then the following statements hold.

  * (i)

If $M$ does not contain the initial cluster variable $A_{k;t_{0}}$ for some
$k$, then we have

$d_{k}=(A_{k;t_{0}}||M)_{d}=(A_{k;t_{0}}||P_{M})_{v}\geq 0.$

  * (ii)

If $M$ is a cluster monomial in non-initial cluster variables, then ${\bf
d}_{M}$ and $M$ are uniquely determined by the numerator polynomial $P_{M}$.

###### Proof.

(i) By Theorem 5.4 (i), we know that $(A_{k;t_{0}}||M)_{v}=0$. Then by Theorem
5.1 (ii), we get $d_{k}=(A_{k;t_{0}}||M)_{d}=(A_{k;t_{0}}||P_{M})_{v}\geq 0$.

(ii) This follows from (i). ∎

Actually, the above result can be extended from cluster monomials to monomials
in cluster variables if we further assume that $\mathcal{U}$ is primitive or
factorial.

######  Proposition 5.6.

Let $\mathcal{U}$ be a full rank, primitive upper cluster algebra with initial
seed $t_{0}$. Let

$M=\frac{P_{M}(A_{1;t_{0}},\ldots,A_{n;t_{0}})}{A_{1;t_{0}}^{d_{1}}\cdots
A_{n;t_{0}}^{d_{n}}}$

be a monomial in cluster variables of $\mathcal{U}$ (not necessarily a cluster
monomial), where ${\bf d}_{M}:=(d_{1},\ldots,d_{n})^{\rm T}$ is the $d$-vector
of $M$ with respect to $A_{t_{0}}$. Then the following statements hold.

  * (i)

If $M$ does not contain the initial cluster variable $A_{k;t_{0}}$ for some
$k$, then we have $(A_{k;t_{0}}||M)_{v}=0$ and
$d_{k}=(A_{k;t_{0}}||M)_{d}=(A_{k;t_{0}}||P_{M})_{v}\geq 0.$

  * (ii)

If $M$ is a monomial in non-initial cluster variables, then ${\bf d}_{M}$ and
$M$ are uniquely determined by the numerator polynomial $P_{M}$.

###### Proof.

(i) Since $M$ is a monomial in cluster variables and it does not contain
$A_{k;t_{0}}$, we can assume that $M=\prod\limits_{i=1}^{s}M_{i}$, where each
$M_{i}$ is a cluster variable of $\mathcal{U}$ different from $A_{k;t_{0}}$.
By Theorem 5.4 (i), we have $(A_{k;t_{0}}||M_{i})_{v}=0$, for $i=1,\ldots,s$.

Since $\mathcal{U}$ is a full rank, primitive upper cluster algebra, we know
that $\mathcal{U}$ is factorial, by Theorem 4.8. Then by Theorem 4.4 (i)(iii),
we know that

$(A_{k;t_{0}}||M)_{v}=\sum\limits_{i=1}^{s}(A_{k;t_{0}}||M_{i})_{v}=0.$

Then by Theorem 5.1 (ii), we get
$d_{k}=(A_{k;t_{0}}||M)_{d}=(A_{k;t_{0}}||P_{M})_{v}\geq 0$.

(ii) This follows from (i). ∎

##  6\. Application to $F$-polynomials

In this section, we always assume that $\mathcal{U}$ is an upper cluster
algebra with principal coefficients at $t_{0}$.

###  6.1. From $F$-polynomials to monomials in non-initial cluster variables

In this subsection, we prove that if $M$ is a monomial in non-initial cluster
variables, then $M$ is uniquely determined by its $F$-polynomial.

Let $M$ be a monomial in cluster variables of $\mathcal{U}$, say
$M=\prod\limits_{i=1}^{s}M_{i}$, where each $M_{i}$ is a cluster variable of
$\mathcal{U}$. The $F$-polynomial $F_{M}$ of $M$ is defined to be the
polynomial $F_{M}:=\prod\limits_{i=1}^{s}F_{M_{i}},$ where $F_{M_{i}}$ is the
$F$-polynomial of the cluster variable $M_{i}$. The $g$-vector of $M$ is
defined to be the vector ${\bf g}_{M}:=\sum\limits_{i=1}^{s}{\bf g}_{M_{i}}$,
where ${\bf g}_{M_{i}}$ is the $g$-vector of $M_{i}$.

######  Theorem 6.1.

Let $\mathcal{U}$ be an upper cluster algebra with principal coefficients at
$t_{0}$. Let $M,N$ be two monomials in non-initial cluster variables of
$\mathcal{U}$ and $F_{M},F_{N}$ their $F$-polynomials. If $F_{M}=F_{N}$, then
$M=N$. In particular, ${\bf g}_{M}={\bf g}_{N}$, where ${\bf g}_{M}$ and ${\bf
g}_{N}$ are $g$-vectors of $M$ and $N$.

###### Proof.

Let ${\bf d}_{M}$ and ${\bf d}_{N}$ be the $d$-vectors of $M$ and $N$ with
respect to $A_{t_{0}}$. We know that $M$ and $N$ have the form

(6.1) $\displaystyle M=P_{M}/A_{t_{0}}^{{\bf d}_{M}}\;\;\text{ and
}\;\;\;N=P_{N}/A_{t_{0}}^{{\bf d}_{N}},$

where
$P_{M},P_{N}\in\mathbb{Z}[Z_{1},\ldots,Z_{n};A_{1;t_{0}},\ldots,A_{n;t_{0}}]$
with $A_{i;t_{0}}\nmid P_{M}$ and $A_{i;t_{0}}\nmid P_{N}$ for any
$i=1,\ldots,n$.

By Theorem 2.26 (i), we know that

$\displaystyle M$ $\displaystyle=$ $\displaystyle A_{t_{0}}^{{\bf
g}_{M}}F_{M}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}}),$ $\displaystyle N$
$\displaystyle=$ $\displaystyle A_{t_{0}}^{{\bf
g}_{N}}F_{N}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}}),$

where
$\hat{X}_{j;t_{0}}=Z_{j}\prod\limits_{i=1}^{n}A_{i;t_{0}}^{b_{ij}^{t_{0}}}$.
So we can get the following equalities:

$\displaystyle F_{M}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}})$
$\displaystyle=$ $\displaystyle P_{M}/A_{t_{0}}^{{\bf d}_{M}+{\bf g}_{M}},$
$\displaystyle F_{N}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}})$
$\displaystyle=$ $\displaystyle P_{N}/A_{t_{0}}^{{\bf d}_{N}+{\bf g}_{N}}.$

Because $P_{M}$ and $P_{N}$ are not divisible by any $A_{i;t_{0}}$, we know
that ${\bf d}_{M}+{\bf g}_{M}$ and ${\bf d}_{N}+{\bf g}_{N}$ are the
$d$-vectors of $F_{M}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}})$ and
$F_{N}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}})$ with respect to $t_{0}$.
By $F_{M}=F_{N}$, we have

$F_{M}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}})=F_{N}(\hat{X}_{1;t_{0}},\ldots,\hat{X}_{n;t_{0}}),$

which implies that ${\bf d}_{M}+{\bf g}_{M}={\bf d}_{N}+{\bf g}_{N}$ and
$P_{M}=P_{N}$.

By $P_{M}=P_{N}$ and applying Proposition 5.6 to $M$ and $N$, we get ${\bf
d}_{M}={\bf d}_{N}$. Then by the equality (6.1), we know that $M=N$. In
particular, we have ${\bf g}_{M}={\bf g}_{N}$. This completes the proof. ∎

###  6.2. $F$-polynomials of non-initial cluster variables are irreducible

In this subsection, we prove that the $F$-polynomials of non-initial cluster
variables are irreducible in $\mathbb{K}[Z_{1},\ldots,Z_{n}]$.

######  Theorem 6.2.

Let $\mathcal{U}$ be an upper cluster algebra with principal coefficients at
$t_{0}$ and $A_{k;t}$ a non-initial cluster variable of $\mathcal{U}$. Then
the $F$-polynomial $F_{k;t}$ of $A_{k;t}$ is irreducible in
$\mathbb{K}[Z_{1},\ldots,Z_{n}]$.

###### Proof.

We claim that $F_{k;t}$ is not a constant. Otherwise, the $f$-compatibility
degree $(A_{i;t_{0}}||A_{k;t})_{f}=0$ for $i=1,\ldots,n$. Then by Proposition
2.30 (ii), we get that $A_{k;t}$ is compatible with any cluster variable in
$A_{t_{0}}$. Thus $A_{k;t}\in A_{t_{0}}$, by Theorem 2.33. This contradicts
that $A_{k;t}$ is a non-initial cluster variable of $\mathcal{U}$. So
$F_{k;t}$ is not a constant.

Since $F_{k;t}$ is not a constant, it can not be invertible in
$\mathbb{K}[Z_{1},\ldots,Z_{n}]$. So we can discuss its irreducibility. Assume
there exist two polynomials $F_{1},F_{2}\in\mathbb{K}[Z_{1},\ldots,Z_{n}]$
such that

(6.2) $\displaystyle
F_{k;t}(Z_{1},\ldots,Z_{n})=F_{1}(Z_{1},\ldots,Z_{n})F_{2}(Z_{1},\ldots,Z_{n}).$

We will show that either $F_{1}\in\mathbb{K}^{\times}$ or
$F_{2}\in\mathbb{K}^{\times}$.

For $1\leq j\leq n$, put
$\hat{X}_{j}=Z_{j}\prod\limits_{i=1}^{n}A_{i;t_{0}}^{b_{ij}^{t_{0}}}$.
Consider the Laurent polynomials $F_{1}(\hat{X}_{1},\ldots,\hat{X}_{n})$ and
$F_{2}(\hat{X}_{1},\ldots,\hat{X}_{n})$ in
$\mathbb{K}[Z_{1},\ldots,Z_{n}][A_{1;t_{0}}^{\pm 1},\ldots,A_{n;t_{0}}^{\pm
1}]$. We choose two vectors ${\bf d}_{1},{\bf d}_{2}$ in $\mathbb{N}^{n}$ such
that

$M_{1}:=A_{t_{0}}^{{\bf
d}_{1}}F_{1}(\hat{X}_{1},\ldots,\hat{X}_{n})\;\;\;\;\text{and}\;\;\;\;M_{2}:=A_{t_{0}}^{{\bf
d}_{2}}F_{2}(\hat{X}_{1},\ldots,\hat{X}_{n})$

are polynomials in
$\mathbb{K}[Z_{1},\ldots,Z_{n}][A_{1;t_{0}},\ldots,A_{n;t_{0}}]\subseteq\mathcal{U}$.
By the local unique factorization property, there exist unique cluster
monomials $N_{1}=A_{t_{0}}^{{\bf v}_{1}},N_{2}=A_{t_{0}}^{{\bf v}_{2}}$ such
that

$L_{1}:=M_{1}/N_{1}=A_{t_{0}}^{{\bf d}_{1}-{\bf
v}_{1}}F_{1}(\hat{X}_{1},\ldots,\hat{X}_{n})$

and

$L_{2}:=M_{2}/N_{2}=A_{t_{0}}^{{\bf d}_{2}-{\bf
v}_{2}}F_{2}(\hat{X}_{1},\ldots,\hat{X}_{n})$

are in $\mathcal{U}$ and satisfy

$(A_{j;t_{0}}||L_{1})_{v}=0=(A_{j;t_{0}}||L_{2})_{v}$

for $j=1,\ldots,n$. Then by Corollary 4.10, we have

$(A_{j;t_{0}}||L_{1}L_{2})_{v}=(A_{j;t_{0}}||L_{1})_{v}+(A_{j;t_{0}}||L_{2})_{v}=0$

for $j=1,\ldots,n$.

By Theorem 2.26 (i), we know that the cluster variable $A_{k;t}$ has the
following form:

$\displaystyle A_{k;t}=A_{t_{0}}^{{\bf g}}\cdot
F_{k;t}(\hat{X}_{1},\ldots,\hat{X}_{n}),$

where ${\bf g}\in\mathbb{Z}^{n}$. By the equality (6.2), we have that

$F_{k;t}(\hat{X}_{1},\ldots,\hat{X}_{n})=F_{1}(\hat{X}_{1},\ldots,\hat{X}_{n})F_{2}(\hat{X}_{1},\ldots,\hat{X}_{n}).$

So we know that

$A_{k;t}/(L_{1}L_{2})=A_{t_{0}}^{{\bf g}+{\bf v}_{1}+{\bf v}_{2}-{\bf
d}_{1}-{\bf d}_{2}}.$

Thus we have $A_{k;t}=A_{t_{0}}^{\bf v}(L_{1}L_{2})$, where ${\bf v}={\bf
g}+{\bf v}_{1}+{\bf v}_{2}-{\bf d}_{1}-{\bf d}_{2}$.

Now we show that ${\bf v}=0$ and $A_{k;t}=L_{1}L_{2}$. We choose two vectors
${\bf v}^{+}$ and ${\bf v}^{-}$ in $\mathbb{N}^{n}$ such that ${\bf v}={\bf
v}^{+}-{\bf v}^{-}$. Then by

$A_{k;t}=A_{t_{0}}^{\bf v}(L_{1}L_{2})=A_{t_{0}}^{{\bf v}^{+}-{\bf
v}^{-}}(L_{1}L_{2}),$

we get that

(6.3) $\displaystyle M:=A_{t_{0}}^{{\bf v}^{-}}A_{k;t}=A_{t_{0}}^{{\bf
v}^{+}}L_{1}L_{2}.$

Since $F_{k;t}$ is not a constant, we know that $A_{k;t}$ is not an initial
cluster variable. Then by Theorem 5.4, we have $(A_{j;t_{0}}||A_{k;t})_{v}=0$
for $j=1,\ldots,n$. On the other hand, we know that $L_{1}L_{2}$ also
satisfies that $(A_{j;t_{0}}||L_{1}L_{2})_{v}=0$ for $j=1,\ldots,n$. So the
equality (6.3) gives two local factorizations of $M$ with respect to $t_{0}$.
Then by the uniqueness in Theorem 3.7, we get that

${\bf v}^{+}={\bf v}^{-}\;\;\text{and}\;\;\;A_{k;t}=L_{1}L_{2}.$

By Theorem 2.35 (ii), we know that $\mathcal{U}^{\times}=\\{\lambda
Z_{1}^{c_{1}}\cdots
Z_{n}^{c_{n}}|\lambda\in\mathbb{K}^{\times},\;c_{1},\ldots,c_{n}\in\mathbb{Z}\\}$
and that $A_{k;t}$ is irreducible in $\mathcal{U}$. So the factorization
$A_{k;t}=L_{1}L_{2}$ implies either $L_{1}=\lambda Z_{1}^{c_{1}}\cdots
Z_{n}^{c_{n}}$ or $L_{2}=\lambda Z_{1}^{c_{1}}\cdots Z_{n}^{c_{n}}$ for some
$\lambda\in\mathbb{K}^{\times}$ and $(c_{1},\ldots,c_{n})\in\mathbb{Z}^{n}$.
So either

$F_{1}(Z_{1},\ldots,Z_{n})=L_{1}|_{A_{1;t_{0}}=\ldots=A_{n;t_{0}}=1}=\lambda
Z_{1}^{c_{1}}\cdots Z_{n}^{c_{n}}$

or

$F_{2}(Z_{1},\ldots,Z_{n})=L_{2}|_{A_{1;t_{0}}=\ldots=A_{n;t_{0}}=1}=\lambda
Z_{1}^{c_{1}}\cdots Z_{n}^{c_{n}}$

for some $\lambda\in\mathbb{K}^{\times}$ and
$(c_{1},\ldots,c_{n})\in\mathbb{Z}^{n}$. Because both $F_{1}$ and $F_{2}$ are
polynomials in $\mathbb{K}[Z_{1},\ldots,Z_{n}]$, we must have
$c_{1},\ldots,c_{n}\geq 0$. Now by Proposition 2.22, we know that
$F_{k;t}=F_{1}F_{2}$ is not divisible by any $Z_{j}$. So we must have
$c_{1}=\ldots=c_{n}=0$. Thus either $F_{1}=\lambda$ or $F_{2}=\lambda$ for
some $\lambda\in\mathbb{K}^{\times}$. So $F_{k;t}$ is irreducible in
$\mathbb{K}[Z_{1},\ldots,Z_{n}]$. ∎

######  Remark 6.3.

Garcia Elsener et al. in [GELS19] prove that the $F$-polynomials of non-
initial cluster variables are irreducible for factorial principal coefficient
cluster algebras. In fact, their proof still works for factorial principal
coefficient upper cluster algebras. Note that the factoriality of principal
coefficient upper cluster algebras is no longer a problem, thanks to Theorem
4.8. Thus one can also use the method in [GELS19] to show the irreducibility
of $F$-polynomials of non-initial cluster variables.

##  7\. Application to combinatorics of cluster Poisson variables

###  7.1. Parametrization of $\mathscr{A}$-exchange pairs by cluster Poisson
variables

In this subsection, we give several equivalent characterizations of when two
cluster Poisson variables are equal. As an application, we prove that the
$\mathscr{A}$-exchange pairs are parameterized by the cluster Poisson
variables.

######  Proposition 7.1.

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and initial seed $t_{0}$. Let $M_{1},M_{2}$ be two cluster
monomials of $\mathcal{U}$ and $N_{1},N_{2}$ two different cluster monomials
in $t_{0}$. If

$p_{1}M_{1}+p_{2}M_{2}=p_{1}^{\prime}N_{1}+p_{2}^{\prime}N_{2}$

holds for some $p_{1},p_{2},p_{1}^{\prime},p_{2}^{\prime}\in\mathbb{P}$, then
either $(M_{1},M_{2},p_{1},p_{2})=(N_{1},N_{2},p_{1}^{\prime},p_{2}^{\prime})$
or $(M_{1},M_{2},p_{1},p_{2})=(N_{2},N_{1},p_{2}^{\prime},p_{1}^{\prime})$
holds.

###### Proof.

Since both $N_{1}$ and $N_{2}$ are cluster monomials in $A_{t_{0}}$, we have

$(A_{i;t_{0}}||p_{1}^{\prime}N_{1})_{d}=(A_{i;t_{0}}||N_{1})_{d}\leq 0$

and

$(A_{i;t_{0}}||p_{2}^{\prime}N_{2})_{d}=(A_{i;t_{0}}||N_{2})_{d}\leq 0$

for $i=1,\ldots,n$. By the positivity of the Laurent phenomenon and since the
coefficients in $p_{1}M_{1}+p_{2}M_{2}$ and
$p_{1}^{\prime}N_{1}+p_{2}^{\prime}N_{2}$ are positive, we have

$\displaystyle{\rm
max}\\{(A_{i;t_{0}}||M_{1})_{d},(A_{i;t_{0}}||M_{2})_{d}\\}$ $\displaystyle=$
$\displaystyle{\rm
max}\\{(A_{i;t_{0}}||p_{1}M_{1})_{d},(A_{i;t_{0}}||p_{2}M_{2})_{d}\\}$
$\displaystyle=$
$\displaystyle(A_{i;t_{0}}||p_{1}M_{1}+p_{2}M_{2})_{d}=(A_{i;t_{0}}||p_{1}^{\prime}N_{1}+p_{2}^{\prime}N_{2})_{d}$
$\displaystyle=$ $\displaystyle{\rm
max}\\{(A_{i;t_{0}}||p_{1}^{\prime}N_{1})_{d},(A_{i;t_{0}}||p_{2}^{\prime}N_{2})_{d}\\}\leq
0,$

for $i=1,\ldots,n$.

We claim that $M_{1}$ and $M_{2}$ are cluster monomials in $t_{0}$. Let
$A_{k;t}$ be a cluster variable appearing in $M_{1}$ or $M_{2}$. By Corollary
2.32 (ii) and

${\rm max}\\{(A_{i;t_{0}}||M_{1})_{d},(A_{i;t_{0}}||M_{2})_{d}\\}\leq 0,$

we know that $A_{k;t}$ is compatible with $A_{i;t_{0}}$ for any
$i=1,\ldots,n$. Then by Theorem 2.33, we get $A_{k;t}\in A_{t_{0}}$. So
$M_{1}$ and $M_{2}$ are cluster monomials in $t_{0}$.

Hence, the polynomial $p_{1}M_{1}+p_{2}M_{2}$ is actually a polynomial in
variables from $A_{t_{0}}$. Since $N_{1}$ and $N_{2}$ are two different
cluster monomials in $t_{0}$, we must have $M_{1}\neq M_{2}$. Otherwise,
$p_{1}^{\prime}N_{1}+p_{2}^{\prime}N_{2}$ can not be equal to
$p_{1}M_{1}+p_{2}M_{2}$. Now the result follows from the algebraic
independence of $A_{1;t_{0}},\ldots,A_{n;t_{0}}$. ∎

######  Proposition 7.2.

Let $\mathcal{U}$ be an upper cluster algebra with coefficient semifield
$\mathbb{P}$ and initial seed $t_{0}$. Let $A_{k;t}$ be a cluster variable of
$\mathcal{U}$. Then the following two statements are equivalent.

  * (i)

$(A_{j;t_{0}},A_{k;t})$ is an $\mathscr{A}$-exchange pair associated with the
mutation $t_{1}:=\mu_{j}(t_{0})$;

  * (ii)

$(A_{j;t_{0}}||A_{k;t})_{d}>0$ and $(A_{i;t_{0}}||A_{k;t})_{d}=0$ for any
$i\neq j$.

###### Proof.

(i)$\Longrightarrow$ (ii): This is clear.

(ii)$\Longrightarrow$ (i): By Proposition 2.30 (4) and the assumption, we get
that $A_{k;t}\notin A_{t_{0}}$ and
$\\{A_{k;t}\\}\cup(A_{t_{0}}\backslash\\{A_{j;t_{0}}\\})$ is a compatible set
with $n$ elements. Then by Theorem 2.33, we know that
$\\{A_{k;t}\\}\cup(A_{t_{0}}\backslash\\{A_{j;t_{0}}\\})$ is a cluster of
$\mathcal{U}$, say
$A_{t_{2}}=\\{A_{k;t}\\}\cup(A_{t_{0}}\backslash\\{A_{j;t_{0}}\\})$. Since the
seeds $t_{2}$ and $t_{0}$ have exactly $n-1$ common cluster variables, we get
that there is an edge between $t_{2}$ and $t_{0}$ in the exchange graph of
$\mathcal{U}$, by Corollary 2.18. Then the result follows. ∎

######  Proposition 7.3.

Let $\mathcal{U}$ and $\mathcal{U}^{\prime}$ be any two upper cluster algebras
with the same initial exchange matrix $B_{t_{0}}$ at $t_{0}$. For any vertex
$t$ of $\mathbb{T}_{n}$, denote $A_{t}$ (respectively, $A_{t}^{\prime}$) be
the cluster of $\mathcal{U}$ (respectively, $\mathcal{U}^{\prime}$) at $t$.
Then we have

  * (i)

$A_{i;t_{1}}=A_{j;t_{2}}$ in $\mathcal{U}$ if and only if
$A_{i;t_{1}}^{\prime}=A_{j;t_{2}}^{\prime}$ in $\mathcal{U}^{\prime}$;

  * (ii)

The map $\rho:A_{k;t}\mapsto A_{k;t}^{\prime}$ gives a bijection from the set
of cluster variables of $\mathcal{U}$ to that of $\mathcal{U}^{\prime}$;

  * (iii)

$\rho$ induces a bijection from the set of $\mathscr{A}$-exchange pairs of
$\mathcal{U}$ to that of $\mathcal{U}^{\prime}$.

###### Proof.

(i) Because $\mathcal{U}$ and $\mathcal{U}^{\prime}$ have the same initial
exchange matrix $B_{t_{0}}$ at $t_{0}$, we know that they have the same
exchange matrix at each vertex $t$ of $\mathbb{T}_{n}$. Notice that both
$(A_{i;t_{1}}||A_{j;t_{2}})_{d}$ and
$(A_{i;t_{1}}^{\prime}||A_{j;t_{2}}^{\prime})_{d}$ are defined by the
$(i,j)$-entry of the $D$-matrix $D_{t_{2}}^{B_{t_{1}};t_{1}}$, which is
independent of the choice of coefficient semifield. So we have
$(A_{i;t_{1}}||A_{j;t_{2}})_{d}=(A_{i;t_{1}}^{\prime}||A_{j;t_{2}}^{\prime})_{d}$.
By Proposition 2.30 (4), we know that $A_{i;t_{1}}=A_{j;t_{2}}$ in
$\mathcal{U}$ if and only if $(A_{i;t_{1}}||A_{j;t_{2}})_{d}=-1$ if and only
if $(A_{i;t_{1}}^{\prime}||A_{j;t_{2}}^{\prime})_{d}=-1$ if and only if
$A_{i;t_{1}}^{\prime}=A_{j;t_{2}}^{\prime}$ in $\mathcal{U}^{\prime}$.

(ii) follows from (i) and (iii) follows from (ii) and the characterization of
$\mathscr{A}$-exchange pairs in Proposition 7.2. ∎

######  Lemma 7.4.

Let $\mathcal{U}_{\rm uc}$ be an upper cluster algebra with universal
coefficient semifield and $t,t_{0}$ two seeds of $\mathcal{U}_{\rm uc}$. Let
$t^{\prime}=\mu_{k}(t)$ and $t_{1}=\mu_{j}(t_{0})$. Let
$\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}$ be the $k$-th
exchange binomial of $t$ and
$\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}$ the
$j$-th exchange binomial of $t_{0}$. If

$\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2},$

then either

$(A_{k;t},A_{k;t^{\prime}})=(A_{j;t_{0}},A_{j;t_{1}})\;\;\text{ or
}\;\;\;(A_{k;t},A_{k;t^{\prime}})=(A_{j;t_{1}},A_{j;t_{0}}).$

###### Proof.

By the assumption, we know that

$A_{k;t}A_{k;t^{\prime}}=\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}.$

The above equality can be viewed as the Laurent expansion of
$A_{k;t}A_{k;t^{\prime}}$ with respect to $A_{t_{0}}$. Clearly, this Laurent
expansion is a polynomial in $A_{t_{0}}$.

In the following proof, we distinguish two cases. Case (a):
$(A_{i;t_{0}}||A_{k;t})_{d}\leq 0$ for any $i=1,\ldots,n$. Case (b): there
exists some $i_{0}$ such that $(A_{i_{0};t_{0}}||A_{k;t})_{d}>0$.

Case (a): If $(A_{i;t_{0}}||A_{k;t})_{d}\leq 0$ for any $i=1,\ldots,n$, then
$A_{k;t}$ is compatible with any cluster variable in $A_{t_{0}}$. By Theorem
2.33, we know that $A_{t_{0}}$ is a maximal compatible set. So $A_{k;t}\in
A_{t_{0}}$, say $A_{k;t}=A_{i_{0};t_{0}}$. We will show that

$i_{0}=j\;\;\text{and}\;\;\;A_{i_{0};t_{0}}=A_{j;t_{0}}.$

We know that

$A_{k;t^{\prime}}=A_{k;t}^{-1}\cdot(\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2})=A_{i_{0};t_{0}}^{-1}\cdot(\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}),$

which is the Laurent expansion of $A_{k;t^{\prime}}$ with respect to
$A_{t_{0}}$. So we have

$(A_{i_{0};t_{0}}||A_{k;t^{\prime}})_{d}=1>0\;\;\text{ and
}\;\;\;(A_{i;t_{0}}||A_{k;t^{\prime}})_{d}=0$

for any $i\neq i_{0}$. Then by Proposition 7.2, we get that

$A_{k;t^{\prime}}=A_{i_{0};t_{2}},$

where $t_{2}=\mu_{i_{0}}(t_{0})$.

Let
$\frac{X_{i_{0};t_{0}}}{1+X_{i_{0};t_{0}}}L_{1}+\frac{1}{1+X_{i_{0};t_{0}}}L_{2}$
be the $i_{0}$-th exchange binomial of $t_{0}$. By Proposition 7.1 and the
following equality,

$\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}=A_{k;t}A_{k;t^{\prime}}=A_{i_{0};t_{0}}A_{i_{0};t_{2}}=\frac{X_{i_{0};t_{0}}}{1+X_{i_{0};t_{0}}}L_{1}+\frac{1}{1+X_{i_{0};t_{0}}}L_{2},$

we get that either $X_{j;t_{0}}=X_{i_{0};t_{0}}$ or
$X_{j;t_{0}}=X_{i_{0};t_{0}}^{-1}$. Since $X_{1;t_{0}},\ldots,X_{n;t_{0}}$ are
algebraically independent, we must have $X_{j;t_{0}}=X_{i_{0};t_{0}}$ and
$j=i_{0}$. So

$A_{k;t}=A_{i_{0};t_{0}}=A_{j;t_{0}}.$

Then by the following equality,

$A_{k;t}A_{k;t^{\prime}}=\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}=A_{j;t_{0}}A_{j;t_{1}},$

we get $A_{k;t^{\prime}}=A_{j;t_{1}}$ and thus
$(A_{k;t},A_{k;t^{\prime}})=(A_{j;t_{0}},A_{j;t_{1}})$.

Case (b): There exists some $i_{0}$ such that
$(A_{i_{0};t_{0}}||A_{k;t})_{d}>0$. By the fact that the expansion of
$A_{k;t}A_{k;t^{\prime}}$ with respect to $A_{t_{0}}$ is a polynomial, we know
that

$0\geq(A_{i_{0};t_{0}}||A_{k;t}A_{k;t^{\prime}})=(A_{i_{0};t_{0}}||A_{k;t})_{d}+(A_{i_{0};t_{0}}||A_{k;t^{\prime}})_{d}.$

So we must have $(A_{i_{0};t_{0}}||A_{k;t^{\prime}})_{d}<0$. Then by
Proposition 2.30 (4), we get

$A_{k;t^{\prime}}=A_{i_{0};t_{0}}.$

Similarly to the proof in Case (a), we can prove that $i_{0}=j$ and thus

$A_{k;t^{\prime}}=A_{i_{0};t_{0}}=A_{j;t_{0}}.$

Then $A_{k;t}=A_{j;t_{1}}$ follows from the following equality

$A_{k;t}A_{k;t^{\prime}}=\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}=A_{j;t_{0}}A_{j;t_{1}}.$

So we have $(A_{k;t},A_{k;t^{\prime}})=(A_{j;t_{1}},A_{j;t_{0}})$. This
completes the proof. ∎

######  Theorem 7.5.

Let $\mathcal{U}_{\rm uc}$ be an upper cluster algebra with universal
coefficient semifield and $t,t_{0}$ two seeds of $\mathcal{U}_{\rm uc}$. Let
$\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}$ be the $k$-th
exchange binomial of $t$ and
$\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}$ the
$j$-th exchange binomial of $t_{0}$. Then the following statements are
equivalent.

  * (i)

$X_{k;t}=X_{j;t_{0}}$;

  * (ii)

$(A_{k;t},M_{1},M_{2})=(A_{j;t_{0}},N_{1},N_{2})$;

  * (iii)

$(A_{k;t},\\{A_{i;t}|b_{ik}^{t}\neq
0\\})=(A_{j;t_{0}},\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\})$;

  * (iv)

$A_{k;t}=A_{j;t_{0}}$ and $\\{A_{i;t}|b_{ik}^{t}\neq 0\\}\subseteq A_{t_{0}}$;

  * (v)

$(A_{k;t},A_{k;t^{\prime}})=(A_{j;t_{0}},A_{j;t_{1}})$, where
$t^{\prime}=\mu_{k}(t)$ and $t_{1}=\mu_{j}(t_{0})$;

  * (vi)

$A_{k;t}=A_{j;t_{0}}$ and
$\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}$.

###### Proof.

(i)$\Longrightarrow$ (ii): By Theorem 2.26 (ii), we know that

$X_{k;t}=X_{t_{0}}^{{\bf
c}_{k;t}^{B_{t_{0}};t_{0}}}\cdot\prod\limits_{i=1}^{n}(F_{i;t}^{B_{t_{0}};t_{0}}(X_{1;t_{0}},\ldots,X_{n;t_{0}}))^{b_{ik}^{t}}.$

By $X_{k;t}=X_{j;t_{0}}$ and Proposition 2.22, we get that ${\bf
c}_{k;t}^{B_{t_{0}};t_{0}}$ is equal to the $j$-th column vector of $I_{n}$
and

$\prod\limits_{i=1}^{n}(F_{i;t}^{B_{t_{0}};t_{0}}(X_{1;t_{0}},\ldots,X_{n;t_{0}}))^{b_{ik}^{t}}=1.$

So we get that

(7.1)
$\displaystyle\prod\limits_{b_{ik}^{t}>0}(F_{i;t}^{B_{t_{0}};t_{0}})^{b_{ik}^{t}}=\prod\limits_{b_{ik}^{t}<0}(F_{i;t}^{B_{t_{0}};t_{0}})^{-b_{ik}^{t}}.$

Let $\overline{\mathcal{U}}$ be the upper cluster algebra with principal
coefficients at $t_{0}$ and with initial exchange matrix $B_{t_{0}}$. We use
$(\overline{B}_{t},\overline{X}_{t},\overline{A}_{t})$ to denote the seed of
$\overline{\mathcal{U}}$ at vertex $t$. By $\overline{B}_{t_{0}}=B_{t_{0}}$,
we know that $\overline{B}_{t}=B_{t}$ for any vertex $t$. For any cluster
monomial $L=A_{1;t}^{v_{1}}\cdots A_{n;t}^{v_{n}}$ of $\mathcal{U}_{\rm uc}$,
we denote by

$\overline{L}=\overline{A}_{1;t}^{\;v_{1}}\cdots\overline{A}_{n;t}^{\;v_{n}}$

the corresponding cluster monomial of $\overline{\mathcal{U}}$ and we denote
by $F_{\overline{L}}^{B_{t_{0}};t_{0}}$ the $F$-polynomial of $\overline{L}$.
By the assumption, we know that
$M_{1}=\prod\limits_{b_{ik}^{t}>0}A_{i;t}^{b_{ik}^{t}}$ and
$M_{2}=\prod\limits_{b_{ik}^{t}<0}A_{i;t}^{-b_{ik}^{t}}$. So we have

$\overline{M}_{1}=\prod\limits_{b_{ik}^{t}>0}\overline{A}_{i;t}^{\;b_{ik}^{t}}\;\;\text{and}\;\;\;\overline{M}_{2}=\prod\limits_{b_{ik}^{t}<0}\overline{A}_{i;t}^{\;-b_{ik}^{t}}.$

We claim that $M_{1}$ and $M_{2}$ are cluster monomials in $A_{t_{0}}$. In
order to prove this claim, we first show that $\overline{M}_{1}$ and
$\overline{M}_{2}$ are cluster monomials in $\overline{A}_{t_{0}}$.

By Theorem 5.4 (iii), we can always write the cluster monomial
$\overline{M}_{s}$ ($s=1,2$) as

$\overline{M}_{s}=\overline{M}_{s}^{\;\prime\prime}\cdot\overline{M}_{s}^{\;\prime},$

where $\overline{M}_{s}^{\;\prime\prime}$ is a cluster monomial in initial
cluster variables of $\overline{\mathcal{U}}$ and
$\overline{M}_{s}^{\;\prime}$ is a cluster monomial in non-initial cluster
variables of $\overline{\mathcal{U}}$. By the equality (7.1), we know that
$F_{\overline{M}_{1}}^{B_{t_{0}};t_{0}}=F_{\overline{M}_{2}}^{B_{t_{0}};t_{0}}$.
Because $\overline{M}_{1}^{\;\prime\prime}$ and
$\overline{M}_{2}^{\;\prime\prime}$ are cluster monomials in initial cluster
variables, we have
$F_{\overline{M}_{1}^{\;\prime\prime}}^{B_{t_{0}};t_{0}}=F_{\overline{M}_{2}^{\;\prime\prime}}^{B_{t_{0}};t_{0}}=1.$
So

$F_{\overline{M}_{1}^{\;\prime}}^{B_{t_{0}};t_{0}}=F_{\overline{M}_{1}}^{B_{t_{0}};t_{0}}=F_{\overline{M}_{2}}^{B_{t_{0}};t_{0}}=F_{\overline{M}_{2}^{\;\prime}}^{B_{t_{0}};t_{0}}.$

Then by Theorem 6.1 and the fact that $\overline{M}_{1}^{\;\prime}$ and
$\overline{M}_{2}^{\;\prime}$ are two cluster monomials in non-initial cluster
variables, we get $\overline{M}_{1}^{\;\prime}=\overline{M}_{2}^{\;\prime}$.
Because $\overline{M}_{1}$ and $\overline{M}_{2}$ have no common factor in
$\mathbb{Z}[\overline{A}_{1;t},\ldots,\overline{A}_{n;t}]$, the same holds for
$\overline{M}_{1}^{\;\prime}$ and $\overline{M}_{2}^{\;\prime}$. So
$\overline{M}_{1}^{\;\prime}=\overline{M}_{2}^{\;\prime}$ implies that

$\overline{M}_{1}^{\;\prime}=\overline{M}_{2}^{\;\prime}=1.$

Thus $\overline{M}_{1}=\overline{M}_{1}^{\;\prime\prime}$ and
$\overline{M}_{2}=\overline{M}_{2}^{\;\prime\prime}$ are cluster monomials in
$\overline{A}_{t_{0}}$. Then by Proposition 7.3 (i), we get that $M_{1}$ and
$M_{2}$ are cluster monomials in $A_{t_{0}}$.

By Proposition 2.9, we know that
$\hat{\mathcal{S}}=\\{(B_{t},\hat{X}_{t})\\}_{t\in\mathbb{T}_{n}}$ forms an
$\mathscr{X}$-seed pattern, where
$\hat{X}_{t}=(\hat{X}_{1;t},\ldots,\hat{X}_{n;t})$ is given by
$\hat{X}_{k;t}:=X_{k;t}\prod\limits_{i=1}^{n}A_{i;t}^{b_{ik}^{t}}$. Since
$\mathcal{U}_{\rm uc}$ is an upper cluster algebra with universal coefficient
semifield, $X_{k;t}=X_{j;t_{0}}$ implies that
$\hat{X}_{k;t}=\hat{X}_{j;t_{0}}$. So we have
$X_{k;t}\prod\limits_{i=1}^{n}A_{i;t}^{b_{ik}^{t}}=X_{j;t_{0}}\prod\limits_{i=1}^{n}A_{i;t_{0}}^{b_{ij}^{t_{0}}}$.
Thus we get that

$\frac{M_{1}}{M_{2}}=\prod\limits_{i=1}^{n}A_{i;t}^{b_{ik}^{t}}=\prod\limits_{i=1}^{n}A_{i;t_{0}}^{b_{ij}^{t_{0}}}=\frac{N_{1}}{N_{2}}.$

Since $M_{1},M_{2},N_{1},N_{2}$ are monomials in
$\mathbb{Z}[A_{1;t_{0}},\ldots,A_{n;t_{0}}]$ and $(M_{1},M_{2})$ and
$(N_{1},N_{2})$ are relatively prime in
$\mathbb{Z}[A_{1;t_{0}},\ldots,A_{n;t_{0}}]$, we get

$M_{1}=N_{1}\;\;\text{ and }\;\;\;M_{2}=N_{2}.$

Then by $X_{k;t}=X_{j;t_{0}}$, we have

$\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}.$

By Lemma 7.4, we get that

$A_{k;t}=A_{j;t_{0}}\;\;\text{ or }\;\;\;A_{k;t}=A_{j;t_{1}},$

where $t_{1}=\mu_{j}(t_{0})$.

Now we rule out the possibility of $A_{k;t}=A_{j;t_{1}}$ with the help of the
condition $X_{k;t}=X_{j;t_{0}}$. Assume by contradiction that
$A_{k;t}=A_{j;t_{1}}$. Then the $k$-th column of the $G$-matrix
$G_{t}^{B_{t_{0}};t_{0}}$ equals the $j$-th column of
$G_{t_{1}}^{B_{t_{0}};t_{0}}$. In particular, we have
$g_{jk;t}^{B_{t_{0};t_{0}}}=g_{jj;t_{1}}^{B_{t_{0};t_{0}}}$. By
$t_{1}=\mu_{j}(t_{0})$, we get that

$g_{jk;t}^{B_{t_{0};t_{0}}}=g_{jj;t_{1}}^{B_{t_{0};t_{0}}}=-1.$

By Theorem 2.28, we know that

(7.2) $\displaystyle S^{-1}(G_{t}^{B_{t_{0}};t_{0}})^{\rm T}S\cdot
C_{t}^{B_{t_{0}};t_{0}}=I_{n},$

where $S=\mathrm{diag}(s_{1},\dots,s_{n})$ is a skew-symmetrizer of
$B_{t_{0}}$. By $X_{k;t}=X_{j;t_{0}}$, we know that the $k$-th column of the
$C$-matrix $C_{t}^{B_{t_{0}};t_{0}}$ equals the $j$-th column of $I_{n}$. By
comparing the $k$-th column of the two sides of the equality (7.2), we get
that the $j$-th column of $S^{-1}(G_{t}^{B_{t_{0}};t_{0}})^{\rm T}S$ equals
the $k$-th column of $I_{n}$. In particular, the $(k,j)$-entry
$s_{k}^{-1}\cdot g_{jk;t}^{B_{t_{0};t_{0}}}\cdot s_{j}$ of
$S^{-1}(G_{t}^{B_{t_{0}};t_{0}})^{\rm T}S$ equals $1$, and thus

$g_{jk;t}^{B_{t_{0};t_{0}}}=\frac{s_{k}}{s_{j}}>0.$

This contradicts $g_{jk;t}^{B_{t_{0};t_{0}}}=-1$. So $A_{k;t}\neq A_{j;t_{1}}$
and thus $A_{k;t}=A_{j;t_{0}}$.

Hence, $X_{k;t}=X_{j;t_{0}}$ implies
$(A_{k;t},M_{1},M_{2})=(A_{j;t_{0}},N_{1},N_{2})$.

(ii)$\Longrightarrow$ (iii): It suffices to show that

$\\{A_{i;t}|b_{ik}^{t}\neq 0\\}=\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\}.$

Let $\mathbb{A}$ be the set of cluster variables of $\mathcal{U}_{\rm uc}$ and
$P\in\mathbb{A}$. We first claim that $(P||M_{1}\cdot M_{2})_{d}<0$ if and
only if $P$ is a cluster variable appearing in the cluster monomial
$M_{1}\cdot M_{2}$.

If $P$ is a cluster variable appearing in the cluster monomial $M_{1}\cdot
M_{2}$, then it is easy to see that $(P||M_{1}\cdot M_{2})_{d}<0$. Conversely,
if $(P||M_{1}\cdot M_{2})_{d}<0$, then by Corollary 2.32 (i), we know that $P$
is a cluster variable appearing in the cluster monomial $M_{1}\cdot M_{2}$.

By the discussion above, we get

$\\{A_{i;t}|b_{ik}^{t}\neq 0\\}=\\{P\in\mathbb{A}|(P||M_{1}\cdot
M_{2})_{d}<0\\}.$

Similarly, we have

$\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\}=\\{P\in\mathbb{A}|(P||N_{1}\cdot
N_{2})_{d}<0\\}.$

By $(A_{k;t},M_{1},M_{2})=(A_{j;t_{0}},N_{1},N_{2})$, we have $M_{1}\cdot
M_{2}=N_{1}\cdot N_{2}$. Thus we get

$\\{A_{i;t}|b_{ik}^{t}\neq 0\\}=\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\}.$

(iii)$\Longrightarrow$ (iv): This is clear.

(iv)$\Longrightarrow$ (v): We know that

$\displaystyle
A_{k;t^{\prime}}=A_{k;t}^{-1}(\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2}).$

By $A_{k;t}=A_{j;t_{0}}$ and the fact that $M_{1},M_{2}$ are monomials in
variables from

$\\{A_{i;t}|b_{ik}^{t}\neq 0\\}\subseteq A_{t_{0}},$

we know that

$A_{k;t^{\prime}}=A_{k;t}^{-1}(\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2})=A_{j;t_{0}}^{-1}(\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2})$

can be viewed as the Laurent expansion of $A_{k;t^{\prime}}$ with respect to
$A_{t_{0}}$. Thus we know that

$(A_{j;t_{0}}||A_{k;t^{\prime}})_{d}=1>0\;\text{ and
}(A_{i;t_{0}}||A_{k;t^{\prime}})_{d}=0,$

for any $i\neq j$. Then by Proposition 7.2, we get
$A_{k;t^{\prime}}=A_{j;t_{1}}$. So

$(A_{k;t},A_{k;t^{\prime}})=(A_{j;t_{0}},A_{j;t_{1}}).$

(v)$\Longrightarrow$ (vi): This is clear.

(vi)$\Longrightarrow$ (i): If the $j$-th column of $B_{t_{0}}$ is zero, then
the same holds for any exchange matrices of $\mathcal{U}_{\rm uc}$. In this
case, the action of $\mu_{j}$ on any seed is very clear. We can see that
$A_{k;t}=A_{j;t_{0}}$ implies that $k=j$ and $X_{k;t}=X_{j;t_{0}}$.

If the $j$-th column of $B_{t_{0}}$ is non-zero, then $N_{1}$ and $N_{2}$ are
two different cluster monomials in $A_{t_{0}}$. Then by Proposition 7.1 and
the following equality,

$\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}N_{1}+\frac{1}{1+X_{j;t_{0}}}N_{2}=\frac{X_{k;t}}{1+X_{k;t}}M_{1}+\frac{1}{1+X_{k;t}}M_{2},$

we get $\frac{X_{k;t}}{1+X_{k;t}}=\frac{X_{j;t_{0}}}{1+X_{j;t_{0}}}$ or
$\frac{X_{k;t}}{1+X_{k;t}}=\frac{1}{1+X_{j;t_{0}}}$. Namely, we get that

$X_{k;t}=X_{j;t_{0}}\;\;\text{ or }\;\;\;X_{k;t}=X_{j;t_{0}}^{-1}.$

Similar to the argument for the statement (i)$\Longrightarrow$ (ii), now we
rule out the possibility of $X_{k;t}=X_{j;t_{0}}^{-1}$ with the help of the
condition $A_{k;t}=A_{j;t_{0}}$. Assume by contradiction that
$X_{k;t}=X_{j;t_{0}}^{-1}$. Then the $k$-th column of
$C_{t}^{B_{t_{0}};t_{0}}$ is the $j$-th column of $-I_{n}$, by Theorem 2.26
(ii). So we have

$c_{jk;t}^{B_{t_{0}};t_{0}}=-1<0.$

By Theorem 2.28, we know that

(7.3) $\displaystyle(G_{t}^{B_{t_{0}};t_{0}})^{\rm
T}(SC_{t}^{B_{t_{0}};t_{0}}S^{-1})=I_{n},$

where $S=\mathrm{diag}(s_{1},\dots,s_{n})$ is a skew-symmetrizer of
$B_{t_{0}}$.

By $A_{k;t}=A_{j;t_{0}}$, we know that the $k$-th column of
$G_{t}^{B_{t_{0}};t_{0}}$ equals the $j$-th column vector of $I_{n}$. So the
$k$-th row of $(G_{t}^{B_{t_{0}};t_{0}})^{\rm T}$ equals the $j$-th row of
$I_{n}$. By comparing the $k$-th row of the two sides of the equality (7.3),
we know that the $j$-th row of $SC_{t}^{B_{t_{0}};t_{0}}S^{-1}$ equals the
$k$-th row of $I_{n}$. In particular, the $(j,k)$-entry
$s_{j}c_{jk;t}^{B_{t_{0}};t_{0}}s_{k}^{-1}$ of
$SC_{t}^{B_{t_{0}};t_{0}}S^{-1}$ equals $1$ and thus

$c_{jk;t}^{B_{t_{0}};t_{0}}=\frac{s_{k}}{s_{j}}>0.$

This contradicts $c_{jk;t}^{B_{t_{0}};t_{0}}=-1<0$. So $X_{k;t}\neq
X_{j;t_{0}}^{-1}$ and thus $X_{k;t}=X_{j;t_{0}}$. ∎

Let $\mathcal{U}$ be an upper cluster algebra and $\mathcal{X}$ a cluster
Poisson algebra. We say that $\mathcal{U}$ and $\mathcal{X}$ have the same
type if they have the same exchange matrix at a vertex $t_{0}$ of
$\mathbb{T}_{n}$.

######  Theorem 7.6.

Let $\mathcal{X}$ be a cluster Poisson algebra and $\mathcal{U}$ an upper
cluster algebra of the same type as $\mathcal{X}$. Denote $\overline{X}_{t}$
the Poisson cluster of $\mathcal{X}$ at $t$ to distinguish the
$\mathscr{X}$-cluster $X_{t}$ of $\mathcal{U}$ at $t$. Each mutation
$t_{k}=\mu_{k}(t)$ gives an $\mathscr{X}$-exchange pair
$(\overline{X}_{k;t},\overline{X}_{k;t_{k}})$ of $\mathcal{X}$ and an
$\mathscr{A}$-exchange pair $(A_{k;t},A_{k;t_{k}})$ of $\mathcal{U}$. Then

  * (i)

The map $\psi:\overline{X}_{k;t}\mapsto(A_{k;t},A_{k;t_{k}})$ gives a
bijection from the set of cluster Poisson variables of $\mathcal{X}$ to the
set of $\mathscr{A}$-exchange pairs of $\mathcal{U}$;

  * (ii)

The map
$\psi_{1}:(\overline{X}_{k;t},\overline{X}_{k;t_{k}})\mapsto(A_{k;t},A_{k;t_{k}})$
gives a bijection from the set of $\mathscr{X}$-exchange pairs of
$\mathcal{X}$ to the set of $\mathscr{A}$-exchange pairs of $\mathcal{U}$.

###### Proof.

(i) Thanks to Proposition 7.3, we can just assume that
$\mathcal{U}=\mathcal{U}_{\rm uc}$ is an upper cluster algebra with universal
coefficient semifield. In this case, the cluster Poisson variables of
$\mathcal{X}$ can be viewed as the $\mathscr{X}$-variables of
$\mathcal{U}=\mathcal{U}_{\rm uc}$. Then (i) follows from Theorem 7.5 (i)(v).

(ii) follows from (i) and the fact that in any $\mathscr{X}$-exchange pair
$(\overline{X}_{k;t},\overline{X}_{k;t_{k}})$, one has
$\overline{X}_{k;t_{k}}=\overline{X}_{k;t}^{-1}$. ∎

######  Remark 7.7.

For (upper) cluster algebras of finite type, a different proof of Theorem 7.6
can be found in [SB19].

###  7.2. Exchange graphs of cluster Poisson algebras

Let $\mathcal{X}$ be a cluster Poisson algebra. In this subsection, we prove
that the seeds of $\mathcal{X}$ whose Poisson clusters contain particular
cluster Poisson variables form a connected subgraph of the exchange graph of
$\mathcal{X}$.

######  Lemma 7.8.

Let $\mathcal{U}_{\rm uc}$ be an upper cluster algebra with universal
coefficient semifield, and let $t_{0}$ and $t_{1}$ be two seeds of
$\mathcal{U}_{\rm uc}$ with $t_{1}=\mu_{k}(t_{0})$. For a given $1\leq j\leq
n$, denote $I(t_{0})=\\{j\\}\sqcup\\{i|b_{ij}^{t_{0}}\neq 0\\}$ and
$I(t_{1})=\\{j\\}\sqcup\\{i|b_{ij}^{t_{1}}\neq 0\\}$. If $k\notin I(t_{0})$,
then we have $X_{j;t_{0}}=X_{j;t_{1}}$ and $I(t_{0})=I(t_{1})$.

###### Proof.

The results follow from $k\notin I(t_{0})$ and the definition of mutation. ∎

######  Lemma 7.9.

Let $\mathcal{U}_{\rm uc}$ be an upper cluster algebra with universal
coefficient semifield and $t,t_{0}$ two seeds of $\mathcal{U}_{\rm uc}$. Then
$X_{j;t_{0}}\in X_{t}$ if and only if
$\\{A_{j;t_{0}}\\}\sqcup\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\}\subseteq
A_{t}$.

###### Proof.

$\Longrightarrow$: This follows from Theorem 7.5 (i)(iii).

$\Longleftarrow$: By $\\{A_{j;t_{0}}\\}\sqcup\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq
0\\}\subseteq A_{t}$, we know that

$U:=\\{A_{j;t_{0}}\\}\sqcup\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\}$

is a common subset of $A_{t}$ and $A_{t_{0}}$. Then by Theorem 2.17, there
exists a sequence of mutations $(\mu_{k_{1}},\ldots,\mu_{k_{s}})$ such that
the $\mathscr{A}$-seed at $t_{s}=\mu_{k_{s}}\cdots\mu_{k_{1}}(t_{0})$ is
equivalent to the $\mathscr{A}$-seed at $t$ and the cluster variables in $U$
remain unchanged when we do each mutation along the sequence of mutations
$(\mu_{k_{1}},\ldots,\mu_{k_{s}})$. Set
$I=\\{j\\}\sqcup\\{i|b_{ij}^{t_{0}}\neq 0\\}$. We know that $k_{\ell}\notin I$
for any $\ell=1,\ldots,s$. Applying Lemma 7.8 iteratively, we get
$X_{j;t_{0}}\in X_{t_{s}}$. Because the two seeds $t$ and $t_{s}$ are
equivalent, we get $X_{j;t_{0}}\in X_{t}$. ∎

######  Theorem 7.10.

Let $\mathcal{X}$ be a cluster Poisson algebra. Then the seeds of
$\mathcal{X}$ whose Poisson clusters contain particular cluster Poisson
variables form a connected subgraph of the exchange graph of $\mathcal{X}$.

###### Proof.

Let $\mathcal{U}=\mathcal{U}_{\rm uc}$ be the upper cluster algebra with
universal coefficient semifield of the same type as $\mathcal{X}$. In this
case, the cluster Poisson variables (respectively, Poisson clusters) of
$\mathcal{X}$ can be viewed as the $\mathscr{X}$-variables (respectively,
$\mathscr{X}$-clusters) of $\mathcal{U}=\mathcal{U}_{\rm uc}$. Now we work on
the upper cluster algebra $\mathcal{U}$.

Fix a subset $V$ of $X_{t_{0}}$. Let $t$ be a seed of $\mathcal{U}$, whose
$\mathscr{X}$-cluster $X_{t}$ satisfies $V\subseteq X_{t}$. It suffices for us
to show that there exists a sequence of mutations
$(\mu_{k_{1}},\ldots,\mu_{k_{s}})$ such that the $\mathscr{X}$-seed at
$t_{s}=\mu_{k_{s}}\cdots\mu_{k_{1}}(t_{0})$ is equivalent to the
$\mathscr{X}$-seed at $t$ and the $\mathscr{X}$-variables in $V$ remain
unchanged when we do each mutation along the sequence of mutations
$(\mu_{k_{1}},\ldots,\mu_{k_{s}})$.

Let $U:=\bigcup\limits_{X_{j;t_{0}}\in
V}(\\{A_{j;t_{0}}\\}\sqcup\\{A_{i;t_{0}}|b_{ij}^{t_{0}}\neq 0\\})$. By Lemma
7.9, we know that for a seed $t^{\prime}$ of $\mathcal{U}$, its
$\mathscr{X}$-cluster $X_{t^{\prime}}$ satisfies $V\subseteq X_{t^{\prime}}$
if and only if its cluster $A_{t^{\prime}}$ satisfies $U\subseteq
A_{t^{\prime}}$.

By $X_{t_{0}}\supseteq V$ and $X_{t}\supseteq V$, we get that
$A_{t_{0}}\supseteq U$ and $A_{t}\supseteq U$. Then by Theorem 2.17, there
exists a sequence of mutations $(\mu_{k_{1}},\ldots,\mu_{k_{s}})$ such that
the $\mathscr{A}$-seed at $t_{s}=\mu_{k_{s}}\cdots\mu_{k_{1}}(t_{0})$ is
equivalent to the $\mathscr{A}$-seed at $t$ and the cluster variables in $U$
remain unchanged when we do each mutation along the sequence of mutations
$(\mu_{k_{1}},\ldots,\mu_{k_{s}})$. So the $\mathscr{X}$-seed at
$t_{s}=\mu_{k_{s}}\cdots\mu_{k_{1}}(t_{0})$ is equivalent to the
$\mathscr{X}$-seed at $t$ and the $\mathscr{X}$-variables in $V$ remain
unchanged when we do each mutation along the sequence of mutations
$(\mu_{k_{1}},\ldots,\mu_{k_{s}})$. This completes the proof. ∎

## References

  * [BFZ05] A. Berenstein, S. Fomin, and A. Zelevinsky.  Cluster algebras III: upper bounds and double Bruhat cells.  Duke Math. J., 126(1):1–52, 2005.
  * [BM06] Aslak Bakke Buan and Robert Marsh.  Cluster-tilting theory.  In Trends in representation theory of algebras and related topics, Contemp. Math., pages 1–30. Amer. Math. Soc., Providence, RI, 2006.
  * [Bri17] T. Bridgeland.  Scattering diagrams, Hall algebras and stability conditions.  Algebr. Geom., 4(5):523–561, 2017.
  * [CC06] P. Caldero and F. Chapoton.  Cluster algebras as Hall algebras of quiver representations.  Comment. Math. Helv., 81(3):595–616, 2006.