#  $F$-invariant in cluster algebras

Peigen Cao  Department of Mathematics, The University of Hong Kong, Pokfulam
Road, Hong Kong<EMAIL_ADDRESS>Dedicated to Professor Fang Li on the
occasion of his 60th birthday

###### Abstract.

In this paper we introduce $F$-invariant in cluster algebras using
tropicalization, which is an analog of the $E$-invariant introduced by
Derksen-Weyman-Zelevinsky in the additive categorification of cluster algebras
and the $\mathfrak{d}$-invariant introduced by Kang-Kashiwara-Kim-Oh in the
monoidal categorification of (quantum) cluster algebras.

###### 2010 Mathematics Subject Classification:

13F60

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on cluster algebras
    1. 2.1 Mutation matrices and their mutations
    2. 2.2 Compatible pairs and their mutations
    3. 2.3 $Y$-pattern and cluster pattern
  3. 3 Tropical points, pointed elements and good elements
    1. 3.1 Universal semi-field and tropical points
    2. 3.2 Compatibly pointed elements and good elements
    3. 3.3 Basic properties of pointed elements
  4. 4 $F$-invariant in cluster algebras
    1. 4.1 Maps between sets of tropical points
    2. 4.2 $F$-invariant and some consequences
  5. 5 Log-canonicality and $F$-compatibility

##  1\. Introduction

Cluster algebras. Cluster algebras were invented by Fomin and Zelevinsky
[fz_2002] as a combinatorial approach to the upper global bases of quantum
groups and to the theory of total positivity in algebraic groups. Such
algebras often arise as the coordinate rings of spaces, such as double Bruhat
cells [bfz_2005], unipotent cells [gls_2011], double Bott-Samelson cells
[Shen-Weng-2021] and arise as the Grothendieck rings of certain monoidal
subcategories of the representations of quantum affine algebras [HL_2010,
HL_2013].

A cluster algebra is a $\mathbb{Z}$-subalgebra of a rational function field
generated by a special set of generators called cluster variables, which are
grouped into overlapping subsets, called clusters. A seed is a pair of a
cluster and a rectangular integer matrix with skew-symmetrizable principal
part. One can obtain new seeds from a given one by a procedure, called
mutations. The sets of cluster variables and clusters of a cluster algebra are
determined by an initial seed and the iterative mutations. A cluster momomial
is a monomial in variables from a single cluster. In the main text, we
actually use a broader definition of cluster monomials, which allows the
presence of the inverse of frozen variables.

$E$-invariant. The $E$-invariant is introduced by Derksen-Weyman-Zelevinsky
[DWZ10] in the additive categorification of cluster algebras using decorated
representations of quivers with potentials. We refer to [DWZ08, Definition
10.1] for the definition of decorated representations. For any two decorated
representations $\mathcal{M}$ and $\mathcal{N}$ of a quiver with potential
$(Q,W)$, Derksen-Weyman-Zelevinsky [DWZ10, Section 7] defined an integer
$E^{\rm inj}(\mathcal{M},\mathcal{N})$, which turns out to take non-negative
values (see [DWZ10, Section 10]). The $E$-invariant $E^{\rm
sym}(\mathcal{M},\mathcal{N})$ associated to $\mathcal{M}$ and $\mathcal{N}$
is defined to be the symmetrized sum

$E^{\rm sym}(\mathcal{M},\mathcal{N})=E^{\rm
inj}(\mathcal{M},\mathcal{N})+E^{\rm inj}(\mathcal{N},\mathcal{M})\geq 0.$

Derksen-Weyman-Zelevinsky [DWZ10, Theorem 7.1] proved that the integer $E^{\rm
sym}(\mathcal{M},\mathcal{N})$ is mutation-invariant in the sense that

$E^{\rm sym}(\mathcal{M},\mathcal{N})=E^{\rm
sym}(\mu_{k}(\mathcal{M}),\mu_{k}(\mathcal{N})),$

where $\mu_{k}(\mathcal{M})$ and $\mu_{k}(\mathcal{N})$ are decorated
representations of the new quiver with potential $\mu_{k}(Q,W)$ obtained by
mutation in direction $k$. As a corollary, they obtained $E^{\rm
sym}(\mathcal{M},\mathcal{M})=0$ for any decorated representation
$\mathcal{M}$ corresponding to a cluster monomial. This result plays a crucial
role in their proofs of [DWZ10, Conjectures 1.3–1.6].

Let $(Q,W)$ be a quiver with potential. There is a dg algebra
$\Gamma=\Gamma_{(Q,W)}$ and a triangulated category $\mathcal{C}_{(Q,W)}$
associated to $(Q,W)$, which are called (complete) Ginzburg dg algebra and
cluster category, cf., [Amiot_2009] and [Keller-Yang_2011]. We remark that
$\Gamma$ can be viewed as an object in $\mathcal{C}_{(Q,W)}$ and it satisfies
$\operatorname{\mathsf{Hom}}_{\mathcal{C}_{(Q,W)}}(\Gamma,\Gamma[1])=0$.
Plamondon [Plamondon_2011b, Section 4.1] proved that there exists a bijection
$\Psi_{(Q,W)}$ from the isomorphism classes of decorated representations of
$(Q,W)$ to the isomorphism classes of the objects in a full subcategory of
$\mathcal{C}_{(Q,W)}$. For two decorated representations $\mathcal{M}$ and
$\mathcal{N}$ of $(Q,W)$, we denote by $M=\Psi_{(Q,W)}(\mathcal{M})$ and
$N=\Psi_{(Q,W)}(\mathcal{N})$ the corresponding objects in the cluster
category $\mathcal{C}_{(Q,W)}$. In [Plamondon_2011b, Proposition 4.15],
Plamondon proved

$E^{\rm inj}(\mathcal{M},\mathcal{N})=\dim(\Gamma[1])(M,N[1]),$

where $(\Gamma[1])(M,N[1])$ denotes the subspace of
$\operatorname{\mathsf{Hom}}_{\mathcal{C}_{(Q,W)}}(M,N[1])$ consisting of
morphisms that factor through an object of
$\operatorname{\mathsf{add}}\nolimits(\Gamma[1])$. Thus we have

$E^{\rm
sym}(\mathcal{M},\mathcal{N})=\dim(\Gamma[1])(M,N[1])+\dim(\Gamma[1])(N,M[1]).$

For the case of $\mathcal{M}=\mathcal{N}$, Plamondon also proved that the
above equality becomes

$E^{\rm
sym}(\mathcal{M},\mathcal{M})=2\dim\operatorname{\mathsf{Hom}}_{\mathcal{C}_{(Q,W)}}(M,M[1]).$

One has that $E^{\rm sym}(\mathcal{M},\mathcal{M})=0$ if and only if the
corresponding object $M$ in the cluster category $\mathcal{C}_{(Q,W)}$ is
rigid, that is, $\operatorname{\mathsf{Hom}}_{\mathcal{C}_{(Q,W)}}(M,M[1])=0$.
Notice that if $\mathcal{M}$ corresponds to a cluster monomial, then $M$ is
rigid in $\mathcal{C}_{(Q,W)}$. Let us mention another very special case. If
$N$ belongs to $\operatorname{\mathsf{add}}\nolimits(\Gamma)$, that is,
$\mathcal{N}$ corresponds to a cluster monomial in initial cluster variables,
then it is easy to see

(1.1) $\displaystyle E^{\rm
sym}(\mathcal{M},\mathcal{N})=\dim\operatorname{\mathsf{Hom}}_{\mathcal{C}_{(Q,W)}}(M,N[1]).$

Recently, tropical $F$-polynomials were studied by Fei [fei_2019a, fei_2019b]
from the viewpoint of representation theory. It turns out that $E^{\rm
inj}(\mathcal{M},\mathcal{N})$ is closely related with the tropical
$F$-polynomials, c.f., [fei_2019b, Theorem 2.6].

$\mathfrak{d}$-invariant. The $\mathfrak{d}$-invariant is introduced by Kang-
Kashiwara-Kim-Oh [kkko-2018] in the monoidal categorification of (quantum)
cluster algebras using representations of quiver Hecke algebras. Here we will
mainly follow [kkko-2018, kk_2019]. Let

$R\text{-gmod}=\bigoplus_{\beta\in Q^{+}}R(\beta)\text{-gmod}$

be the category of finite-dimensional graded modules with homomorphisms of
degree $0$ given in [kk_2019, Page 2270], which is a monoidal category with
the tensor product $\Huge{\circ}$ given there. Denote by $q$ the grading shift
functor in $R\text{-gmod}$: for $M\in R\text{-gmod}$, we have
$(qM)_{i}=M_{i-1}$ for $i\in\mathbb{Z}$ by definition. For each pair $(M,N)$
with $0\neq M\in R(\beta)\text{-gmod}$ and $0\neq N\in R(\gamma)\text{-gmod}$,
one has the (renormalized) $R$-matrix ${\bf r}_{M,N}$ (see [kkko-2018,
Definition 2.2.1]), which by construction is a morphism from $M\Huge{\circ}N$
to $q^{c}N\Huge{\circ}M$ in $R\text{-gmod}$ for some integer $c$. We denote by

$\Lambda(M,N)=-c.$

The $\mathfrak{d}$-invariant $\mathfrak{d}(M,N)$ associated to $(M,N)$ is
defined to be

$\mathfrak{d}(M,N)=\frac{1}{2}(\Lambda(M,N)+\Lambda(N,M)).$

By [kkko-2018, Lemma 3.2.1], $\mathfrak{d}(M,N)$ is a non-negative integer. A
simple object $S$ in $R\text{-gmod}$ is said to be real, if the tensor product
$S{\huge\circ}S$ is still simple. Kang-Kashiwara-Kim-Oh proved that if $M$ is
a real simple object in $R\text{-gmod}$, then $\mathfrak{d}(M,M)=0$, cf.
[kkko-2018, Lemma 3.2.3]. Notice that in the monoidal categorification of
cluster algebras, each cluster monomial corresponds to a real simple object.
In particular, the $\mathfrak{d}$-invariant associated to a “cluster monomial”
is zero. This is an analog of the corresponding result in the additive
categorification case.

Recently, Kashiwara-Kim-Oh-Park [kkop-2020] introduced the similar invariants
$\Lambda(M,N)$ and $\mathfrak{d}(M,N)$ into the monoidal categorification of
cluster algebras using representations of quantum affine algebras. In
particular, in this setting, they proved that (i) $\mathfrak{d}(M,N)$ takes
non-negative values; (ii) if $M$ is a real simple object, then
$\mathfrak{d}(M,M)=0$, cf. [kkop-2020, Proposition 3.16, Corollary 3.17].

Results. In this paper, we introduce $F$-invariant in cluster algebras using
tropicalization, which is an analog of the $E$-invariant introduced by
Derksen-Weyman-Zelevinsky in the additive categorification of cluster algebras
and the $\mathfrak{d}$-invariant introduced by Kang-Kashiwara-Kim-Oh in the
monoidal categorification of (quantum) cluster algebras.

Decorated representations of $(Q,W)$ and finite-dimensional graded modules in
$R\text{-gmod}$ are replaced by “good elements” in upper cluster algebras. A
good element is a compatibly pointed element in the sense of [qin_2019]
satisfying the universally positive property, see Definition 3.5 for details.

Given two good elements $u$ and $u^{\prime}$. For any choice of a seed $({\bf
x}_{t},\widetilde{B}_{t})$, we define an integer $\langle
u,u^{\prime}\rangle_{t}$ in Definition 4.5 (i). We prove in Theorem 4.8 that
the integer $\langle u,u^{\prime}\rangle_{t}$ only depends on $u$ and
$u^{\prime}$, not depend on the choice of $({\bf x}_{t},\widetilde{B}_{t})$.
The key reason is that the integer $\langle u,u^{\prime}\rangle_{t}$ can be
interpreted as the evaluation of the positive function $u$ (on cluster
$\mathscr{A}$-variety in the sense of [FG-2009]) at a tropical point
(associated to the same cluster $\mathscr{A}$-variety) determined by
$u^{\prime}$ (see the proof of Theorem 4.8). Different choices of seeds
provide different charts to calculate this evaluation.

Hence, we get an invariant $\langle u,u^{\prime}\rangle:=\langle
u,u^{\prime}\rangle_{t}$ associated to any pair of good elements. We remark
that the invariant $\langle u,u^{\prime}\rangle=\langle
u,u^{\prime}\rangle_{t}$ is an analog of the invariant $\Lambda(M,N)$ in the
monoidal categorification of cluster algebras, because both $\langle
u,u^{\prime}\rangle$ and $\Lambda(M,N)$ are closely related to the
quantization matrices appearing in compatible pairs that are used to define
quantum cluster algebras in the sense of [bz-2005], cf. Remark 4.10.

The $F$-invariant $(u\mid\mid u^{\prime})_{F}$ associated to two good elements
$u$ and $u^{\prime}$ is defined to be

(1.2) $\displaystyle(u\mid\mid u^{\prime})_{F}:=\langle
u,u^{\prime}\rangle+\langle u^{\prime},u\rangle=\langle
u,u^{\prime}\rangle_{t}+\langle u^{\prime},u\rangle_{t}.$

The $F$-invariant above is an analog of the $E$-invariant $E^{\rm
sym}(\mathcal{M},\mathcal{N})$ in the additive categorification and
$\mathfrak{d}$-invariant $\mathfrak{d}(M,N)$ in the monoidal categorification.
Strictly speaking, the $F$-invariant $(u\mid\mid u^{\prime})_{F}$ is analogous
to $2\mathfrak{d}(M,N)$, because in (1.2) we did not divide out $``2"$ from
the symmetrized sum $\langle u,u^{\prime}\rangle_{t}+\langle
u^{\prime},u\rangle_{t}$.

By our construction, the $F$-invariant $(u\mid\mid u^{\prime})_{F}$ takes non-
negative values for a very clear reason, see Remark 4.6. One can easily show
that $(u\mid\mid u)_{F}=0$ for any cluster monomial $u$, because we can freely
choose a chart (i.e., a seed) to calculate the value $(u\mid\mid u)_{F}$. We
remark that good elements $u$ satisfying $(u\mid\mid u)_{F}=0$ are analogous
to the decorated representations $\mathcal{M}$ satisfying $E^{\rm
sym}(\mathcal{M},\mathcal{M})=0$ and to the finite-dimension graded modules
$M\in R\text{-gmod}$ satisfying $\mathfrak{d}(M,M)=0$.

As mentioned in [kkop-2020, Pages 1040–1041], the integer-valued invariants
$\Lambda(M,N)$ and $\mathfrak{d}(M,N)$ provide important information in the
representation theory of quiver Hecke algebra. However, in general, computing
those values are quite difficult. We hope our invariant $\langle
u,u^{\prime}\rangle$ and $F$-invariant $(u\mid\mid u^{\prime})_{F}$ can shed
some light on the computation of the integer-valued invariants $\Lambda(M,N)$
and $\mathfrak{d}(M,N)$ in the monoidal categorification of cluster algebras.

We should mention that $F$-invariant is closely related with the
$f$-compatibility degree introduced by Fu-Gyoda in [Fu-Gyoda] (see Remark
4.13) and the tropical $F$-polynomials studied by Fei in [fei_2019a,
fei_2019b] (see Remark 4.7).

One of the interesting consequences of $F$-invariant is that it provides a
very concrete and easily computable numerical invariant to distinguish whether
the product of two cluster monomials is still a cluster monomial (see
Proposition 4.17).

Since the $F$-invariant $(u\mid\mid u^{\prime})_{F}$ always takes non-negative
values, it is natural to ask when $(u\mid\mid u^{\prime})_{F}=0$ holds. Our
next main result is to give a necessary condition for $(u\mid\mid
u^{\prime})_{F}=0$. Before stating the result, we need some preparations.

In this paper, we use $m$ to denote the number of cluster variables (including
frozen ones and unfrozen ones) in a cluster and we use $n$ to denote the
number of unfrozen cluster variables in a cluster. By Definition 3.5, each
good element $u_{[{\bf g}]}$ correspond to a tropical point $[{\bf g}]=\\{{\bf
g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ (associated to cluster
$\mathscr{X}$-variety in the sense of [FG-2009]). (Actually, it also
determines a tropical point $[{\bf a}]=\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ associated to cluster $\mathscr{A}$-variety, thanks to
Proposition 4.4.) Two tropical points $[{\bf g}]=\\{{\bf
g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ and $[{\bf
g}^{\prime}]=\\{{\bf g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ are said to be compatible if for any vertex
$t\in\mathbb{T}_{n}$ and any $k\in[1,n]$, we have $g_{k;t}\cdot
g_{k;t}^{\prime}\geq 0$, where $g_{k;t}$ and $g_{k;t^{\prime}}$ are the $k$th
components of ${\bf g}_{t}$ and ${\bf g}_{t}^{\prime}$. This is a sign-
coherent condition on the level of tropical points. Two good elements
$u_{[{\bf g}]}$ and $u_{[{\bf g}^{\prime}]}$ are said to be $F$-compatible, if
$(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}=0$.

We prove in Theorem 4.25 that if two good elements $u_{[{\bf g}]}$ and
$u_{[{\bf g}^{\prime}]}$ are $F$-compatible, then the tropical points $[{\bf
g}]$ and $[{\bf g}^{\prime}]$ are compatible and the product $u_{[{\bf
g}]}u_{[{\bf g}^{\prime}]}$ is still a good element in the sense of Definition
3.5. This gives a necessary condition for $(u_{[{\bf g}]}\mid\mid u_{[{\bf
g}^{\prime}]})_{F}=0$.

In this paper, we also have a discussion on the link between the compatible
Poisson structure of (upper) cluster algebra and the $F$-compatibility. We
prove in Theorem 5.2 that if a good element $u$ and an unfrozen cluster
variable $x$ are log-canonical with respect to the given compatible Poisson
bracket, then they are $F$-compatible. We also prove that if a good element
$u$ is log-canonical with any unfrozen cluster variable $x_{i;w}$ with
$i\in\\{1,\ldots,n\\}\backslash\\{k\\}$ for some seed $({\bf
x}_{w},\widetilde{B}_{w})$, then $u$ is either a cluster monomial in seed
$({\bf x}_{w},\widetilde{B}_{w})$ or a cluster monomial in seed $\mu_{k}({\bf
x}_{w},\widetilde{B}_{w})$. This result is analogous to [kk_2019, Theorem
4.10] by Kashiwara-Kim.

Organization of the paper. In Section 2, we recall some well-known notations
in cluster algebras, including, $Y$-pattern, cluster pattern, compatible
pairs, (upper) cluster algebras. In Section 3, we recall the definition of
tropical points for $Y$-pattern (equivalently, cluster $\mathscr{X}$-variety)
and for cluster pattern (equivalently, cluster $\mathscr{A}$-variety), and we
also recall the (compatibly) pointed elements introduced by Qin [Qin_2017,
qin_2019]. We introduce the notation of good elements in upper cluster
algebras. The $F$-invariant will be defined for a pair of good elements.

In Section 4.1, we give two maps between the set of tropical points. Let
$(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair in the sense of
Definition 2.6. The first map is from the set of tropical points associated to
the cluster pattern $\mathcal{S}_{X}$ to the set of tropical points associated
to the $Y$-pattern $\mathcal{S}_{Y}$ and the second map is from the set of
tropical points associated to the $Y$-pattern $\mathcal{S}_{Y}$ to the set of
tropical points associated to the cluster pattern $\mathcal{S}_{X}$. The
second map will play an important role in the proof Theorem 4.8 which says
that the pairing $\langle u,u^{\prime}\rangle_{t}$ is independent of the
choice of a seed $({\bf x}_{t},\widetilde{B}_{t})$. In Section 4.2, we give
the definition of $F$-invariant and the analogous results for $E$-invariant
and $\mathfrak{d}$-invariant in the categorification of cluster algebras. The
main results in this subsection are Theorem 4.8 and Theorem 4.25. In Section
5, we discuss the link between the compatible Poisson structure of (upper)
cluster algebra and the $F$-compatibility. The main result in this section is
Theorem 5.2.

##  2\. Preliminaries on cluster algebras

Throughout this paper, we fix a pair $(n,m)$ of integers with $m\geq n>0$. For
any positive integer $r$, we denote $[1,r]:=\\{1,\ldots,r\\}$. The matrices in
this paper are always integer matrices.

###  2.1. Mutation matrices and their mutations

Recall that an $n\times n$ integer matrix $B$ is said to be skew-
symmetrizable, if there exists a diagonal integer matrix
$S=diag(s_{1},\ldots,s_{n})$ with $s_{i}>0$ ($i\in[1,n]$) such that $SB$ is
skew-symmetric. Such a diagonal matrix $S$ is called a skew-symmetrizer of
$B$.

An $m\times n$ integer matrix $\widetilde{B}=\begin{bmatrix}B\\\
P\end{bmatrix}=(b_{ij})$ is called a mutation matrix, if its top $n\times n$
submatrix $B$ is skew-symmetrizable. The submatrix $B$ is called the principal
part of $\widetilde{B}$.

Let $\widetilde{B}=\begin{bmatrix}B\\\ P\end{bmatrix}=(b_{ij})$ be an $m\times
n$ mutation matrix. The mutation of $\widetilde{B}$ in direction $k\in[1,n]$
is defined to be the new integer matrix
$\mu_{k}(\widetilde{B})=\widetilde{B}^{\prime}=\begin{bmatrix}B^{\prime}\\\
P^{\prime}\end{bmatrix}=(b_{ij}^{\prime})$ given by

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

where $[a]_{+}:=\max\\{a,0\\}$ for any $a\in\mathbb{R}$.

The following statements are well-known, c.f., [fz_2002, Proposition 4.5],
[bfz_2005, Lemma 3.2].

  * •

$\mu_{k}(\widetilde{B})=\widetilde{B}^{\prime}$ is still a mutation matrix,
namely, the submatrix $B^{\prime}$ of $\widetilde{B}^{\prime}$ is skew-
symmetrizable;

  * •

$B$ and $B^{\prime}$ share the same skew-symmetrizer;

  * •

$\mu_{k}$ is an involution, i.e., $\mu_{k}^{2}=id$;

  * •

The rank of $\widetilde{B}^{\prime}=\mu_{k}(\widetilde{B})$ is equal to the
rank of $\widetilde{B}$.

As shown in [bfz_2005, (3.2)], the matrix
$\widetilde{B}^{\prime}=\mu_{k}(\widetilde{B})$ can be written as

(2.2)
$\displaystyle\widetilde{B}^{\prime}=\mu_{k}(\widetilde{B})=E_{k,\varepsilon}^{\widetilde{B}}\widetilde{B}F_{k,\varepsilon}^{\widetilde{B}},$

where $\varepsilon\in\\{\pm\\}$ and

  * •

$E_{k,\varepsilon}^{\widetilde{B}}$ is the $m\times m$ matrix obtained by
replacing the $k$th column of $I_{m}$ by the column vector
$(e_{1},\ldots,e_{m})^{T}$, where

$e_{i}=\begin{cases}-1,&\text{if }i=k;\\\ [-\varepsilon b_{ik}]_{+},&\text{if
}i\neq k.\end{cases}$

  * •

$F_{k,\varepsilon}^{\widetilde{B}}$ is the $n\times n$ matrix obtained by
replacing the $k$th row of $I_{n}$ by the row vector $(f_{1},\ldots,f_{n})$,
where

$f_{i}=\begin{cases}-1,&\text{if }i=k;\\\ [\varepsilon b_{ki}]_{+},&\text{if
}i\neq k.\end{cases}$

It turns out that $\widetilde{B}^{\prime}$ is independent of the choice of a
sign $\varepsilon\in\\{\pm\\}$ in (2.2). The matrices
$E_{k,\varepsilon}^{\widetilde{B}}$ and $F_{k,\varepsilon}^{\widetilde{B}}$
defined above coincide with the ones used in [bz-2005, (3.2), (3.3)]. It is
easy to check

(2.3)
$\displaystyle(E_{k,\varepsilon}^{\widetilde{B}})^{2}=I_{m}\;\;\;\text{and}\;\;\;(F_{k,\varepsilon}^{\widetilde{B}})^{2}=I_{n}.$

###  2.2. Compatible pairs and their mutations

Let $\Lambda=(\lambda_{ij})$ be an $m\times m$ skew-symmetric integer matrix,
and $\widetilde{B}=\begin{bmatrix}B\\\ P\end{bmatrix}=(b_{ij})$ an $m\times n$
integer matrix, where $B$ is the top $n\times n$ submatrix of $\widetilde{B}$.
Here we do not assume that $B$ is skew-symmetrizable.

The notation of compatible pairs are introduced in [bz-2005] to define quantum
cluster algebras. In this paper, we will use the notation of compatible pairs,
but we will not go to the quantum setting.

######  Definition 2.1 (Compatible pair, [bz-2005]).

Keep the above setting. Let $S=diag(s_{1},\ldots,s_{n})$ be a diagonal integer
matrix with $s_{j}>0$ ($j\in[1,n]$). The pair $(\widetilde{B},\Lambda)$ is
called a compatible pair of type $S$, if for any $j\in[1,n]$ and $i\in[1,m]$,
we have

(2.4) $\displaystyle\sum_{k=1}^{m}b_{kj}\lambda_{ki}=\delta_{i,j}s_{j}.$

In other words, $\widetilde{B}^{T}\Lambda=(S\mid{\bf 0})$, where ${\bf 0}$ is
the $n\times(m-n)$ zero matrix.

Notice that the diagonal matrix $S$ is uniquely determined by the compatible
pair $(\widetilde{B},\Lambda)$. The matrix $\Lambda$ in a compatible pair
$(\widetilde{B},\Lambda)$ is called a quantization matrix.

######  Proposition 2.2.

[bz-2005, Proposition 3.3] Let $(\widetilde{B},\Lambda)$ be a compatible pair
of type $S$ and $B$ the top $n\times n$ submatrix of $\widetilde{B}$. Then the
matrix $\widetilde{B}$ has full rank $n$ and
$SB=\widetilde{B}^{T}\Lambda\widetilde{B}$. In particular, $B$ is skew-
symmetrizable.

Conversely, by [gsv-2003, Theorem 1.4], whenever $\widetilde{B}$ is a mutation
matrix of full rank, we can find a skew-symmetric integer matrix $\Lambda$
such that $(\widetilde{B},\Lambda)$ is a compatible pair.

Let $(\widetilde{B},\Lambda)$ be a compatible pair of type $S$ and
$k\in[1,n]$. We set
$\widetilde{B}^{\prime}:=\mu_{k}(\widetilde{B})=E_{k,\varepsilon}^{\widetilde{B}}\widetilde{B}F_{k,\varepsilon}^{\widetilde{B}}$
and

$\Lambda^{\prime}:=(E_{k,\varepsilon}^{\widetilde{B}})^{T}\Lambda
E_{k,\varepsilon}^{\widetilde{B}}.$

######  Proposition 2.3.

[bz-2005, Proposition 3.4] (i) The matrix $\Lambda^{\prime}$ is independent of
the choice of $\varepsilon\in\\{\pm\\}$.

(ii) The pair $(\widetilde{B}^{\prime},\Lambda^{\prime})$ is still a
compatible pair of type $S$.

Keep the above setting. We call $(\widetilde{B}^{\prime},\Lambda^{\prime})$
the mutation of the compatible pair $(\widetilde{B},\Lambda)$ in direction $k$
and denote
$(\widetilde{B}^{\prime},\Lambda^{\prime})=\mu_{k}(\widetilde{B},\Lambda)$. It
is known from [bz-2005, Proposition 3.6] that
$\mu_{k}(\widetilde{B}^{\prime},\Lambda^{\prime})=(\widetilde{B},\Lambda)$,
i.e., $\mu_{k}$ is an involution.

The following proposition will be not used in this paper, but it is very
useful to construct many examples of compatible pairs from any given skew-
symmetrizable matrix.

######  Proposition 2.4.

[zelevinsky-2005, Example 0.5] Let $B$ be an $n\times n$ skew-symmetrizable
integer matrix and set $\widetilde{B}=\begin{bmatrix}B\\\ I_{n}\end{bmatrix}$.
Then the skew-symmetric matrices $\Lambda$ compatible with $\widetilde{B}$ in
the sense of (2.4) are those of the form

$\Lambda=\begin{bmatrix}\Lambda_{0}&-S-\Lambda_{0}B\\\
S-B^{T}\Lambda_{0}&-SB+B^{T}\Lambda_{0}B\end{bmatrix},$

where $\Lambda_{0}$ is an arbitrary skew-symmetric integer $n\times n$ matrix
and $S$ is a arbitrary skew-symmetrizer of $B$.

###  2.3. $Y$-pattern and cluster pattern

Recall that we fixed a pair $(n,m)$ of integers with $m\geq n>0$. Let
$\mathbb{F}$ be the field of rational functions over $\mathbb{Q}$ in $m$
variables.

A $Y$-seed of rank $n$ in $\mathbb{F}$ is a pair $({\bf y},\widehat{B})$,
where

  * •

${\bf y}=(y_{1},\ldots,y_{m})$ satisfies that $\\{y_{1},\ldots,y_{m}\\}$ is a
free generating set of $\mathbb{F}$ over $\mathbb{Q}$;

  * •

$\widehat{B}=(B\mid Q)=(\hat{b}_{ij})$ is an $n\times m$ integer matrix such
that its leftmost $n\times n$ submatrix $B$ is skew-symmetrizable.

The variables $y_{1},\ldots,y_{m}$ are called the $y$-variables of $({\bf
y},\widehat{B})$.

Let $({\bf y},\widehat{B})$ be a $Y$-seed of rank $n$ in $\mathbb{F}$. The
mutation of $({\bf y},\widehat{B})$ in direction $k\in[1,n]$ is the pair
$({\bf y}^{\prime},\widehat{B}^{\prime}):=\mu_{k}({\bf y},\widehat{B})$ given
as follows:

(2.5) $\displaystyle y_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\begin{cases}y_{k}^{-1},&\text{if}\;i=k;\\\
y_{i}y_{k}^{[\hat{b}_{ki}]_{+}}(1+y_{k})^{-\hat{b}_{ki}},&\text{otherwise}.\end{cases}$
$\displaystyle\hat{b}_{ij}^{\prime}$ $\displaystyle=$
$\displaystyle\begin{cases}-\hat{b}_{ij},&\text{if}\;i=k\;\text{or}\;j=k;\\\
\hat{b}_{ij}+[\hat{b}_{ik}]_{+}[\hat{b}_{kj}]_{+}-[-\hat{b}_{ik}]_{+}[-\hat{b}_{kj}]_{+},&\text{otherwise}.\end{cases}$

One can check that the new pair $({\bf
y}^{\prime},\widehat{B}^{\prime})=\mu_{k}({\bf y},\widehat{B})$ is still a
$Y$-seed of rank $n$ and $({\bf y},\widehat{B})=\mu_{k}({\bf
y}^{\prime},\widehat{B}^{\prime})$.

A cluster seed or simply a seed of rank $n$ in $\mathbb{F}$ is a pair $({\bf
x},\widetilde{B})$, where

  * •

${\bf x}=(x_{1},\ldots,x_{m})$ satisfies that $\\{x_{1},\ldots,x_{m}\\}$ is a
free generating set of $\mathbb{F}$ over $\mathbb{Q}$;

  * •

$\widetilde{B}=\begin{bmatrix}B\\\ P\end{bmatrix}=(b_{ij})$ is an $m\times n$
mutation matrix.

The tuple ${\bf x}$ is called the cluster of $({\bf x},\widetilde{B})$.
Elements in ${\bf x}$ are called cluster variables. More precisely, we call
$x_{1},\ldots,x_{n}$ unfrozen cluster variables and $x_{n+1},\ldots,x_{m}$
frozen cluster variables. The matrices $B$ and $P$ are respectively called the
exchange matrix, coefficient matrix of $({\bf x},\widetilde{B})$.

Let $({\bf x},\widetilde{B})$ be a seed. We denote by $\widehat{\bf
y}:=(\hat{y}_{1},\ldots,\hat{y}_{n})$, where
$\hat{y}_{k}:=\prod_{j=1}^{m}x_{j}^{b_{jk}}$. The variables
$\hat{y}_{1},\ldots,\hat{y}_{n}$ are called $\hat{y}$-variables associated to
$({\bf x},\widetilde{B})$.

Let $({\bf x},\widetilde{B})$ be a seed of rank $n$ in $\mathbb{F}$. The
mutation of $({\bf x},\widetilde{B})$ in direction $k\in[1,n]$ is the pair
$({\bf x}^{\prime},\widetilde{B}^{\prime})=\mu_{k}({\bf x},\widetilde{B})$
given by $\widetilde{B}^{\prime}=\mu_{k}(\widetilde{B})$ and

(2.6) $\displaystyle x_{i}^{\prime}=\begin{cases}x_{i},&\text{if}\;i\neq k;\\\
x_{k}^{-1}\cdot(\prod_{j=1}^{m}x_{j}^{[b_{jk}]_{+}}+\prod_{j=1}^{m}x_{j}^{[-b_{jk}]_{+}}),&\text{if}\;i=k.\end{cases}$

One can check that the new pair $({\bf
x}^{\prime},\widetilde{B}^{\prime})=\mu_{k}({\bf x},\widetilde{B})$ is still a
seed of rank $n$ and $({\bf x},\widetilde{B})=\mu_{k}({\bf
x}^{\prime},\widetilde{B}^{\prime})$.

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

######  Definition 2.5.

(i) A $Y$-pattern $\mathcal{S}_{Y}=\\{({\bf y}_{t},\widehat{B}_{t})\mid
t\in\mathbb{T}_{n}\\}$ of rank $n$ is an assignment of a $Y$-seed $({\bf
y}_{t},\widehat{B}_{t})$ of rank $n$ to every vertex $t$ of $\mathbb{T}_{n}$
such that $({\bf y}_{t^{\prime}},\widehat{B}_{t^{\prime}})=\mu_{k}({\bf
y}_{t},\widehat{B}_{t})$ whenever
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in $\mathbb{T}_{n}$.

(ii) A cluster pattern $\mathcal{S}_{X}=\\{({\bf x}_{t},\widetilde{B}_{t})\mid
t\in\mathbb{T}_{n}\\}$ of rank $n$ is an assignment of a cluster seed $({\bf
x}_{t},\widetilde{B}_{t})$ of rank $n$ to every vertex $t$ of $\mathbb{T}_{n}$
such that $({\bf x}_{t^{\prime}},\widetilde{B}_{t^{\prime}})=\mu_{k}({\bf
x}_{t},\widetilde{B}_{t})$ whenever
$\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in $\mathbb{T}_{n}$.

We usually write ${\bf y}_{t}=(y_{1;t},\ldots,y_{m;t})$,
$\widehat{B}_{t}=(B_{t}\mid Q_{t})=(\hat{b}_{ij;t})$, ${\bf
x}_{t}=(x_{1;t},\ldots,x_{m;t})$ and

$\widetilde{B}_{t}=\begin{pmatrix}B_{t}\\\ P_{t}\end{pmatrix}=(b_{ij;t}).$

We write $\widehat{\bf y}_{t}=(\hat{y}_{1;t},\ldots,\hat{y}_{n;t})$ for the
collection of $\hat{y}$-variables associated to seed $({\bf
x}_{t},\widetilde{B}_{t})$, where

$\hat{y}_{k;t}=\prod_{j=1}^{m}x_{j;t}^{b_{jk;t}}.$

In this paper, we usually fix a vertex $t_{0}$ as the rooted vertex of the
$n$-regular tree $\mathbb{T}_{n}$. Clearly, both $Y$-pattern and cluster
pattern are uniquely determined by the data at the rooted vertex $t_{0}$.

######  Definition 2.6.

Let $\mathcal{S}_{X}=\\{({\bf x}_{t},\widetilde{B}_{t})\mid
t\in\mathbb{T}_{n}\\}$ be a cluster pattern, $\mathcal{S}_{Y}=\\{({\bf
y}_{t},\widehat{B}_{t})\mid t\in\mathbb{T}_{n}\\}$ a $Y$-pattern, and let
$\Lambda=\\{\Lambda_{t}\mid t\in\mathbb{T}_{n}\\}$ be a collection of $m\times
m$ skew-symmetric matrices indexed by the vertices in $\mathbb{T}_{n}$.

  * (i)

The pair $(\mathcal{S}_{X},\mathcal{S}_{Y})$ is called a Langlands dual pair,
if $\widehat{B}_{t_{0}}=-\widetilde{B}_{t_{0}}^{T}$ holds for the rooted
vertex $t_{0}$;

  * (ii)

The triple $(\mathcal{S}_{X},\mathcal{S}_{Y},\Lambda)$ is called a Langlands-
Poisson triple, if it satisfies that

    * –

$(\mathcal{S}_{X},\mathcal{S}_{Y})$ forms a Langlands dual pair;

    * –

$\\{(\widetilde{B}_{t},\Lambda_{t})\mid t\in\mathbb{T}_{n}\\}$ forms a
collection of compatible pairs and
$(\widetilde{B}_{t^{\prime}},\Lambda_{t^{\prime}})=\mu_{k}(\widetilde{B}_{t},\Lambda_{t})$
whenever $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$.

The above definition is purely for convenience in this paper. The following
result can be checked easily.

######  Corollary 2.7.

Let $(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair. Then
$\widehat{B}_{t}=-\widetilde{B}_{t}^{T}$ holds for any vertex $t$ of
$\mathbb{T}_{n}$.

Given a Langlands-Poisson triple $(\mathcal{S}_{X},\mathcal{S}_{Y},\Lambda)$.
One can define a Poisson bracket $\\{-,-\\}$ on the ambient field $\mathbb{F}$
using the $m\times m$ skew-symmetric matrix
$\Lambda_{t_{0}}=(\lambda_{ij;t_{0}})$ at the rooted vertex $t_{0}$:

$\\{x_{i;t_{0}},x_{j;t_{0}}\\}:=\lambda_{ij;t_{0}}\cdot
x_{i;t_{0}}x_{j;t_{0}}.$

It turns out this Poisson bracket is compatible with the cluster pattern
$\mathcal{S}_{X}$, that is, for any cluster ${\bf x}_{t}$ of
$\mathcal{S}_{X}$, one has

$\\{x_{i;t},x_{j;t}\\}=\lambda_{ij;t}\cdot x_{i;t}x_{j;t},$

where $\lambda_{ij;t}$ is the $(i,j)$-entry of $\Lambda_{t}$, cf. [gsv-2003],
[bz-2005, Remark 4.6].

Finally, let us give the definition of (upper) cluster algebras.

######  Definition 2.8.

Let $\mathcal{S}_{X}=\\{({\bf x}_{t},\widetilde{B}_{t})\mid
t\in\mathbb{T}_{n}\\}$ be a cluster pattern of rank $n$ in $\mathbb{F}$.

  * (i)

The cluster algebra $\mathcal{A}$ associated to $\mathcal{S}_{X}$ is the
$\mathbb{Z}$-subalgebra of $\mathbb{F}$ given by

$\mathcal{A}:=\mathbb{Z}[x_{1;t},\ldots,x_{n;t},x_{n+1;t}^{\pm
1},\ldots,x_{m;t}^{\pm 1}|\;t\in\mathbb{T}_{n}].$

  * (ii)

The upper cluster algebra $\mathcal{U}$ associated to $\mathcal{S}_{X}$ is the
$\mathbb{Z}$-subalgebra of $\mathbb{F}$ given by the following intersections

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

where $\mathcal{L}(t):=\mathbb{Z}[x_{1;t}^{\pm 1},\ldots,x_{m;t}^{\pm 1}]$.

######  Theorem 2.9.

[fz_2002]*Laurent phenomenon For any cluster variable $z$ and any seed $({\bf
x}_{t},\widetilde{B}_{t})$ of $\mathcal{A}$, we have
$z\in\mathbb{Z}[x_{1;t}^{\pm 1},\ldots,x_{n;t}^{\pm
1},x_{n+1;t},\ldots,x_{m;t}]$. In particular, $\mathcal{A}$ is contained in
$\mathcal{U}$.

##  3\. Tropical points, pointed elements and good elements

###  3.1. Universal semi-field and tropical points

Recall that $(\mathbb{P},\cdot,\oplus)$ is called a semi-field 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. For
example, $\mathbb{Z}^{t}:=(\mathbb{Z},+,\oplus=\min\\{-,-\\})$ and
$\mathbb{Z}^{T}:=(\mathbb{Z},+,\oplus=\max\\{-,-\\})$ are semi-field, which
are called tropical semi-field.

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

From now on, we fix $\mathcal{F}:=\mathbb{Q}(u_{1},\ldots,u_{m})$ and
$\mathcal{F}_{>0}:=\mathbb{Q}_{\rm sf}(u_{1},\ldots,u_{m})$. For any semi-
field $\mathbb{P}$ and ${\bf p}=(p_{1},\ldots,p_{m})^{T}\in\mathbb{P}^{m}$,
there exists a unique semi-field homomorphism

$\pi_{\bf p}:\mathcal{F}_{>0}\rightarrow\mathbb{P}$

induced by $u_{i}\mapsto p_{i}$ for any $i$. The map $\pi:{\bf
p}\mapsto\pi_{\bf p}$ induces a bijection from $\mathbb{P}^{m}$ to the set
$\operatorname{\mathsf{Hom}}_{\rm sf}(\mathcal{F}_{>0},\mathbb{P})$ of
semifield homomorphisms from $\mathcal{F}_{>0}$ to $\mathbb{P}$.

In this paper, we often take $\mathbb{P}=\mathbb{Z}^{T}=(\mathbb{Z},+,\max)$.
The elements in $\operatorname{\mathsf{Hom}}_{\rm
sf}(\mathcal{F}_{>0},\mathbb{Z}^{T})$ are called tropical
$\mathbb{Z}^{T}$-points or simply tropical points, which are in bijection with
the points in $\mathbb{Z}^{m}$.

######  Definition 3.1.

A chart on $\mathcal{F}_{>0}$ is $m$-tuple ${\bf
u}^{\prime}=(u_{1}^{\prime},\ldots,u_{m}^{\prime})$ of elements in
$\mathcal{F}_{>0}=\mathbb{Q}_{\rm sf}(u_{1},\ldots,u_{m})$ that generate
$\mathcal{F}_{>0}$ as a semifield.

######  Proposition 3.2.

Let $\mathcal{C}=\\{{\bf u}_{t}\mid t\in\mathbb{T}\\}$ be a collection of
charts on $\mathcal{F}_{>0}=\mathbb{Q}_{\rm sf}(u_{1},\ldots,u_{m})$ indexed
by any set $\mathbb{T}$. Then the following statements hold.

  * (i)

For any chart ${\bf u}_{t}=(u_{1;t},\ldots,u_{m;t})$, denote by $u_{i}({\bf
u}_{t})$ the expression of $u_{i}$ as an element in
$\mathcal{F}_{>0;t}:=\mathbb{Q}_{\rm sf}(u_{1;t},\ldots,u_{m;t})$. The map
$\varphi_{t}:u_{i}\rightarrow u_{i}({\bf u}_{t})$ induces an isomorphism from
the semifield $\mathcal{F}_{>0}$ to $\mathcal{F}_{>0;t}:=\mathbb{Q}_{\rm
sf}(u_{1;t},\ldots,u_{m;t})$;

  * (ii)

Let $\nu$ be a tropical point in $\operatorname{\mathsf{Hom}}_{\rm
sf}(\mathcal{F}_{>0},\mathbb{Z}^{T})$ and ${\bf
u}_{t}=(u_{1;t},\ldots,u_{m;t})$ a chart on $\mathcal{F}_{>0}$. Set

${\bf q}_{t}(\nu):=(\nu(u_{1;t}),\ldots,\nu(u_{m;t}))^{T}\in\mathbb{Z}^{m}.$

Then the following diagram commutes.

---
$\textstyle{\mathcal{F}_{>0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{t}}$$\scriptstyle{\nu}$$\textstyle{\mathcal{F}_{>0;t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\bf
q}_{t}(\nu);t}}$$\textstyle{\mathbb{Z}^{T}}$

where $\pi_{{\bf q}_{t}(\nu);t}$ is the unique semifield homomorphism from
$\mathcal{F}_{>0;t}$ to $\mathbb{Z}^{T}$ induced by
$u_{i;t}\mapsto\nu(u_{i;t})$ for any $i$.

  * (iii)

The map $\nu\mapsto{\bf q}_{t}(\nu)$ induces a bijection from the tropical
points in $\operatorname{\mathsf{Hom}}_{\rm
sf}(\mathcal{F}_{>0},\mathbb{Z}^{T})$ and the points in $\mathbb{Z}^{m}$.

###### Proof.

Obviously. ∎

Keep the setting in the above proposition. We call the vector ${\bf
q}_{t}(\nu)$ the coordinate vector of the tropical point $\nu$ under the chart
${\bf u}_{t}$. When a collection $\mathcal{C}=\\{{\bf u}_{t}\mid
t\in\mathbb{T}\\}$ of charts on $\mathcal{F}_{>0}$ is given, we often identify
a tropical point $\nu\in\operatorname{\mathsf{Hom}}_{\rm
sf}(\mathcal{F}_{>0},\mathbb{Z}^{T})$ with the collection $\\{{\bf
q}_{t}(\nu)\mid t\in\mathbb{T}\\}$ of coordinate vectors of $\nu$ under the
charts.

Now let us go back to the situations that we are interested in.

  * •

If we work on a $Y$-pattern $\mathcal{S}_{Y}=\\{({\bf
y}_{t},\widehat{B}_{t})\mid t\in\mathbb{T}_{n}\\}$, we take $\mathcal{F}_{>0}$
to be the universal semifield generated by the initial $y$-variables
$y_{1;t_{0}},\ldots,y_{m;t_{0}}$. Then $\mathcal{C}:=\\{{\bf y}_{t}\mid
t\in\mathbb{T}_{n}\\}$ is a collection of charts on $\mathcal{F}_{>0}$.

  * •

If we work on a cluster pattern $\mathcal{S}_{X}=\\{({\bf
x}_{t},\widetilde{B}_{t})\mid t\in\mathbb{T}_{n}\\}$, we take
$\mathcal{F}_{>0}$ to be the universal semifield generated by the initial
cluster variables $x_{1;t_{0}},\ldots,x_{m;t_{0}}$. Then $\mathcal{C}:=\\{{\bf
x}_{t}\mid t\in\mathbb{T}_{n}\\}$ is a collection of charts on
$\mathcal{F}_{>0}$.

Since the transition maps between any two adjacent charts are clear in both
$Y$-pattern and cluster pattern, the corresponding tropical points (after
identifying with the corresponding coordinate vectors) can be defined using
the tropical version of the transition maps in (2.5) and (2.6).

######  Definition 3.3 (Tropical points).

Let $\mathcal{S}_{Y}=\\{({\bf y}_{t},\widehat{B}_{t})\mid
t\in\mathbb{T}_{n}\\}$ be a $Y$-pattern of rank $n$ in $\mathbb{F}$ and
$\mathcal{S}_{X}=\\{({\bf x}_{t},\widetilde{B}_{t})\mid t\in\mathbb{T}_{n}\\}$
a cluster pattern of rank $n$ in $\mathbb{F}$.

  * (i)

A tropical point $[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ associated to the $Y$-pattern $\mathcal{S}_{Y}$ is an
assignment of a (column) vector ${\bf g}_{t}=(g_{1;t},\ldots,g_{m;t})^{T}$ in
$\mathbb{Z}^{m}$ to each vertex $t$ of $\mathbb{T}_{n}$ such that

(3.1) $\displaystyle
g_{i;t^{\prime}}=\begin{cases}-g_{k;t},&\text{if}\;i=k;\\\
g_{i;t}+[\hat{b}_{ki;t}]_{+}g_{k;t}+(-\hat{b}_{ki;t})[g_{k;t}]_{+},&\text{if}\;i\neq
k.\end{cases}$

whenever $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$. We denote by $\mathcal{S}_{Y}(\mathbb{Z}^{T})$ the set of
tropical points associated to the $Y$-pattern $\mathcal{S}_{Y}$.

  * (ii)

A tropical point $[{\bf a}]=\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ associated to the cluster pattern $\mathcal{S}_{X}$ is
an assignment of a (column) vector ${\bf a}_{t}=(a_{1;t},\ldots,a_{m;t})^{T}$
in $\mathbb{Z}^{m}$ to each vertex $t$ of $\mathbb{T}_{n}$ such that

(3.2) $\displaystyle
a_{i;t^{\prime}}=\begin{cases}-a_{k;t}+\max\\{\sum_{j=1}^{m}[b_{jk;t}]_{+}a_{j;t},\;\sum_{j=1}^{m}[-b_{jk;t}]_{+}a_{j;t}\\},&\text{if}\;i=k;\\\
a_{i;t},&\text{if}\;i\neq k.\end{cases}$

whenever $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$. We denote by $\mathcal{S}_{X}(\mathbb{Z}^{T})$ the set of
tropical points associated to the cluster pattern $\mathcal{S}_{X}$.

######  Remark 3.4.

The relations in (3.1) and (3.2) are obtained from mutation relations (2.5)
and (2.6) by tropicalization over the tropical semifield
$\mathbb{Z}^{T}=(\mathbb{Z},+,\max)$. Namely, we make the following
replacements.

  * •

Replace the multiplication and addition in (2.5) and (2.6) by “+” and
“max{–,–}” over $\mathbb{Z}$, respectively.

  * •

Replace “1” in (2.5) by “0”;

###  3.2. Compatibly pointed elements and good elements

In this subsection, we fix a full rank upper cluster algebra $\mathcal{U}$,
which means its initial mutation matrix $\widetilde{B}_{t_{0}}$ has the full
rank $n$, equivalently, any mutation matrix $\widetilde{B}_{t}$ of
$\mathcal{U}$ has the full rank $n$.

Since $\mathcal{U}$ is of full rank, each seed $({\bf
x}_{t},\widetilde{B}_{t})$ of $\mathcal{U}$ defines a partial order
$\preceq_{t}$ on $\mathbb{Z}^{m}$. For two vectors ${\bf g},{\bf
g}^{\prime}\in\mathbb{Z}^{m}$, we write ${\bf g}^{\prime}\preceq_{t}{\bf g}$
if there exists some vector ${\bf
v}=(v_{1},\ldots,v_{n})^{T}\in\mathbb{N}^{n}$ such that

${\bf g}^{\prime}={\bf g}+\widetilde{B}_{t}{\bf v},$

equivalently, ${\bf x}_{t}^{{\bf g}^{\prime}}={\bf x}_{t}^{\bf
g}\cdot\widehat{\bf y}_{t}^{\bf v}$, where $\widehat{\bf y}_{t}^{\bf v}={\bf
x}_{t}^{\widetilde{B}_{t}{\bf v}}$. We denote by ${\bf
g}^{\prime}\prec_{t}{\bf g}$ for the case ${\bf g}^{\prime}\preceq_{t}{\bf g}$
and ${\bf g}^{\prime}\neq{\bf g}$.

The partial order $\preceq_{t}$ above is known as dominance order associated
to seed $({\bf x}_{t},\widetilde{B}_{t})$ in [Qin_2017]. The dominance order
plays an important role in study of the bases problem of (upper) cluster
algebras.

For any seed $({\bf x}_{t},\widetilde{B}_{t})$ of $\mathcal{U}$, we denote by
$R_{t}$ the collection of the formal Laurent series

$u=\sum_{{\bf h}\in\mathbb{Z}^{m}}b_{\bf h}{\bf x}_{t}^{\bf h},\;\;\;b_{\bf
h}\in\mathbb{Z},$

such that the set $\\{{\bf h}\in\mathbb{Z}^{m}\mid b_{\bf h}\neq 0\\}$ has
finitely many maximal elements with respect to the dominance order
$\preceq_{t}$ on $\mathbb{Z}^{m}$. One can see that the collection $R_{t}$ has
a ring structure.

Now we recall the definition of pointed elements introduced in [Qin_2017].
Given any

$u=\sum_{{\bf h}\in\mathbb{Z}^{m}}b_{\bf h}{\bf x}_{t}^{\bf h}\in R_{t}.$

We say that $u$ is pointed for the seed $({\bf x}_{t},\widetilde{B}_{t})$ if
the following two conditions are satisfied.

  * •

The set $\\{{\bf h}\in\mathbb{Z}^{m}\mid b_{\bf h}\neq 0\\}$ has a unique
maximal element ${\bf g}$ under the dominance order $\preceq_{t}$ on
$\mathbb{Z}^{m}$. We denote by $\operatorname{\mathsf{deg}}^{t}(u):={\bf g}$
and call it the degree of $u$ with respect to the seed $({\bf
x}_{t},\widetilde{B}_{t})$.

  * •

The coefficient corresponds to the degree term is $1$, i.e., $b_{\bf g}=1$ for
${\bf g}=\operatorname{\mathsf{deg}}^{t}(u)$.

By definition, each pointed element $u$ for the seed $({\bf
x}_{t},\widetilde{B}_{t})$ takes the following form.

$u={\bf x}_{t}^{\bf g}+\sum_{{\bf h}\prec_{t}{\bf g}}b_{\bf h}{\bf x}_{t}^{\bf
h}={\bf x}_{t}^{\bf g}F(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where $b_{\bf h}\in\mathbb{Z}$ and $F\in\mathbb{Z}[[y_{1},\ldots,y_{n}]]$ is a
polynomial series with constant term $1$.

In cluster algebras, we are often interested in the pointed elements which can
be controlled by tropical points associated to a $Y$-pattern. Such elements
correspond to compatibly pointed elements introduced in [qin_2019].

######  Definition 3.5 (Compatibly pointed elements and good elements).

Let $(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair and
$\mathcal{U}$ the upper cluster algebra associated to the cluster pattern
$\mathcal{S}_{X}$. We assume that $\mathcal{U}$ is of full rank.

  * (i)

An element $u\in\mathcal{U}$ is said to be compatibly pointed, if $u$ is
pointed for any seed $({\bf x}_{t},\widetilde{B}_{t})$ of $\mathcal{U}$ and
the collection $[{\bf
g}]:=\\{\operatorname{\mathsf{deg}}^{t}(u)\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ forms a tropical point in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$. In this case, we also say that $u$ is
$[{\bf g}]$-pointed.

  * (ii)

A $[{\bf g}]$-pointed element $u\in\mathcal{U}$ is said to be $[{\bf
g}]$-good, if it is universally positive. In other words, the Laurent
expansion of $u$ with respect to any seed $({\bf x}_{t},\widetilde{B}_{t})$ of
$\mathcal{U}$ can be written as

(3.3) $\displaystyle u={\bf x}_{t}^{{\bf
g}_{t}}F_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where ${\bf g}_{t}=\operatorname{\mathsf{deg}}^{t}(u)$ is the value of $[{\bf
g}]$ at vertex $t$ and $F_{t}$ is a polynomial in $\mathbb{Z}_{\geq
0}[y_{1},\ldots,y_{n}]$ with constant term $1$.

Consider the Laurent expansion of a $[{\bf g}]$-pointed element $u$ with
respect to any seed $({\bf x}_{t},\widetilde{B}_{t})$ of $\mathcal{U}$. It
takes the form:

(3.4) $\displaystyle u={\bf x}_{t}^{{\bf
g}_{t}}F_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where ${\bf g}_{t}=\operatorname{\mathsf{deg}}^{t}(u)$ is the value of $[{\bf
g}]$ at vertex $t$ and $F_{t}$ is a polynomial in
$\mathbb{Z}[y_{1},\ldots,y_{n}]$ with constant term $1$.

  * •

The polynomial $F_{t}$ is called the $F$-polynomial of $u$ with respect to
vertex $t$;

  * •

For $k\in[1,n]$, we denote by $f_{k;t}$ the highest degree of $y_{k}$ in
$F_{t}$. The vector

${\bf f}_{t}:=(f_{1;t},\ldots,f_{n;t})^{T}\in\mathbb{N}^{n}$

is called the $f$-vector of $u$ with respect to vertex $t$;

  * •

The $[{\bf g}]$-pointed element $u$ is said to be $[{\bf g}]$-bipointed, if
the monomial $y_{1}^{f_{1;t}}\ldots y_{n}^{f_{n;t}}$ appears in $F_{t}$ and it
has coefficient $1$, for any vertex $t\in\mathbb{T}_{n}$;

  * •

The $[{\bf g}]$-pointed element $u$ is said to be $[{\bf g}]$-bigood, if it is
$[{\bf g}]$-bipointed and universally positive.

Cluster monomials provide a class of important examples of bigood elements.
Recall that a cluster monomial in $\mathcal{U}$ is a Laurent monomial of the
form

${\bf x}_{w}^{\bf v}:=\prod_{j=1}^{m}x_{j;w}^{v_{j}}$

for some cluster ${\bf x}_{w}$ of $\mathcal{U}$ and some vector ${\bf
v}=(v_{1},\ldots,v_{m})^{T}\in\mathbb{Z}^{m}$ with $v_{i}\geq 0$ for any
$i\in[1,n]$. Notice that we allow $v_{j}$ to be negative for $j\in[n+1,m]$.

Given a vertex $w\in\mathbb{T}_{n}$ and a vector ${\bf v}\in\mathbb{Z}^{m}$,
we denote by $[({\bf v},w)]$ the unique tropical point $\\{{\bf
g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$ determined by the condition ${\bf
g}_{w}={\bf v}$.

######  Proposition 3.6.

Let ${\bf x}_{w}^{\bf v}$ be a cluster monomial in $\mathcal{U}$. Then ${\bf
x}_{w}^{\bf v}$ is $[({\bf v},w)]$-bigood in $\mathcal{U}$.

###### Proof.

By the results in [GHKK18], we know that ${\bf x}_{w}^{\bf v}$ is $[({\bf
v},w)]$-good in $\mathcal{U}$. Let $F_{t}$ and ${\bf
f}_{t}=(f_{1;t},\ldots,f_{n;t})^{T}$ be the $F$-polynomial and $f$-vector of
${\bf x}_{w}^{\bf v}$ with respect to vertex $t\in\mathbb{T}_{n}$. Since the
$F$-polynomials of cluster variables have constant term $1$ and by
[fomin_zelevinsky_2007, Proposition 5.3], we know that the monomial
$y_{1}^{f_{1;t}}\ldots y_{n}^{f_{n;t}}$ appears in $F_{t}$. Hence, ${\bf
x}_{w}^{\bf v}$ is $[({\bf v},w)]$-bigood in $\mathcal{U}$. ∎

###  3.3. Basic properties of pointed elements

As before, we fix a full rank upper cluster algebra $\mathcal{U}$. Although we
mainly care about the elements in the upper cluster algebra $\mathcal{U}$,
sometimes our calculations will be carried out in $R_{t}$. Let us recall some
basic properties of pointed elements in $R_{t}$.

######  Lemma 3.7.

[qin_2019]*Lemmas 3.2.5, 3.2.6 Suppose that $u,u^{\prime}\in R_{t}$ are two
pointed elements for seed $({\bf x}_{t},\widetilde{B}_{t})$. Then the
following statements hold.

  * (i)

The product $u\cdot u^{\prime}\in R_{t}$ is pointed for seed $({\bf
x}_{t},\widetilde{B}_{t})$ and we have

$\operatorname{\mathsf{deg}}^{t}(u\cdot
u^{\prime})=\operatorname{\mathsf{deg}}^{t}(u)+\operatorname{\mathsf{deg}}^{t}(u^{\prime}).$

  * (ii)

$u$ has a multiplicative inverse $u^{-1}$ in $R_{t}$. Moreover, $u^{-1}$ is
pointed for $({\bf x}_{t},\widetilde{B}_{t})$ and

$\operatorname{\mathsf{deg}}^{t}(u^{-1})=-\operatorname{\mathsf{deg}}^{t}(u).$

Let $({\bf x}_{t},\widetilde{B}_{t})$ and $({\bf x}_{w},\widetilde{B}_{w})$ be
two seeds of $\mathcal{U}$ at vertices $t,w\in\mathbb{T}_{n}$. By Proposition
3.6, we know that any cluster variable $x_{j;t}$ ($j\in[1,m]$) is pointed for
$({\bf x}_{w},\widetilde{B}_{w})$. We denote by $\widetilde{G}_{t}^{w}$ the
$m\times m$ matrix whose $j$th column is given by
$\operatorname{\mathsf{deg}}^{w}(x_{j;t})\in\mathbb{Z}^{m}$. Notice that
$\widetilde{G}_{t}^{w}$ takes the form

$\widetilde{G}_{t}^{w}=\begin{bmatrix}G_{t}^{w}&0\\\
\ast&I_{m-n}\end{bmatrix},$

where $G_{t}^{w}$ is the $n\times n$ submatrix of $\widetilde{G}_{t}^{w}$. We
call $\widetilde{G}_{t}^{w}$ the extended $G$-matrix and $G_{t}^{w}$ the
$G$-matrix of vertex $t$ with respect to $w$.

Let $\overset{\leftarrow}{\mu}$ be the mutation sequence corresponding to the
unique path from vertex $w$ to $t$ in $\mathbb{T}_{n}$. We apply the mutation
sequence $\overset{\leftarrow}{\mu}$ to $\begin{pmatrix}B_{w}\\\
I_{n}\end{pmatrix}$, where $B_{w}$ is the principal part of
$\widetilde{B}_{w}$. The resulting matrix
$\overset{\leftarrow}{\mu}\begin{pmatrix}B_{w}\\\ I_{n}\end{pmatrix}$ takes
the form $\begin{pmatrix}B_{t}\\\ C_{t}^{w}\end{pmatrix}$. We call the
$n\times n$ matrix $C_{t}^{w}$ the $C$-matrix of the vertex $t$ with respect
to $w$.

######  Lemma 3.8.

[GHKK18] We have $\det(\widetilde{G}_{t}^{w})=\pm 1$ and $\det(C_{t}^{w})=\pm
1$.

######  Lemma 3.9.

Let $\mathcal{U}$ be a full rank upper cluster algebra and $t,w$ two vertices
of $\mathbb{T}_{n}$. Let $\widetilde{G}_{t}^{w}$ and $C_{t}^{w}$ be the
extended $G$-matrix and $C$-matrix of vertex $t$ with respect to $w$. Then the
following statements hold.

  * (i)

[qin_2019]*Lemma 3.3.6 Any Laurent monomial ${\bf x}_{t}^{\bf h}$ is pointed
for the seed $({\bf x}_{w},\widetilde{B}_{w})$ and we have

$\operatorname{\mathsf{deg}}^{w}({\bf x}_{t}^{\bf h})=\tilde{G}_{t}^{w}{\bf
h}=\operatorname{\mathsf{deg}}^{w}({\bf x}_{w}^{\tilde{G}_{t}^{w}{\bf h}}).$

  * (ii)

[qin_2019]*Proposition 3.3.10 Let $\widehat{\bf y}_{t}$ and $\widehat{\bf
y}_{w}$ be the collections of $\hat{y}$-variables at vertices $t$ and $w$.
Then for any ${\bf v}\in\mathbb{Z}^{n}$, the Laurent monomial $\widehat{\bf
y}_{t}^{\bf v}={\bf x}_{t}^{\widetilde{B}_{t}{\bf v}}$ is pointed for the seed
$({\bf x}_{w},\widetilde{B}_{w})$ and

$\operatorname{\mathsf{deg}}^{w}(\widehat{\bf y}_{t}^{\bf
v})=\widetilde{G}_{t}^{w}\widetilde{B}_{t}{\bf
v}=\widetilde{B}_{w}C_{t}^{w}{\bf
v}=\operatorname{\mathsf{deg}}^{w}(\widehat{\bf y}_{w}^{C_{t}^{w}{\bf v}}).$

Notice that any element in $R_{w}$ can be written as a finite
$\mathbb{Z}$-linear combinations of pointed elements for seed $({\bf
x}_{w},\widetilde{B}_{w})$. By the above lemma, any Laurent monomial ${\bf
x}_{t}^{\bf h}$ in $\mathcal{L}(t):=\mathbb{Z}[x_{1;t}^{\pm
1},\ldots,x_{m;t}^{\pm 1}]$ is pointed for the seed $({\bf
x}_{w},\widetilde{B}_{w})$. Hence, $\mathcal{L}(t)$ is contained in $R_{w}$.

######  Corollary 3.10.

Let $\mathcal{U}$ be a full rank upper cluster algebra and $t,w$ two vertices
of $\mathbb{T}_{n}$. Let $\widetilde{G}_{t}^{w}$ and $C_{t}^{w}$ be the
extended $G$-matrix and $C$-matrix of vertex $t$ with respect to $w$. Let

$u=\sum\limits_{{\bf h}\in\mathbb{Z}^{m}}b_{\bf h}{\bf x}_{t}^{\bf h}$

be a Laurent polynomial in $\mathcal{L}(t)=\mathbb{Z}[x_{1;t}^{\pm
1},\ldots,x_{m;t}^{\pm 1}]$. Then the following statements hold.

  * (i)

$u$ is pointed for the seed $({\bf x}_{w},\widetilde{B}_{w})$ if and only if
there exists a (unique) ${\bf g}\in\mathbb{Z}^{m}$ with $b_{\bf g}=1$ such
that $\operatorname{\mathsf{deg}}^{w}({\bf x}_{t}^{\bf
h})\preceq_{w}\operatorname{\mathsf{deg}}^{w}({\bf x}_{t}^{\bf g})$ for any
${\bf h}\in\mathbb{Z}^{m}$ with $b_{\bf h}\neq 0$, equivalently,

$\widetilde{G}_{t}^{w}{\bf h}\preceq_{w}\widetilde{G}_{t}^{w}{\bf g}$

for any ${\bf h}\in\mathbb{Z}^{m}$ with $b_{\bf h}\neq 0$.

  * (ii)

Suppose that $u$ is pointed for the seed $({\bf x}_{t},\widetilde{B}_{t})$,
that is, $u$ can be written as

$u={\bf x}_{t}^{\bf g}\cdot\sum\limits_{{\bf v}\in\mathbb{N}^{n}}c_{\bf
v}\widehat{\bf y}_{t}^{\bf v},$

where $c_{\bf v}\in\mathbb{Z}$ and $c_{0}=1$. Then $u$ is pointed for the new
seed $({\bf x}_{w},\widetilde{B}_{w})$ if and only if there exists a (unique)
${\bf u}\in\mathbb{N}^{n}$ with $c_{\bf u}=1$ such that
$\operatorname{\mathsf{deg}}^{w}(\widehat{\bf y}_{t}^{\bf
v})\preceq_{w}\operatorname{\mathsf{deg}}^{w}(\widehat{\bf y}_{t}^{\bf u})$
for any ${\bf v}\in\mathbb{N}^{n}$ with $c_{\bf v}\neq 0$, equivalently,

$C_{t}^{w}{\bf v}-C_{t}^{w}{\bf u}=C_{t}^{w}({\bf v}-{\bf
u})\in\mathbb{N}^{n}$

for any ${\bf v}\in\mathbb{N}^{n}$ with $c_{\bf v}\neq 0$.

###### Proof.

Thanks to Lemma 3.8 and by Lemma 3.9, we know that
$\operatorname{\mathsf{deg}}^{w}({\bf x}_{t}^{\bf
h})=\operatorname{\mathsf{deg}}^{w}({\bf x}_{t}^{{\bf h}^{\prime}})$ if and
only if ${\bf h}={\bf h}^{\prime}$. By the similar argument and since
$\widetilde{B}_{w}$ has full rank, we know that
$\operatorname{\mathsf{deg}}^{w}(\widehat{\bf y}_{t}^{\bf
v})=\operatorname{\mathsf{deg}}^{w}(\widehat{\bf y}_{t}^{{\bf v}^{\prime}})$
if and only if ${\bf v}={\bf v}^{\prime}$.

Based on the above observations, the results in (i) and (ii) follow from the
definition of pointed elements and Lemma 3.9. ∎

Let $(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair and
$\mathcal{U}$ the upper cluster algebra associated to the cluster pattern
$\mathcal{S}_{X}$. We still assume that $\mathcal{U}$ is of full rank.

######  Corollary 3.11.

Keep the above setting. Let $\textstyle{w}$$\textstyle{s}$$\scriptstyle{k}$ be
an edge in $\mathbb{T}_{n}$. Let $\widetilde{G}_{s}^{w}$ and $C_{s}^{w}$ be
the extended $G$-matrix and $C$-matrix of vertex $s$ with respect to $w$. Let
$[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ be a
tropical point in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$, and write ${\bf
g}_{t}=(g_{1;t},\ldots,g_{m;t})^{T}$, where $t\in\mathbb{T}_{n}$. Then the
following statements hold.

  * (i)

$\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf g}_{s}}\cdot\widehat{\bf
y}_{s}^{[-g_{k;s}]_{+}{\bf e}_{k}})={\bf g}_{w}$, where ${\bf e}_{k}$ is the
$k$th column of $I_{n}$.

  * (ii)

Suppose that $u\in\mathcal{U}$ is a $[{\bf g}]$-pointed element. Let $u={\bf
x}_{s}^{{\bf g}_{s}}\cdot\sum\limits_{{\bf v}\in\mathbb{N}^{n}}c_{\bf
v}\widehat{\bf y}_{s}^{\bf v}$ be the Laurent expansion of $u$ with respect to
seed $({\bf x}_{s},\widetilde{B}_{s})$. Then the coefficient of $\widehat{\bf
y}_{s}^{[-g_{k;s}]_{+}{\bf e}_{k}}$ in $\sum\limits_{{\bf
v}\in\mathbb{N}^{n}}c_{\bf v}\widehat{\bf y}_{s}^{\bf v}$ is $1$ and for any
integer $d$ with $d>[-g_{k;s}]_{+}$, the coefficient of $\widehat{\bf
y}_{s}^{d{\bf e}_{k}}$ in $\sum\limits_{{\bf v}\in\mathbb{N}^{n}}c_{\bf
v}\widehat{\bf y}_{s}^{\bf v}$ is zero.

###### Proof.

(i) By Lemma 3.7 and Lemma 3.9, we have

$\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf g}_{s}}\cdot\widehat{\bf
y}_{s}^{[-g_{k;s}]_{+}{\bf e}_{k}})=\widetilde{G}_{s}^{w}{\bf
g}_{s}+\widetilde{B}_{w}C_{s}^{w}\cdot([-g_{k;s}]_{+}{\bf e}_{k}).$

Since the two vertices $w$ and $s$ are jointed in $\mathbb{T}_{n}$ by an edge
labelled by $k$, we know that $\widetilde{G}_{s}^{w}=(g_{ij;s})$ and
$C_{s}^{w}=(c_{ij;s})$ are given as follows:

$\displaystyle g_{ij;s}$ $\displaystyle=$
$\displaystyle\begin{cases}-1,&i=j=k;\\\
[-b_{ik;w}]_{+}=[b_{ik;s}]_{+},&j=k,\;i\neq k;\\\
\delta_{ij},&\text{else},\end{cases}$ $\displaystyle c_{ij;s}$
$\displaystyle=$ $\displaystyle\begin{cases}-1,&i=j=k;\\\
[b_{kj;w}]_{+}=[-b_{kj;s}]_{+},&i=k,\;j\neq k;\\\
\delta_{ij},&\text{else}.\end{cases}$

Because the $k$-th column of $C_{s}^{w}$ equals $-{\bf e}_{k}$, we have

$\widetilde{G}_{s}^{w}{\bf
g}_{s}+\widetilde{B}_{w}C_{s}^{w}\cdot([-g_{k;s}]_{+}{\bf
e}_{k})=\widetilde{G}_{s}^{w}{\bf
g}_{s}-\widetilde{B}_{w}\cdot([-g_{k;s}]_{+}{\bf e}_{k}).$

Now we look at the $i$th component $g_{i}^{\prime}$ of the vector

${\bf g}^{\prime}:=\widetilde{G}_{s}^{w}{\bf
g}_{s}-\widetilde{B}_{w}\cdot([-g_{k;s}]_{+}{\bf e}_{k}).$

We have

$g_{i}^{\prime}=\begin{cases}-g_{k;s}-0=-g_{k;s},&i=k;\\\
(g_{i;s}+[b_{ik;s}]_{+}g_{k;s})-b_{ik;w}[-g_{k;s}]_{+}=g_{i;s}+[b_{ik;s}]_{+}g_{k;s}+b_{ik;s}[-g_{k;s}]_{+},&i\neq
k.\end{cases}$

Since $\widehat{B}_{s}=-\widetilde{B}_{s}^{T}$, we can rewrite the above
equality as follows:

$g_{i}^{\prime}=\begin{cases}-g_{k;s},&i=k;\\\
g_{i;s}+[-\hat{b}_{ki;s}]_{+}g_{k;s}-\hat{b}_{ki;s}[-g_{k;s}]_{+},&i\neq
k.\end{cases}$

Using the equality $[b]_{+}g-[-b]_{+}g=bg=b[g]_{+}-b[-g]_{+}$ for any
$b,g\in\mathbb{R}$, one can see the above equality is the same as

$g_{i}^{\prime}=\begin{cases}-g_{k;s},&i=k;\\\
g_{i;s}+[\hat{b}_{ki;s}]_{+}g_{k;s}-\hat{b}_{ki;s}[g_{k;s}]_{+},&i\neq
k.\end{cases}$

Then by (3.1), we get ${\bf g}_{w}={\bf g}^{\prime}=\widetilde{G}_{s}^{w}{\bf
g}_{s}-\widetilde{B}_{w}\cdot([-g_{k;s}]_{+}{\bf e}_{k})$. Hence,
$\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf g}_{s}}\cdot\widehat{\bf
y}_{s}^{[-g_{k;s}]_{+}{\bf e}_{k}})={\bf g}_{w}$.

(ii) Since $u$ is $[{\bf g}]$-pointed, we know that it is pointed for the seed
$({\bf x}_{w},\widetilde{B}_{w})$. By Corollary 3.10 (ii), there exists a
unique ${\bf u}\in\mathbb{N}^{n}$ with $c_{\bf u}=1$ such that

$\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf g}_{s}}\cdot\widehat{\bf
y}_{s}^{\bf v})\preceq_{w}\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf
g}_{s}}\cdot\widehat{\bf y}_{s}^{\bf
u})=\operatorname{\mathsf{deg}}^{w}(u)={\bf g}_{w}$

for any ${\bf v}\in\mathbb{N}^{n}$ with $c_{\bf v}\neq 0$. By the result in
(i), we know that ${\bf u}=[-g_{k;s}]_{+}{\bf e}_{k}$.

Notice that $\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf
g}_{s}}\cdot\widehat{\bf y}_{s}^{\bf
v})\preceq_{w}\operatorname{\mathsf{deg}}^{w}({\bf x}_{s}^{{\bf
g}_{s}}\cdot\widehat{\bf y}_{s}^{\bf u})$ holds if and only if the following
relation holds.

$C_{s}^{w}{\bf v}-C_{s}^{w}{\bf u}=C_{s}^{w}({\bf v}-{\bf
u})\in\mathbb{N}^{n}.$

Since the $k$th column of $C_{s}^{w}$ is $-{\bf e}_{k}$, we obtain that
$C_{s}^{w}(d{\bf e}_{k}-[-g_{k;s}]_{+}{\bf e}_{k})$ belongs to
$\mathbb{N}^{n}$ if and only if $d\leq[-g_{k;s}]_{+}$. Hence, for any integer
$d$ with $d>[-g_{k;s}]_{+}$, the coefficient of $\widehat{\bf y}_{s}^{d{\bf
e}_{k}}$ in $\sum\limits_{{\bf v}\in\mathbb{N}^{n}}c_{\bf v}\widehat{\bf
y}_{s}^{\bf v}$ is zero.

∎

######  Corollary 3.12.

Let $[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ be a
tropical point in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$ and $u$ a $[{\bf
g}]$-pointed element in $\mathcal{U}$. Let ${\bf
f}_{t}=(f_{1;t},\ldots,f_{n;t})^{T}$ be the $f$-vector of $u$ with respect to
vertex $t\in\mathbb{T}_{n}$. The following statements hold.

  * (i)

If $f_{k;t}=0$ for some $k\in[1,n]$, then the $k$th component $g_{k;t}$ of
${\bf g}_{t}=\operatorname{\mathsf{deg}}^{t}(u)$ is non-negative;

  * (ii)

If $f_{i;t}=0$ for any $i\in[1,n]$, then $u$ is a cluster monomial in ${\bf
x}_{t}$;

  * (iii)

Let $k$ be an integer in $[1,n]$ and let $({\bf
x}_{t^{\prime}},\widetilde{B}_{t^{\prime}})=\mu_{k}({\bf
x}_{t},\widetilde{B}_{t})$. Suppose that $f_{k;t}\neq 0$ but $f_{i;t}=0$ for
any $i\in[1,n]\backslash\\{k\\}$. Then $u$ is a cluster monomial in ${\bf
x}_{t^{\prime}}$.

###### Proof.

Since $u$ is $[{\bf g}]$-pointed in $\mathcal{U}$, the Laurent expansion of
$u$ with respect to ${\bf x}_{t}$ takes the form:

$u={\bf x}_{t}^{{\bf g}_{t}}F_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where ${\bf g}_{t}$ is the value of $[{\bf g}]$ at vertex $t$ and $F_{t}$ is a
polynomial in $\mathbb{Z}[y_{1},\ldots,y_{n}]$ with constant term $1$.

(i) By Corollary 3.11 (ii), we know that $y_{k}^{[-g_{k;t}]_{+}}$ must be a
monomial appearing in $F_{t}\in\mathbb{Z}[y_{1},\ldots,y_{n}]$. Recall that
$f_{k;t}$ is the highest degree of $y_{k}$ in $F_{t}$. So we have
$f_{k;t}\geq[-g_{k;t}]_{+}\geq 0$. Hence, if $f_{k;t}=0$, then
$[-g_{k;t}]_{+}=0$. So $g_{k;t}$ is non-negative.

(ii) Since $f_{i;t}=0$ for any $i\in[1,n]$, we get $F_{t}=1$ and $u={\bf
x}_{t}^{{\bf g}_{t}}$. On the other hand, by (i), we have $g_{i;t}\geq 0$ for
any $i\in[1,n]$. Hence, $u={\bf x}_{t}^{{\bf g}_{t}}$ is a cluster monomial in
${\bf x}_{t}$.

(iii) Since $f_{i;t}=0$ holds for any $k\neq i\in[1,n]$, we get that $F_{t}$
is a polynomial in the single variable $y_{k}$. In this case, Corollary 3.11
(ii) implies $f_{k;t}=[-g_{k;t}]_{+}$. Since $f_{k;t}\neq 0$, we have
$g_{k;t}<0$ and thus $f_{k;t}=-g_{k;t}$.

Now $u$ has the following form:

$u=x_{1;t}^{g_{1;t}}\ldots
x_{m;t}^{g_{m;t}}(1+\sum_{1<d<-g_{k;t}}c_{d}\hat{y}_{k;t}^{d}+\hat{y}_{k;t}^{-g_{k;t}}),$

where $c_{d}\in\mathbb{Z}$. By the exchange relations, we have
$x_{j;t^{\prime}}=x_{j;t}$ for any $j\neq k$ and $x_{k;t}\cdot
x_{k;t^{\prime}}=M\cdot(1+\hat{y}_{k;t})$, where
$M=\prod_{j=1}^{m}x_{j;t}^{[-b_{jk;t}]_{+}}$. By substituting these relations
into the above equality and using the fact
$u\in\mathbb{Z}[x_{1;t^{\prime}}^{\pm 1},\ldots,x_{m;t^{\prime}}^{\pm 1}]$,
one deduces that

$(1+\sum_{1<d<-g_{k;t}}c_{d}\hat{y}_{k;t}^{d}+\hat{y}_{k;t}^{-g_{k;t}})=(1+\hat{y}_{k;t})^{-g_{k;t}}$

and the Laurent expansion of $u$ with respect to seed $({\bf
x}_{t^{\prime}},\widetilde{B}_{t^{\prime}})$ is a Laurent monomial in
$\mathbb{Z}[x_{1;t^{\prime}}^{\pm 1},\ldots,x_{m;t^{\prime}}^{\pm 1}]$. Thus
the $F$-polynomial $F_{t^{\prime}}$ of $u$ with respect to vertex $t^{\prime}$
is $1$. In particular, the $f$-vector ${\bf f}_{t^{\prime}}$ of $u$ with
respect to vertex $t^{\prime}$ is zero. By (ii), we know that $u$ is a cluster
monomial in ${\bf x}_{t^{\prime}}$.

∎

##  4\. $F$-invariant in cluster algebras

###  4.1. Maps between sets of tropical points

Let $(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair. In this
subsection, we give two maps. The first map is from the tropical points in
$\mathcal{S}_{X}(\mathbb{Z}^{T})$ to those in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$ and the second one is from the tropical
points in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$ to those in
$\mathcal{S}_{X}(\mathbb{Z}^{T})$.

######  Lemma 4.1.

Let $(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair and $\\{{\bf
g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ a collection of vectors in
$\mathbb{Z}^{m}$ indexed by the vertices in $\mathbb{T}_{n}$. Then the
collection $\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ is a
tropical point in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$ if and only if for any
edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$, one has ${\bf
g}_{t^{\prime}}=E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}{\bf g}_{t}$,
where

$\varepsilon(g_{k;t})=\begin{cases}-,&\text{if }g_{k;t}\geq 0;\\\ +,&\text{if
}g_{k;t}<0,\end{cases}$

and $E_{k,\pm}^{\widetilde{B}_{t}}$ is as defined in (2.2).

###### Proof.

By definition, the collection $\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ is a tropical point in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$ if and only if the relations in (3.1) hold.
Since $(\mathcal{S}_{X},\mathcal{S}_{Y})$ is a Langlands dual pair, we have
$\widehat{B}_{t}=-\widetilde{B}_{t}^{T}$. Thus $\hat{b}_{ij;t}=-b_{ji;t}$ for
any $i\in[1,n]$ and $j\in[1,m]$. Then the relations in (3.1) can be rewritten
as

$\displaystyle g_{i;t^{\prime}}$ $\displaystyle=$
$\displaystyle\begin{cases}-g_{k;t},&\text{if}\;i=k;\\\
g_{i;t}+[-b_{ik;t}]_{+}g_{k;t}+b_{ik;t}[g_{k;t}]_{+},&\text{if}\;i\neq
k.\end{cases}$ $\displaystyle=$
$\displaystyle\begin{cases}-g_{k;t},&\text{if}\;i=k;\\\
g_{i;t}+[b_{ik;t}]_{+}g_{k;t},&\text{if}\;i\neq k\;\text{and }\;g_{k;t}\geq
0;\\\ g_{i;t}+[-b_{ik;t}]_{+}g_{k;t},&\text{if}\;i\neq k\;\text{and
}\;g_{k;t}<0.\end{cases}$

This is exactly the same with ${\bf
g}_{t^{\prime}}=E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}{\bf g}_{t}$. ∎

######  Lemma 4.2.

Let $\mathcal{S}_{X}$ be a cluster pattern of rank $n$ in $\mathbb{F}$ and
$\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ a collection of
vectors in $\mathbb{Z}^{m}$ indexed by the vertices of $\mathbb{T}_{n}$. Then
the collection $\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ is
a tropical point in $\mathcal{S}_{X}(\mathbb{Z}^{T})$ if and only if for any
edge $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ in
$\mathbb{T}_{n}$, one has ${\bf
a}_{t^{\prime}}=(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}{\bf a}_{t}$,
where

$\varepsilon_{0}=\begin{cases}-,&\text{if }\sum_{j=1}^{m}b_{jk;t}a_{j;t}\geq
0;\\\ +,&\text{if }\sum_{j=1}^{m}b_{jk;t}a_{j;t}<0,\end{cases}$

and $E_{k,\pm}^{\widetilde{B}_{t}}$ is as defined in (2.2).

###### Proof.

By definition, the collection $\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ is a tropical point in
$\mathcal{S}_{X}(\mathbb{Z}^{T})$ if and only if the relations in (3.2) hold.
The relations in (3.2) can be rewritten as follows:

$a_{i;t^{\prime}}=\begin{cases}-a_{k;t}+\sum_{j=1}^{m}[b_{jk;t}]_{+}a_{j;t},&\text{if}\;i=k\;\text{and
}\;\sum_{j=1}^{m}b_{jk;t}a_{j;t}\geq 0;\\\
-a_{k;t}+\sum_{j=1}^{m}[-b_{jk;t}]_{+}a_{j;t},&\text{if}\;i=k\;\text{and
}\;\sum_{j=1}^{m}b_{jk;t}a_{j;t}<0;\\\ a_{i;t},&\text{if}\;i\neq
k.\end{cases}$

This is exactly the same with ${\bf
a}_{t^{\prime}}=(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}{\bf a}_{t}$. ∎

Let $(\mathcal{S}_{X},\mathcal{S}_{Y})$ be a Langlands dual pair. We extend
$\widetilde{B}_{t_{0}}$ to an $m\times m$ skew-symmetrizable matrix
$\widetilde{B}_{t_{0}}^{\operatorname{\mathsf{sq}}}=(\widetilde{B}_{t_{0}}\mid
M)$, where $M$ is an $m\times(m-n)$ matrix. Denote by
$\\{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}\mid t\in\mathbb{T}_{n}\\}$
the collection of matrices obtained from
$\widetilde{B}_{t_{0}}^{\operatorname{\mathsf{sq}}}$ by any sequence of
mutations in directions $1,\ldots,n$. Let
$\widetilde{S}=diag(s_{1},\ldots,s_{m})$ be a skew-symmetrizer of the
transposed matrix $(\widetilde{B}_{t_{0}}^{\operatorname{\mathsf{sq}}})^{T}$.
In particular,
$\widetilde{S}(\widetilde{B}_{t_{0}}^{\operatorname{\mathsf{sq}}})^{T}$ is a
skew-symmetric matrix.

######  Proposition 4.3.

Keep the above setting. Then

$[{\bf a}]:=\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}\mapsto\\{\widetilde{S}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}{\bf
a}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$

defines a map from the tropical points in $\mathcal{S}_{X}(\mathbb{Z}^{T})$ to
those in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$.

###### Proof.

Denote by ${\bf
g}_{t}:=\widetilde{S}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}{\bf
a}_{t}=(g_{1;t},\ldots,g_{m;t})^{T}$ for any vertex $t\in\mathbb{T}_{n}$. We
will use Lemma 4.1 to show the collection $\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ is a tropical point in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$.

Let $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ be an edge in
$\mathbb{T}_{n}$. Since $[{\bf a}]=\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ is a tropical point in
$\mathcal{S}_{X}(\mathbb{Z}^{T})$ and by Lemma 4.2, we have ${\bf
a}_{t^{\prime}}=(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}{\bf a}_{t}$,
where

$\varepsilon_{0}=\begin{cases}-,&\text{if }\sum_{j=1}^{m}b_{jk;t}a_{j;t}\geq
0;\\\ +,&\text{if }\sum_{j=1}^{m}b_{jk;t}a_{j;t}<0.\end{cases}$

By (2.2), we have
$\widetilde{B}_{t^{\prime}}^{\operatorname{\mathsf{sq}}}=E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}}\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}F_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}}$.
Thus
$(\widetilde{B}_{t^{\prime}}^{\operatorname{\mathsf{sq}}})^{T}=(F_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})^{T}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})^{T}$.
Since $\widetilde{S}$ is a skew-symmetrizer of
$(\widetilde{B}_{t_{0}}^{\operatorname{\mathsf{sq}}})^{T}$, we have

$\widetilde{S}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}=-\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}\widetilde{S}$

for any vertex $t\in\mathbb{T}_{n}$. In the following proofs, we will use the
following obvious relations:

$\widetilde{S}(F_{k,\pm}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})^{T}=(E_{k,\pm}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})\widetilde{S},\;\;\;E_{k,\pm}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}}=E_{k,\pm}^{\widetilde{B}_{t}},\;\;\;\text{and}\;\;\;(E_{k,\pm}^{\widetilde{B}_{t}})^{2}=I_{m}.$

Now we have

$\displaystyle{\bf g}_{t^{\prime}}$ $\displaystyle=$
$\displaystyle\widetilde{S}(\widetilde{B}_{t^{\prime}}^{\operatorname{\mathsf{sq}}})^{T}{\bf
a}_{t^{\prime}}=\widetilde{S}\left((F_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})^{T}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})^{T}\right)\cdot(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}{\bf
a}_{t}$ $\displaystyle=$
$\displaystyle\widetilde{S}(F_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})^{T}\cdot(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}\cdot(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}\cdot{\bf
a}_{t}$ $\displaystyle=$
$\displaystyle(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}}})\widetilde{S}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}{\bf
a}_{t}$ $\displaystyle=$
$\displaystyle(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}}){\bf g}_{t}.$

Now we prove $\varepsilon_{0}=\varepsilon(g_{k;t})$, where
$\varepsilon(g_{k;t})$ is as given in Lemma 4.1. By definitions of
$\varepsilon(g_{k;t})$ and $\varepsilon_{0}$, it suffices to show that
$g_{k;t}\geq 0$ if and only if $\sum_{j=1}^{m}b_{jk;t}a_{j;t}\geq 0$. This is
clear, because ${\bf
g}_{t}=\widetilde{S}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}{\bf
a}_{t}$ implies that $g_{k;t}=s_{k}\cdot(\sum_{j=1}^{m}b_{jk;t}a_{j;t})$.

Hence, ${\bf g}_{t^{\prime}}=(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}}){\bf
g}_{t}=(E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}){\bf g}_{t}$. By Lemma
4.1, we know the collection

$\\{{\bf
g}_{t}=\widetilde{S}(\widetilde{B}_{t}^{\operatorname{\mathsf{sq}}})^{T}{\bf
a}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$

is a tropical point in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$. This completes the
proof. ∎

######  Proposition 4.4.

Let $(\mathcal{S}_{X},\mathcal{S}_{Y},\Lambda)$ be a Langlands-Poisson triple.
Then

$[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}\mapsto\\{\Lambda_{t}{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$

defines a map from the tropical points in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$ to
those in $\mathcal{S}_{X}(\mathbb{Z}^{T})$.

###### Proof.

Denote by ${\bf a}_{t}:=\Lambda_{t}{\bf g}_{t}=(a_{1;t},\ldots,a_{m;t})^{T}$
for any vertex $t\in\mathbb{T}_{n}$. We will use Lemma 4.2 to show the
collection $\\{{\bf a}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ is a
tropical point in $\mathcal{S}_{X}(\mathbb{Z}^{T})$.

Let $\textstyle{t}$$\textstyle{t^{\prime}}$$\scriptstyle{k}$ be an edge in
$\mathbb{T}_{n}$. Since $[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ is a tropical point in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$ and by Lemma 4.1, we have ${\bf
g}_{t^{\prime}}=E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}{\bf g}_{t}$,
where

$\varepsilon(g_{k;t})=\begin{cases}-,&\text{if }g_{k;t}\geq 0;\\\ +,&\text{if
}g_{k;t}<0.\end{cases}$

By Proposition 2.3 (i), we have
$\Lambda_{t^{\prime}}=(E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}})^{T}\Lambda_{t}E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}$.
Hence,

$\displaystyle{\bf a}_{t^{\prime}}$ $\displaystyle\xlongequal{\text{by def.}}$
$\displaystyle\Lambda_{t^{\prime}}{\bf
g}_{t^{\prime}}=\left((E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}})^{T}\Lambda_{t}E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}\right)\cdot\left(E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}}{\bf
g}_{t}\right)$
$\displaystyle\xlongequal{(E_{k,\pm}^{\widetilde{B}_{t}})^{2}=I_{m}}$
$\displaystyle(E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}})^{T}(\Lambda_{t}{\bf
g}_{t})=(E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}})^{T}{\bf a}_{t}.$

Now we prove $\varepsilon(g_{k;t})=\varepsilon_{0}$, where $\varepsilon_{0}$
is as given in Lemma 4.2. By definitions of $\varepsilon(g_{k;t})$ and
$\varepsilon_{0}$, it suffices to show that $g_{k;t}\geq 0$ if and only if
$\sum_{j=1}^{m}b_{jk;t}a_{j;t}\geq 0$. Since $(\widetilde{B}_{t},\Lambda_{t})$
is a compatible pair, we know that $\widetilde{B}_{t}^{T}\Lambda_{t}$ has the
form

$\widetilde{B}_{t}^{T}\Lambda_{t}=(S\mid{\bf 0}),$

where $S=diag(s_{1},\ldots,s_{n})$ is a diagonal matrix with $s_{i}>0$
($i\in[1,n]$) and ${\bf 0}$ is the $n\times(m-n)$ zero matrix. Thus we have
$\widetilde{B}_{t}^{T}{\bf a}_{t}=\widetilde{B}_{t}^{T}\Lambda_{t}{\bf
g}_{t}=(S\mid{\bf 0}){\bf g}_{t}$. By comparing the $k$th component on both
sides of the equality, we get

$\sum_{j=1}^{m}b_{jk;t}a_{j;t}=s_{k}g_{k;t}.$

Since $s_{k}$ is a positive integer, we get $g_{k;t}\geq 0$ if and only if
$\sum_{j=1}^{m}b_{jk;t}a_{j;t}\geq 0$. So we have
$\varepsilon(g_{k;t})=\varepsilon_{0}$.

Hence, ${\bf
a}_{t^{\prime}}=(E_{k,\varepsilon(g_{k;t})}^{\widetilde{B}_{t}})^{T}{\bf
a}_{t}=(E_{k,\varepsilon_{0}}^{\widetilde{B}_{t}})^{T}{\bf a}_{t}$. By Lemma
4.2, we know the collection

$\\{{\bf a}_{t}=\Lambda_{t}{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$

is a tropical point in $\mathcal{S}_{X}(\mathbb{Z}^{T})$. This completes the
proof. ∎

###  4.2. $F$-invariant and some consequences

We fix a Langlands-Poisson triple $(\mathcal{S}_{X},\mathcal{S}_{Y},\Lambda)$,
and let $\mathcal{U}$ be the upper cluster algebra associated to the cluster
pattern $\mathcal{S}_{X}$. Denote by $S=diag(s_{1},\ldots,s_{n})$ the type of
the compatible pair $(\widetilde{B}_{t_{0}},\Lambda_{t_{0}})$, namely,
$\widetilde{B}_{t_{0}}^{T}\Lambda_{t_{0}}=(S\mid{\bf 0})$, where ${\bf 0}$ is
the $n\times(m-n)$ zero matrix.

By Proposition 2.2, the mutation matrices of $\mathcal{U}$ have full rank. We
denote by $\mathcal{U}^{[{\bf g}]}$ the set of $[{\bf g}]$-good elements in
$\mathcal{U}$ and by

$\mathcal{U}^{\operatorname{\mathsf{good}}}:=\bigcup_{[{\bf
g}]\in\mathcal{S}_{Y}(\mathbb{Z}^{T})}\mathcal{U}^{[\bf g]}$

the set of all good elements in $\mathcal{U}$. Notice that the union
$\bigcup_{[{\bf g}]\in\mathcal{S}_{Y}(\mathbb{Z}^{T})}\mathcal{U}^{[\bf g]}$
is disjoint.

Notation: Given a polynomial $F=\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf v}}{\bf
y}^{{\bf v}}\in\mathbb{Z}[y_{1},\ldots,y_{n}]$ and a vector ${\bf
h}\in\mathbb{Z}^{n}$, we denote by

$F[{\bf h}]:=\max\\{{\bf v}^{T}{\bf h}\mid c_{{\bf v}}\neq 0\\}\in\mathbb{Z}.$

If the polynomial $F$ has constant term $1$, then by definition, $F[{\bf
h}]\geq 0$ for any ${\bf h}\in\mathbb{Z}^{n}$.

######  Definition 4.5 ($F$-invariant).

Keep the setting at the beginning of this subsection.

(i) Any vertex $t\in\mathbb{T}_{n}$ gives rise to a pairing

(4.1)
$\displaystyle\langle-,-\rangle_{t}:\mathcal{U}^{\operatorname{\mathsf{good}}}\times\mathcal{U}^{\operatorname{\mathsf{good}}}\rightarrow\mathbb{Z}$

defined as follows: for $u_{[{\bf g}]}={\bf x}_{t}^{{\bf
g}_{t}}F_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t})\in\mathcal{U}^{[{\bf g}]}$
and $u_{[{\bf g}^{\prime}]}={\bf x}_{t}^{{\bf
g}_{t}^{\prime}}F_{t}^{\prime}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t})\in\mathcal{U}^{[{\bf
g}^{\prime}]}$,

$\langle u_{[{\bf g}]},u_{[{\bf g}^{\prime}]}\rangle_{t}:={\bf
g}_{t}^{T}\Lambda_{t}{\bf g}_{t}^{\prime}+F_{t}[(S\mid{\bf 0}){\bf
g}_{t}^{\prime}].$

Notice that $\widetilde{B}_{t}^{T}\Lambda_{t}=(S\mid{\bf 0})$.

(ii) We also define the symmetrized version of the above pairing as follows:

$\displaystyle(u_{[{\bf g}]},u_{[{\bf g}^{\prime}]})_{t}=\langle u_{[{\bf
g}]},u_{[{\bf g}^{\prime}]}\rangle_{t}+\langle u_{[{\bf g}^{\prime}]},u_{[{\bf
g}]}\rangle_{t}=F_{t}[(S\mid{\bf 0}){\bf
g}_{t}^{\prime}]+F_{t}^{\prime}[(S\mid{\bf 0}){\bf g}_{t}].$

Namely,

$(u_{[{\bf g}]},u_{[{\bf g}^{\prime}]})_{t}=F_{t}[S({\bf
g}_{t}^{\prime})^{\circ}]+F_{t}^{\prime}[S{\bf g}_{t}^{\circ}],$

where ${\bf r}^{\circ}:=(r_{1},\ldots,r_{n})^{T}$ denotes the principal part
of a vector ${\bf r}=(r_{1},\ldots,r_{m})^{T}$ in $\mathbb{Z}^{m}$.

(iii) We will show the pairing in (4.1) is independent of the choice of
$t\in\mathbb{T}_{n}$. Hence, we can get rid of the vertex $t$ and denote

$\displaystyle\langle u_{[{\bf g}]},u_{[{\bf g}^{\prime}]}\rangle$
$\displaystyle:=$ $\displaystyle\langle u_{[{\bf g}]},u_{[{\bf
g}^{\prime}]}\rangle_{t},$ (4.2) $\displaystyle(u_{[{\bf g}]}\mid\mid u_{[{\bf
g}^{\prime}]})_{F}$ $\displaystyle:=$ $\displaystyle(u_{[{\bf g}]},u_{[{\bf
g}^{\prime}]})_{t}=F_{t}[S({\bf g}_{t}^{\prime})^{\circ}]+F_{t}^{\prime}[S{\bf
g}_{t}^{\circ}].$

We call $(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}$ the $F$-invariant
of $u_{[{\bf g}]}$ and $u_{[{\bf g}^{\prime}]}$.

(iv) Two good elements $u_{[{\bf g}]}$ and $u_{[{\bf g}^{\prime}]}$ are said
to be $F$-compatible, if

$(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}=0.$

######  Remark 4.6.

Since both $F_{t}$ and $F_{t}^{\prime}$ are polynomials with constant term
$1$, we have

$F_{t}[{\bf h}],F_{t}^{\prime}[{\bf h}]\in\mathbb{Z}_{\geq 0}$

for any ${\bf h}\in\mathbb{Z}^{n}$. So we have

$(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}\in\mathbb{Z}_{\geq 0}.$

It would be very interesting to discuss when $(u_{[{\bf g}]}\mid\mid u_{[{\bf
g}^{\prime}]})_{F}=0$, i.e., when $u_{[{\bf g}]}$ and $u_{[{\bf g}^{\prime}]}$
are $F$-compatible. A necessary condition can be found in Theorem 4.25.

######  Remark 4.7.

Recently, Fei introduced and studied the tropical $F$-polynomials in
[fei_2019a, fei_2019b] from the viewpoint of representation theory. One can
see that the righthand of (4.5) is the sum of evaluations of two tropical
$F$-polynomials at vectors $S({\bf g}_{t}^{\prime})^{\circ}$ and $S{\bf
g}_{t}^{\circ}$ respectively.

######  Theorem 4.8.

The pairing in (4.1) is independent of the choice of $t\in\mathbb{T}_{n}$.

###### Proof.

Let $u_{[{\bf g}]}$ be a $[{\bf g}]$-good element in $\mathcal{U}$ and
$u_{[{\bf g}^{\prime}]}$ a $[{\bf g}^{\prime}]$-good element in $\mathcal{U}$.
We need to show

$\langle u_{[{\bf g}]},u_{[{\bf g}^{\prime}]}\rangle_{w}=\langle u_{[{\bf
g}]},u_{[{\bf g}^{\prime}]}\rangle_{s}$

for any two vertices $w,s\in\mathbb{T}_{n}$.

Since $[{\bf g}^{\prime}]$ is a tropical point in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$, we can write it as $[{\bf
g}^{\prime}]=\\{{\bf g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$. By Proposition 4.4, the collection $[{\bf a}]:=\\{{\bf
a}_{t}=\Lambda_{t}{\bf g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ is a tropical point in
$\mathcal{S}_{X}(\mathbb{Z}^{T})$. We write ${\bf a}_{t}=\Lambda_{t}{\bf
g}_{t}^{\prime}=(a_{1;t},\ldots,a_{m;t})^{T}$ for any $t\in\mathbb{T}_{n}$. We
know that the tropical point $[{\bf a}]\in\mathcal{S}_{X}(\mathbb{Z}^{T})$
corresponds to the semifield homorphism

$\nu:\mathcal{F}_{>0}:=\mathbb{Q}_{\rm
sf}(x_{1;t_{0}},\ldots,x_{m;t_{0}})\rightarrow\mathbb{Z}^{T}$

induced by $x_{j;t_{0}}\mapsto a_{j;t_{0}}$ for any $j\in[1,m]$. We also know
that the coordinate vector of the tropical point
$\nu\in\operatorname{\mathsf{Hom}}_{\rm sf}(\mathcal{F}_{>0},\mathbb{Z}^{T})$
under any chart ${\bf x}_{t}$ is the vector ${\bf a}_{t}=\Lambda_{t}{\bf
g}_{t}^{\prime}=(a_{1;t},\ldots,a_{m;t})^{T}$. Namely, we have
$\nu(x_{j;t})=a_{j;t}$ for any $j\in[1,m]$.

By Proposition 3.2 (ii), we have the following commutative diagram.

---
$\textstyle{\mathcal{F}_{>0;w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\bf
a}_{w};w}}$$\textstyle{\mathcal{F}_{>0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{s}}$$\scriptstyle{\nu}$$\scriptstyle{\varphi_{w}}$$\textstyle{\mathcal{F}_{>0;s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{{\bf
a}_{s};s}}$$\textstyle{\mathbb{Z}^{T}}$

where $\mathcal{F}_{>0;w}=\mathbb{Q}_{\rm sf}(x_{1;w},\ldots,x_{m;w})$, and
$\varphi_{w}:\mathcal{F}_{>0}\rightarrow\mathcal{F}_{>0;w}$ is the unique
semifield isomorphism induced by $x_{j;t_{0}}\mapsto x_{j;t_{0}}({\bf
x}_{t_{w}})$ (the expression of $x_{j;t_{0}}$ as an element in
$\mathcal{F}_{>0;w}$) for any $j\in[1,m]$, and $\pi_{{\bf
a}_{w};w}:\mathcal{F}_{>0;w}\rightarrow\mathbb{Z}^{T}$ is the unique semifield
homomorphism induced by $x_{j;w}\mapsto a_{j;w}$ for any $j\in[1,m]$.
Similarly, $\mathcal{F}_{>0;s},\varphi_{s}$ and $\pi_{{\bf a}_{s};s}$ are
defined. We have the equality

$\pi_{{\bf a}_{w};w}\circ\varphi_{w}=\nu=\pi_{{\bf a}_{s};s}\circ\varphi_{s}.$

Since ${\bf u}_{[{\bf g}]}$ is $[{\bf g}]$-good in $\mathcal{U}$, the Laurent
expansion of ${\bf u}_{[{\bf g}]}$ with respect to seed $({\bf
x}_{w},\widetilde{B}_{w})$ has the form

$u_{[{\bf g}]}={\bf x}_{w}^{{\bf
g}_{w}}F_{w}(\hat{y}_{1;w},\ldots,\hat{y}_{n;w}),$

where $F_{w}$ is a polynomial in $\mathbb{Z}_{\geq 0}[y_{1},\ldots,y_{n}]$
with constant term $1$ and $\hat{y}_{k;w}=\prod_{j=1}^{m}x_{j;w}^{b_{jk;w}}$.
Notice that $\varphi_{w}$ maps ${\bf u}_{[{\bf g}]}\in\mathcal{F}_{>0}$ to
${\bf x}_{w}^{{\bf g}_{w}}F_{w}(\hat{y}_{1;w},\ldots,\hat{y}_{n;w})$, the
expansion of ${\bf u}_{[{\bf g}]}$ using the chart ${\bf x}_{w}$. Hence,

$\nu(u_{[{\bf g}]})=\pi_{{\bf a}_{w};w}\circ\varphi_{w}(u_{[{\bf
g}]})=\pi_{{\bf a}_{w};w}({\bf x}_{w}^{{\bf
g}_{w}}F_{w}(\hat{y}_{1;w},\ldots,\hat{y}_{n;w})).$

We write $F_{w}=\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf v}}{\bf y}^{{\bf
v}}\in\mathbb{Z}_{\geq 0}[y_{1},\ldots,y_{n}]$. Thus
$F_{w}(\hat{y}_{1;w},\ldots,\hat{y}_{n;w})=\sum_{{\bf
v}\in\mathbb{N}^{n}}c_{{\bf v}}{\bf x}_{w}^{\widetilde{B}_{w}{\bf v}}$. Since
$\pi_{{\bf a}_{w};w}(x_{j;w})=a_{j;w}$ for any $j\in[1,m]$, i.e., $\pi_{{\bf
a}_{w};w}({\bf x}_{w})={\bf a}_{w}^{T}=(\Lambda_{w}{\bf g}_{w}^{\prime})^{T}$,
we have

$\displaystyle\pi_{{\bf a}_{w};w}({\bf x}_{w}^{{\bf
g}_{w}}F_{w}(\hat{y}_{1;w},\ldots,\hat{y}_{n;w}))$ $\displaystyle=$
$\displaystyle{\bf a}_{w}^{T}{\bf g}_{w}+\max\\{{\bf
a}_{w}^{T}\widetilde{B}_{w}{\bf v}\mid c_{\bf v}\neq 0\\}$ $\displaystyle=$
$\displaystyle{\bf g}_{w}^{T}{\bf a}_{w}+\max\\{{\bf
v}^{T}\tilde{B}_{w}^{T}{\bf a}_{w}\mid c_{\bf v}\neq 0\\}$ $\displaystyle=$
$\displaystyle{\bf g}_{w}^{T}\Lambda_{w}{\bf g}_{w}^{\prime}+\max\\{{\bf
v}^{T}\tilde{B}_{w}^{T}\Lambda_{w}{\bf g}_{w}^{\prime}\mid c_{\bf v}\neq 0\\}$
$\displaystyle=$ $\displaystyle{\bf g}_{w}^{T}\Lambda_{w}{\bf
g}_{w}^{\prime}+\max\\{{\bf v}^{T}(S\mid{\bf 0}){\bf g}_{w}^{\prime}\mid
c_{\bf v}\neq 0\\}$ $\displaystyle=$ $\displaystyle{\bf
g}_{w}^{T}\Lambda_{w}{\bf g}_{w}^{\prime}+F_{w}[(S\mid{\bf 0}){\bf
g}_{w}^{\prime}]$ $\displaystyle=$ $\displaystyle\langle u_{[{\bf
g}]},u_{[{\bf g}^{\prime}]}\rangle_{w}.$

Hence, $\nu(u_{[{\bf g}]})=\langle u_{[{\bf g}]},u_{[{\bf
g}^{\prime}]}\rangle_{w}$. By the same arguments, we have $\nu(u_{[{\bf
g}]})=\langle u_{[{\bf g}]},u_{[{\bf g}^{\prime}]}\rangle_{s}$. Hence,
$\langle u_{[{\bf g}]},u_{[{\bf g}^{\prime}]}\rangle_{w}=\langle u_{[{\bf
g}]},u_{[{\bf g}^{\prime}]}\rangle_{s}$. So the paring in (4.1) is independent
of the choice of $t\in\mathbb{T}_{n}$. ∎

######  Remark 4.9.

From the proof above, we see that the integer $\langle u_{[{\bf g}]},u_{[{\bf
g}^{\prime}]}\rangle_{t}$ can be interpreted as the evaluation of the positive
function $u_{[{\bf g}]}$ (on cluster $\mathscr{A}$-variety in the sense of
[FG-2009]) at the tropical point $[{\bf a}]:=\\{{\bf a}_{t}=\Lambda_{t}{\bf
g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ (associated to the
same cluster $\mathscr{A}$-variety) determined by $u_{[{\bf g}^{\prime}]}$.

######  Remark 4.10.

If we take $u_{[{\bf g}]}=x_{i;t}$ and $u_{[{\bf g}^{\prime}]}=x_{j;t}$, then
by definition, we have

$\langle u_{[{\bf g}]},u_{[{\bf g}^{\prime}]}\rangle=\langle
x_{i;t},x_{j;t}\rangle=\langle x_{i;t},x_{j;t}\rangle_{t}={\bf
e}_{i}^{T}\Lambda_{t}{\bf e}_{j}+0=\lambda_{ij;t},$

where ${\bf e}_{k}$ is the $k$th column of $I_{m}$ for $k\in[1,m]$ and
$\lambda_{ij;t}$ is the $(i,j)$-entry of $\Lambda_{t}$. So the invariant
$\langle x_{i;t},x_{j;t}\rangle$ is closely related with the quantization
matrix $\Lambda_{t}$. This is analogous to [kkko-2018, Proposition 7.1.2 (a)]
for the $\Lambda$-invariant $\Lambda(M,N)$ in the monoidal categorification of
cluster algebras.

######  Proposition 4.11.

Let $x$ be a cluster variable of $\mathcal{U}$, say $x=x_{i;t}$ for some
$i\in[1,m]$ and $t\in\mathbb{T}_{n}$. Let $u$ be a $[{\bf g}]$-good element in
$\mathcal{U}$ and ${\bf f}_{t}=(f_{1;t},\ldots,f_{n;t})^{T}$ the $f$-vector of
$u$ with respect to vertex $t$. Then

$(x||u)_{F}=\begin{cases}s_{i}f_{i;t},&i\in[1,n];\\\
0,&i\in[n+1,m],\end{cases}$

where $s_{i}$ is the $(i,i)$-entry of the diagonal matrix
$S=diag(s_{1},\ldots,s_{n})$.

###### Proof.

Since the $F$-invariant $(x\mid\mid u)_{F}$ between $x$ and $u$ is independent
of a choice of a cluster chart, we can use the chart ${\bf x}_{t}$ to
calculate it. Then the result follows from $x=x_{i;t}$ and the formula in
(4.5). ∎

The $F$-invariant can be viewed as an analog of the $E$-invariant introduced
by Derksen-Weyman-Zelevinsky [DWZ10] in the additive categorification of
cluster algebras and the $\mathfrak{d}$-invariant introduced by Kang-
Kashiwara-Kim-Oh [kkko-2018] in the monoidal categorification of cluster
algebras. We provide some reasons in Remark 4.12 and Remark 4.19.

######  Remark 4.12.

(i) By Proposition 4.11, we know that the $F$-invariant $(x||u)_{F}$ is
closely related with the components of $f$-vectors;

(ii) By the equality (1.1) in the introduction part, the $E$-invariant is
closely related with the extension dimension in the cluster categories. Such
dimensions are used to categorify the components of $f$-vectors;

(iii) The components of $f$-vectors measure the distance between the highest
degree of some $y_{k}$ and the lowest degree of $y_{k}$ in $F$-polynomials,
while the $\mathfrak{d}$-invariant has the similar meaning, cf., [kkko-2018,
Corollary 4.2.4] and [qin-2020, Lemma 5.1].

######  Remark 4.13.

(i) The $F$-invariant
$(-\mid\mid-)_{F}:\mathcal{U}^{\operatorname{\mathsf{good}}}\times\mathcal{U}^{\operatorname{\mathsf{good}}}\rightarrow\mathbb{Z}_{\geq
0}$ is a great generalization of the $f$-compatibility degree introduced by
Fu-Gyoda in [Fu-Gyoda]. The $f$-compatibility degree, defined using components
of $f$-vectors of cluster variables, is a map

$(-||-)_{f}:\mathfrak{X}\times\mathfrak{X}\rightarrow\mathbb{Z}_{\geq 0},$

where $\mathfrak{X}$ is the set of unfrozen cluster variables in
$\mathcal{U}$. When restriced to $\mathfrak{X}$, the $F$-invariant and the
$f$-compatibility degree are related as follows:

$(x_{i;t}\mid\mid x_{j;t^{\prime}})_{F}=s_{i}(x_{i;t}\mid\mid
x_{j;t^{\prime}})_{f}.$

(ii) By construction, $F$-invariant is symmetric, i.e., $(u_{[{\bf
g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}=(u_{[{\bf g}^{\prime}]}\mid\mid
u_{[{\bf g}]})_{F}$ for any two good elements in $\mathcal{U}$. Applying to
the case $u_{[{\bf g}]}=x_{i;t}$ and $u_{[{\bf g}^{\prime}]}=x_{j;t^{\prime}}$
with $i,j\in[1,n]$, we get

$s_{i}(x_{i;t}\mid\mid x_{j;t^{\prime}})_{f}=(x_{i;t}\mid\mid
x_{j;t^{\prime}})_{F}=(x_{j;t^{\prime}}\mid\mid
x_{i;t})_{F}=s_{j}(x_{j;t^{\prime}}\mid\mid x_{i;t})_{f},$

which recovers the result in [Fu-Gyoda, Proposition 4.14 (2)].

######  Corollary 4.14.

Let $u$ be a $[{\bf g}]$-good element in $\mathcal{U}$ and ${\bf x}_{t}$ a
cluster of $\mathcal{U}$. The following statements hold.

  * (i)

If $(x_{k;t}\mid\mid u)_{F}=0$ for some $k\in[1,n]$, then the $k$th component
$g_{k;t}$ of ${\bf g}_{t}=\operatorname{\mathsf{deg}}^{t}(u)$ is non-negative.

  * (ii)

If $(x_{i;t}\mid\mid u)_{F}=0$ holds for any $i\in[1,n]$, then $u$ is a
cluster monomial in ${\bf x}_{t}$.

  * (iii)

Let $k$ be an integer in $[1,n]$ and let $({\bf
x}_{t^{\prime}},\widetilde{B}_{t^{\prime}})=\mu_{k}({\bf
x}_{t},\widetilde{B}_{t})$. Suppose that $(x_{k;t}\mid\mid u)_{F}\neq 0$ but
$(x_{i;t}\mid\mid u)_{F}=0$ for any $i\in[1,n]\backslash\\{k\\}$. Then
$(x_{i;t^{\prime}}\mid\mid u)_{F}=0$ for any $i\in[1,n]$. In particular, $u$
is a cluster monomial in ${\bf x}_{t^{\prime}}$.

###### Proof.

The results follow from Proposition 4.11 and Corollary 3.12. ∎

######  Proposition 4.15.

Let $u$ be a $[{\bf g}]$-bigood element in $\mathcal{U}$, and let ${\bf
x}_{t}^{\bf h}=\prod_{j=1}^{m}x_{j;t}^{h_{j}}$ be a cluster monomial of
$\mathcal{U}$. Then

$(u\mid\mid{\bf x}_{t}^{\bf h})_{F}=\sum_{j=1}^{n}h_{j}\cdot(u\mid\mid
x_{j;t})_{F}.$

###### Proof.

Since the $F$-invariant $(u\mid\mid{\bf x}_{t}^{\bf h})_{F}$ between $u$ and
${\bf x}_{t}^{\bf h}$ is independent of a choice of a cluster chart, we can
use the chart ${\bf x}_{t}$ to calculate it. Because $u$ is $[{\bf g}]$-bigood
in $\mathcal{U}$, the expansion of $u$ with respect to seed $({\bf
x}_{t},\widetilde{B}_{t})$ has the form:

$u={\bf x}_{t}^{{\bf g}_{t}}F_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where ${\bf g}_{t}$ is the value of $[{\bf g}]$ at vertex $t$ and

$F_{t}=1+\text{middle terms }+y_{1}^{f_{1;t}}\ldots y_{n}^{f_{n;t}}$

is the $F$-polynomial of $u$ with respect to vertex $t$. Here ${\bf
f}_{t}=(f_{1;t},\ldots,f_{n;t})^{T}$ is the $f$-vector of $u$ with respect to
vertex $t$.

Since ${\bf x}_{t}^{\bf h}$ is a cluster momomial, we have
$(h_{1},\ldots,h_{n})\in\mathbb{Z}_{\geq 0}^{n}$. Then one obtains

$\displaystyle(u\mid\mid{\bf x}_{t}^{\bf h})_{F}$ $\displaystyle=$
$\displaystyle F_{t}[(S\mid{\bf 0}){\bf
h}]+0=F_{t}[(s_{1}h_{1},\ldots,s_{n}h_{n})^{T}]$ $\displaystyle=$
$\displaystyle
f_{1;t}s_{1}h_{1}+\ldots+f_{n;t}s_{n}h_{n}=\sum_{j=1}^{n}h_{j}\cdot(u\mid\mid
x_{j;t})_{F}.$

∎

######  Lemma 4.16.

Let $\\{z_{1},\ldots,z_{p}\\}$ be a subset of unfrozen cluster variables of
$\mathcal{U}$. If $(z_{i}\mid\mid z_{j})_{F}=0$ for any $i,j\in[1,p]$, then
$\\{z_{1},\ldots,z_{p}\\}$ is a subset of some cluster of $\mathcal{U}$.

###### Proof.

Since $(z_{i}\mid\mid z_{j})_{F}=0$ for any $i,j\in[1,p]$ and by Remark 4.13,
the $f$-compatibility degree $(z_{i}\mid\mid z_{j})_{f}$ is zero for any
$i,j\in[1,p]$. Then the result follows from [Fu-Gyoda, Theorem 4.18]. ∎

The following result says that $F$-invariant is a very concrete and easily
computable numerical invariant to distinguish whether the product of two
cluster monomials is still a cluster monomial.

######  Proposition 4.17.

Let $u$ and $u^{\prime}$ be two cluster monomials in $\mathcal{U}$ and $t$ a
vertex of $\mathbb{T}_{n}$. Let ${\bf g}_{t},F_{t}$ (respectively, ${\bf
g}_{t}^{\prime},F_{t}^{\prime}$) be the degree and $F$-polynomial of $u$
(respectively, $u^{\prime}$) with respect to vertex $t$. Then the product
$uu^{\prime}$ is still a cluster monomial in $\mathcal{U}$ if and only if

$F_{t}[(S\mid{\bf 0}){\bf g}_{t}^{\prime}]+F_{t}^{\prime}[(S\mid{\bf 0}){\bf
g}_{t}]=0,$

i.e., $(u\mid\mid u^{\prime})_{F}=0$.

###### Proof.

Suppose that the product $uu^{\prime}$ is still a cluster monomial in
$\mathcal{U}$. Say, $uu^{\prime}={\bf x}_{w}^{\bf h}$ for some vertex
$w\in\mathbb{T}_{n}$. Then both $u$ and $u^{\prime}$ can be viewed as cluster
monomials in ${\bf x}_{w}$. We use the chart ${\bf x}_{w}$ to calculate the
$F$-invariant $(u\mid\mid u^{\prime})_{F}$, then one easily gets $(u\mid\mid
u^{\prime})_{F}=0$. Since the $F$-invariant $(u\mid\mid u^{\prime})_{F}$ is
independent of the choice of a cluster chart, we get

$F_{t}[(S\mid{\bf 0}){\bf g}_{t}^{\prime}]+F_{t}^{\prime}[(S\mid{\bf 0}){\bf
g}_{t}]=(u\mid\mid u^{\prime})_{F}=0.$

Conversely, suppose $F_{t}[(S\mid{\bf 0}){\bf
g}_{t}^{\prime}]+F_{t}^{\prime}[(S\mid{\bf 0}){\bf g}_{t}]=(u\mid\mid
u^{\prime})_{F}=0$. We will show that $uu^{\prime}$ is a cluster monomial in
$\mathcal{U}$. Since $u$ and $u^{\prime}$ are cluster monomials in
$\mathcal{U}$, we can assume $u={\bf x}_{s}^{\bf h}$ and $u^{\prime}={\bf
x}_{s^{\prime}}^{\bf r}$, where $s,s^{\prime}\in\mathbb{T}_{n}$ and ${\bf
h}=(h_{1},\ldots,h_{m})^{T},{\bf r}=(r_{1},\ldots,r_{m})^{T}\in\mathbb{Z}^{m}$
with $h_{i},r_{i}\geq 0$ for any $i\in[1,n]$.

By Proposition 3.6 and Proposition 4.15, we have

$(u\mid\mid
u^{\prime})_{F}=\sum_{i=1}^{n}\sum_{j=1}^{n}h_{i}r_{j}(x_{i;s}\mid\mid
x_{j;s^{\prime}})_{F}.$

Since all the numbers $h_{i},r_{j}$ and $(x_{i;s}\mid\mid
x_{j;s^{\prime}})_{F}$ are non-negative, $i,j\in[1,n]$, the assumption
$(u\mid\mid u^{\prime})_{F}=0$ implies $h_{i}r_{j}(x_{i;s}\mid\mid
x_{j;s^{\prime}})_{F}=0$ for any $i,j\in[1,n]$. Hence, $(x_{i;s}\mid\mid
x_{j;s^{\prime}})_{F}=0$ for any $i,j\in[1,n]$ with $h_{i}r_{j}\neq 0$.

Let $V=\\{x_{i;s}\mid i\in[1,n],h_{i}\neq 0\\}$ and
$V^{\prime}=\\{x_{j;s^{\prime}}\mid j\in[1,n],r_{j}\neq 0\\}$. We have
$(z\mid\mid z^{\prime})_{F}=0$ for any $z\in V$ and $z^{\prime}\in
V^{\prime}$. Since $V$ and $V^{\prime}$ are subsets of clusters of
$\mathcal{U}$, we have

$(z_{1}\mid\mid z_{2})_{F}=0\;\;\;\text{and}\;\;\;(z_{1}^{\prime}\mid\mid
z_{2}^{\prime})_{F}=0$

for any $z_{1},z_{2}\in V$ and $z_{1}^{\prime},z_{2}^{\prime}\in V^{\prime}$.

Hence, $(z\mid\mid z^{\prime})_{F}=0$ holds for any two cluster variables
$z,z^{\prime}$ in $V\cup V^{\prime}$. Then by Lemma 4.16, we know that $V\cup
V^{\prime}$ is a subset of some cluster ${\bf x}_{w}$ of $\mathcal{U}$. Hence,
the product $uu^{\prime}$ is a cluster monomial in ${\bf x}_{w}$.

∎

######  Corollary 4.18.

Let $u$ be a cluster monomial in $\mathcal{U}$. Then $(u\mid\mid u)_{F}=0$.

###### Proof.

If $u$ is a cluster monomial, so is $u^{2}$. Then the result follows from
Proposition 4.17. ∎

######  Remark 4.19.

(i) The corollary above is analogous to [DWZ10, Corollary 7.2] for the
$E$-invariant in the additive categorification of cluster algebras and to
[kkko-2018, Lemma 3.2.3] for the $\mathfrak{d}$-invariant in the monoidal
categorification of cluster algebras.

(ii) The good elements satisfying $(u\mid\mid u)_{F}=0$ are analogous to the
rigid objects in cluster categories and to the real simples in the monoidal
categorification of cluster algebras.

Recall that the Newton polytope of a polynomial $F=\sum_{{\bf
v}\in\mathbb{N}^{n}}c_{{\bf v}}{\bf y}^{{\bf
v}}\in\mathbb{Z}[y_{1},\ldots,y_{n}]$ is the convex hull of $\\{{\bf
v}\in\mathbb{N}^{n}\mid c_{\bf v}\neq 0\\}$ in $\mathbb{R}^{n}$.

######  Corollary 4.20.

Let $u$ and $u^{\prime}$ be two cluster monomials in $\mathcal{U}$ and $t$ a
vertex of $\mathbb{T}_{n}$. Let ${\bf g}_{t},F_{t}$ (respectively, ${\bf
g}_{t}^{\prime},F_{t}^{\prime}$) be the degree and $F$-polynomial of $u$
(respectively, $u^{\prime}$) with respect to vertex $t$. If $F_{t}$ and
$F_{t}^{\prime}$ have the same Newton polytope, then the product $uu^{\prime}$
is still a cluster monomial.

###### Proof.

Since $u$ and $u^{\prime}$ are cluster monomials and by Corollary 4.18, we
have

$\displaystyle(u\mid\mid u)_{F}$ $\displaystyle=$ $\displaystyle
2F_{t}[(S\mid{\bf 0}){\bf g}_{t}]=0,$ $\displaystyle(u^{\prime}\mid\mid
u^{\prime})_{F}$ $\displaystyle=$ $\displaystyle 2F_{t}^{\prime}[(S\mid{\bf
0}){\bf g}_{t}^{\prime}]=0.$

We get $F_{t}[(S\mid{\bf 0}){\bf g}_{t}]=0=F_{t}^{\prime}[(S\mid{\bf 0}){\bf
g}_{t}^{\prime}]$. Since $F_{t}$ and $F_{t}^{\prime}$ have the same Newton
polytope, we have $F_{t}[{\bf h}]=F_{t}^{\prime}[{\bf h}]$ for any ${\bf
h}\in\mathbb{Z}^{n}$. Hence, we have

$\displaystyle(u\mid\mid u^{\prime})_{F}$ $\displaystyle=$ $\displaystyle
F_{t}[(S\mid{\bf 0}){\bf g}_{t}^{\prime}]+F_{t}^{\prime}[(S\mid{\bf 0}){\bf
g}_{t}]$ $\displaystyle=$ $\displaystyle F_{t}^{\prime}[(S\mid{\bf 0}){\bf
g}_{t}^{\prime}]+F_{t}[(S\mid{\bf 0}){\bf g}_{t}]=0.$

Then by Proposition 4.17, the product $uu^{\prime}$ is still a cluster
monomial. ∎

Corollary 4.18 says that $(u\mid\mid u)_{F}=0$ for any cluster monomial $u$.
In general, there exist good elements $u$ in $\mathcal{U}$ such that
$(u\mid\mid u)_{F}=0$ but $u$ is not a cluster monomial.

######  Proposition 4.21.

Let $[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ be a
tropical point in $S_{Y}(\mathbb{Z}^{T})$ and $u$ a $[{\bf g}]$-good element
in $\mathcal{U}$. Suppose there exists some vertex $w\in\mathbb{T}_{n}$ such
that the principal part $(g_{1;w},\ldots,g_{n;w})^{T}$ of ${\bf g}_{w}$ lies
in $\mathbb{Z}_{\leq 0}^{n}$. Then $(u\mid\mid u)_{F}=0$.

###### Proof.

Using the chart ${\bf x}_{w}$ to calculate $(u\mid\mid u)_{F}$, we have

$(u\mid\mid u)_{F}=F_{w}[(S\mid{\bf 0}){\bf g}_{w}]+F_{w}[(S\mid{\bf 0}){\bf
g}_{w}].$

Since the principal part $(g_{1;w},\ldots,g_{n;w})^{T}$ of ${\bf g}_{w}$ lies
in $\mathbb{Z}_{\leq 0}^{n}$, we have ${\bf v}^{T}(S\mid{\bf 0}){\bf
g}_{w}\leq 0$ for any ${\bf v}\in\mathbb{N}^{n}$. Hence, $(u\mid\mid
u)_{F}=2F_{w}[(S\mid{\bf 0}){\bf g}_{w}]=0$. ∎

Using the above proposition, one can construct examples such that $(u\mid\mid
u)_{F}=0$ but $u$ is not a cluster monomial.

######  Example 4.22.

Now we take $\mathcal{U}$ to be the Markov cluster algebra with principal
coefficients. Namely, the initial mutation matrix $\widetilde{B}_{t_{0}}$ is
given by $\widetilde{B}_{t_{0}}=\begin{bmatrix}B_{t_{0}}\\\
I_{3}\end{bmatrix}$, where

$B_{t_{0}}=\begin{bmatrix}0&2&-2\\\ -2&0&2\\\ 2&-2&0\end{bmatrix}.$

Now we consider the tropical point $[{\bf g}]=\\{{\bf
g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ determined by ${\bf
g}_{t_{0}}=(-1,-1,-1,0,0,0)^{T}$. By the results in [zhou-2020], we know that
the theta function $\theta_{[{\bf g}]}$ associated to $[{\bf g}]$ (constructed
in [GHKK18]) is contained in $\mathcal{U}$. The general properties of theta
functions imply that $\theta_{[{\bf g}]}$ is $[{\bf g}]$-good. By Proposition
4.21, we have $(\theta_{[{\bf g}]}\mid\mid\theta_{[{\bf g}]})_{F}=0$.

On the other hand, by [Plamondon_2013, Example 4.3], we know that there exist
no cluster monomials $u$ such that
$\operatorname{\mathsf{deg}}^{t_{0}}(u)={\bf g}_{t_{0}}=(-1,-1,-1,0,0,0)^{T}$.
Hence, $\theta_{[{\bf g}]}$ satisfies $(\theta_{[{\bf
g}]}\mid\mid\theta_{[{\bf g}]})_{F}=0$ but it is not a cluster monomial in
$\mathcal{U}$.

######  Definition 4.23.

Two tropical points $[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ and $[{\bf g}^{\prime}]=\\{{\bf
g}^{\prime}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$ are said to be compatible, if for any vertex
$t\in\mathbb{T}_{n}$ and any $k\in[1,n]$, we have $g_{k;t}\cdot
g_{k;t}^{\prime}\geq 0$, where $g_{k;t}$ and $g_{k;t^{\prime}}$ are $k$th
components of ${\bf g}_{t}$ and ${\bf g}_{t}^{\prime}$.

######  Lemma 4.24.

Let $[{\bf g}]=\\{{\bf g}_{t}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ and
$[{\bf g}^{\prime}]=\\{{\bf g}^{\prime}_{t}\in\mathbb{Z}^{m}\mid
t\in\mathbb{T}_{n}\\}$ be two tropical points in
$\mathcal{S}_{Y}(\mathbb{Z}^{T})$. If $[{\bf g}]$ and $[{\bf g}^{\prime}]$ are
compatible, then the collection $\\{{\bf g}_{t}+{\bf
g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ is still a
tropical point in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$. We denote it by $[{\bf
g}\uplus{\bf g}^{\prime}]$.

###### Proof.

It suffices to show that the collection $\\{{\bf g}_{t}+{\bf
g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ satisfies the
relations in (3.1). This is clear. ∎

######  Theorem 4.25.

Let $u_{[{\bf g}]}$ be a $[{\bf g}]$-good element in $\mathcal{U}$ and
$u_{[{\bf g}^{\prime}]}$ a $[{\bf g}^{\prime}]$-good element in $\mathcal{U}$.
Suppose that $(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}=0$. Then the
following statements hold.

  * (i)

The two tropical points $[{\bf g}]$ and $[{\bf g}^{\prime}]$ are compatible.
In particular, the tropical point $[{\bf g}\uplus{\bf g}^{\prime}]$ can be
defined.

  * (ii)

The product $u_{[{\bf g}]}u_{[{\bf g}^{\prime}]}$ is a $[{\bf g}\uplus{\bf
g}^{\prime}]$-good element in $\mathcal{U}$.

###### Proof.

(i) We write ${\bf g}_{t}=(g_{1;t},\ldots,g_{m;t})^{T}$ and ${\bf
g}_{t}^{\prime}=(g_{1;t}^{\prime},\ldots,g_{m;t}^{\prime})^{T}$ for the values
of the tropical points $[{\bf g}]$ and $[{\bf g}^{\prime}]$ at vertex
$t\in\mathbb{T}_{n}$. Assume by contradiction that the two tropical points
$[{\bf g}]$ and $[{\bf g}^{\prime}]$ are not compatible. Then there exist some
vertex $s\in\mathbb{T}_{n}$ and some $k\in[1,n]$ such that $g_{k;s}\cdot
g_{k;s}^{\prime}<0$. Without loss of generality, we can assume $g_{k;s}<0$ and
$g_{k;s}^{\prime}>0$. Denote by $F_{s},F_{s}^{\prime}\in\mathbb{Z}_{\geq
0}[y_{1},\ldots,y_{n}]$ the $F$-polynomials of $u_{[{\bf g}]}$ and $u_{[{\bf
g}^{\prime}]}$ with respect to vertex $s$.

By Corollary 3.11, we know that the monomial ${\bf y}^{[-g_{k;s}]_{+}{\bf
e}_{k}}=y_{k}^{[-g_{k;s}]_{+}}$ appears in $F_{s}$ and it has coefficient $1$.
Thus we have

$F_{s}[(S\mid{\bf 0}){\bf g}_{s}^{\prime}]\geq([-g_{k;s}]_{+}{\bf
e}_{k})^{T}(S\mid{\bf 0}){\bf g}_{s}^{\prime}=[-g_{k;s}]_{+}\cdot
s_{k}g_{k;s}^{\prime}.$

Then by $g_{k;s}<0$, $g_{k;s}^{\prime}>0$ and $s_{k}>0$, we get
$F_{s}[(S\mid{\bf 0}){\bf g}_{s}^{\prime}]>0$.

Using the chart ${\bf x}_{s}$ to calculate the $F$-invariant $(u_{[{\bf
g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}$, we have

$(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}=F_{s}[(S\mid{\bf 0}){\bf
g}_{s}^{\prime}]+F_{s}^{\prime}[(S\mid{\bf 0}){\bf g}_{s}]\geq
F_{s}[(S\mid{\bf 0}){\bf g}_{s}^{\prime}]>0,$

which contradicts $(u_{[{\bf g}]}\mid\mid u_{[{\bf g}^{\prime}]})_{F}=0$.
Hence, the two tropical points $[{\bf g}]$ and $[{\bf g}^{\prime}]$ must be
compatible.

(ii) For any vertex $t$, the Laurent expansion of $u_{[{\bf g}]}$ and
$u_{[{\bf g}]}$ with respect to vertex $t$ take the form:

$u_{[{\bf g}]}={\bf x}_{t}^{{\bf
g}_{t}}F_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),\;\;\;u_{[{\bf
g}^{\prime}]}={\bf x}_{t}^{{\bf
g}_{t}^{\prime}}F_{t}^{\prime}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where $F_{t}$ and $F_{t}^{\prime}$ are polynomials in $\mathbb{Z}_{\geq
0}[y_{1},\ldots,y_{n}]$ with constant term $1$. Hence, we have

$u_{[{\bf g}]}u_{[{\bf g}^{\prime}]}={\bf x}_{t}^{{\bf g}_{t}+{\bf
g}_{t}^{\prime}}H_{t}(\hat{y}_{1;t},\ldots,\hat{y}_{n;t}),$

where $H_{t}=F_{t}F_{t}^{\prime}$ is a polynomial in $\mathbb{Z}_{\geq
0}[y_{1},\ldots,y_{n}]$ with constant term $1$.

By (i) and Lemma 4.24, we know that $[{\bf g}\uplus{\bf g}^{\prime}]:=\\{{\bf
g}_{t}+{\bf g}_{t}^{\prime}\in\mathbb{Z}^{m}\mid t\in\mathbb{T}_{n}\\}$ forms
a tropical point in $\mathcal{S}_{Y}(\mathbb{Z}^{T})$. Then by the definition
of good elements, we know that the product $u_{[{\bf g}]}u_{[{\bf
g}^{\prime}]}$ is a $[{\bf g}\uplus{\bf g}^{\prime}]$-good element in
$\mathcal{U}$. ∎

##  5\. Log-canonicality and $F$-compatibility

In this section, we discuss the link between the compatible Poisson structure
of (upper) cluster algebra and the $F$-compatibility.

As before, we fix a Langlands-Poisson triple
$(\mathcal{S}_{X},\mathcal{S}_{Y},\Lambda)$, and let $\mathcal{U}$ be the
upper cluster algebra associated to the cluster pattern $\mathcal{S}_{X}$.
Denote by $S=diag(s_{1},\ldots,s_{n})$ the type of the compatible pair
$(\widetilde{B}_{t_{0}},\Lambda_{t_{0}})$, namely,
$\widetilde{B}_{t_{0}}^{T}\Lambda_{t_{0}}=(S\mid{\bf 0})$, where ${\bf 0}$ is
the $n\times(m-n)$ zero matrix.

Recall that we can define a Poisson bracket $\\{-,-\\}$ on the ambient field
$\mathbb{F}$ using the $m\times m$ skew-symmetric matrix
$\Lambda_{t_{0}}=(\lambda_{ij;t_{0}})$ at the rooted vertex $t_{0}$:

$\\{x_{i;t_{0}},x_{j;t_{0}}\\}:=\lambda_{ij;t_{0}}\cdot
x_{i;t_{0}}x_{j;t_{0}}.$

This Poisson bracket is compatible with the cluster pattern $\mathcal{S}_{X}$,
that is, for any cluster ${\bf x}_{t}$ of $\mathcal{S}_{X}$, we have

$\\{x_{i;t},x_{j;t}\\}=\lambda_{ij;t}\cdot x_{i;t}x_{j;t},$

where $\lambda_{ij;t}$ is the $(i,j)$-entry of $\Lambda_{t}$, cf. [gsv-2003],
[bz-2005, Remark 4.6].

Two elements $f_{1},f_{2}$ in $\mathbb{F}$ are said to be log-canonical with
respect to the Poisson bracket $\\{-,-\\}$, if

$\\{f_{1},f_{2}\\}=cf_{1}f_{2}$

for some $c\in\mathbb{Q}$.

Clearly, any two cluster variables in the same cluster are log-canonical with
respect to $\\{-,-\\}$. It is natural to ask whether the converse is true. The
answer is yes. See Theorem 5.2 (ii).

######  Lemma 5.1.

Let $w$ be a vertex of $\mathbb{T}_{n}$ and $u$ an element in $\mathbb{F}$
with the form

$u={\bf x}_{w}^{\bf g}F(\hat{y}_{1;w},\ldots,\hat{y}_{n;w}),$

where $F=\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf v}}{\bf y}^{{\bf v}}$ is a
polynomial in $\mathbb{Z}[y_{1},\ldots,y_{n}]$ with constant term $1$. For
$i\in[1,n]$, denote by $f_{i;w}$ the highest degree of $y_{i}$ in $F$. If $u$
and $x_{k;w}$ are log-canonical with respect to $\\{-,-\\}$ for some
$k\in[1,n]$, then $f_{k;w}=0$.

###### Proof.

We know that $\Lambda_{w}=(\lambda_{ij;w})$ is the coefficient matrix of
$\\{-,-\\}$ with respect to the cluster ${\bf x}_{w}$. Since
$(\widetilde{B}_{w},\Lambda_{w})$ is a compatible pair of type $S$, we have
$(\widetilde{B}_{w})^{T}\Lambda_{w}=(S\mid{\bf 0})$. For $i\in[1,n]$ and
$j\in[1,m]$, we denote by ${\bf e}_{i}$ the $i$th column of $I_{n}$ and ${\bf
e}_{j}^{\prime}$ the $j$th column of $I_{m}$. We have

$\displaystyle\\{\hat{y}_{i;w},x_{j;w}\\}$ $\displaystyle=$
$\displaystyle\\{{\bf x}_{w}^{\widetilde{B}_{w}{\bf e}_{i}},{\bf x}_{w}^{{\bf
e}_{j}^{\prime}}\\}=\left((\widetilde{B}_{w}{\bf e}_{i})^{T}\Lambda_{w}{\bf
e}_{j}^{\prime}\right)\hat{y}_{i;w}x_{j;w}$ $\displaystyle=$
$\displaystyle\left({\bf e}_{i}^{T}(\widetilde{B}_{w})^{T}\Lambda_{w}{\bf
e}_{j}^{\prime}\right)\hat{y}_{i;w}x_{j;w}=\left({\bf e}_{i}^{T}(S\mid{\bf
0}){\bf e}_{j}^{\prime}\right)\hat{y}_{i;w}x_{j;w}$ $\displaystyle=$
$\displaystyle\begin{cases}s_{j}\cdot\hat{y}_{j;w}x_{j;w},&i=j;\\\ 0,&i\neq
j.\end{cases}$

For ${\bf v}=(v_{1},\ldots,v_{n})^{T}\in\mathbb{N}^{n}$, we have

$\\{\widehat{\bf y}_{w}^{\bf v},x_{k;w}\\}=s_{k}(v_{k}\widehat{\bf y}_{w}^{\bf
v})x_{k;w}.$

For simplicity, we write $F(\widehat{\bf
y}_{w})=F(\hat{y}_{1;w},\ldots,\hat{y}_{n;w})$. We have

$\\{F(\widehat{\bf y}_{w}),x_{k;w}\\}=\\{\sum_{{\bf
v}\in\mathbb{N}^{n}}c_{{\bf v}}\widehat{\bf y}_{w}^{{\bf
v}},x_{k;w}\\}=s_{k}(\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf
v}}v_{k}\widehat{\bf y}_{w}^{\bf v})x_{k;w}.$

Now let us calculate $\\{u,x_{k;w}\\}=\\{{\bf x}_{w}^{\bf g}F(\widehat{\bf
y}_{w}),x_{k;w}\\}$.

$\displaystyle\\{{\bf x}_{w}^{\bf g}F(\widehat{\bf y}_{w}),x_{k;w}\\}$
$\displaystyle=$ $\displaystyle\\{{\bf x}_{w}^{\bf g},x_{k;w}\\}F(\widehat{\bf
y}_{w})+{\bf x}_{w}^{\bf g}\\{F(\widehat{\bf y}_{w}),x_{k;w}\\}$
$\displaystyle=$ $\displaystyle({\bf g}^{T}\Lambda_{w}{\bf
e}_{k}^{\prime})\cdot{\bf x}_{w}^{\bf g}\cdot x_{k;w}\cdot F(\widehat{\bf
y}_{w})+{\bf x}_{w}^{\bf g}\cdot s_{k}(\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf
v}}v_{k}\widehat{\bf y}_{w}^{\bf v})x_{k;w}.$

Since $u$ and $x_{k;w}$ are log-canonical with respect to $\\{-,-\\}$, there
exists some $c\in\mathbb{Q}$ such that

$\\{u,x_{k;w}\\}=c\cdot ux_{k;w}=c\cdot{\bf x}_{w}^{\bf g}\cdot F(\widehat{\bf
y}_{w})\cdot x_{k;w}.$

Thus we have the following equality in $\mathbb{F}$.

$({\bf g}^{T}\Lambda_{w}{\bf e}_{k}^{\prime})\cdot{\bf x}_{w}^{\bf g}\cdot
x_{k;w}\cdot F(\widehat{\bf y}_{w})+{\bf x}_{w}^{\bf g}\cdot s_{k}(\sum_{{\bf
v}\in\mathbb{N}^{n}}c_{{\bf v}}v_{k}\widehat{\bf y}_{w}^{\bf
v})x_{k;w}=c\cdot{\bf x}_{w}^{\bf g}\cdot F(\widehat{\bf y}_{w})\cdot
x_{k;w}.$

We get

(5.1) $\displaystyle s_{k}(\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf
v}}v_{k}\widehat{\bf y}_{w}^{\bf v})=(c-{\bf g}^{T}\Lambda_{w}{\bf
e}_{k}^{\prime})\cdot F(\widehat{\bf y}_{w}).$

Assume by contradiction that the highest degree $f_{k;w}$ of $y_{k}$ in
$F=\sum_{{\bf v}\in\mathbb{N}^{n}}c_{{\bf v}}{\bf y}^{{\bf v}}$ is non-zero.
Then the left side of (5.1) is non-zero, which implies $c-{\bf
g}^{T}\Lambda_{w}{\bf e}_{k}^{\prime}\neq 0$. Now we compare the constant term
on both sides of (5.1). The constant term on the left side is $0$ while the
constant term on the right side is $(c-{\bf g}^{T}\Lambda_{w}{\bf
e}_{k}^{\prime})\cdot 1=c-{\bf g}^{T}\Lambda_{w}{\bf e}_{k}^{\prime}\neq 0$.
This is a contradiction. Hence, $f_{k;w}=0$. ∎

######  Theorem 5.2.

Let $u$ be a $[{\bf g}]$-good element in $\mathcal{U}$ and $w$ a vertex of
$\mathbb{T}_{n}$. The following statements hold.

  * (i)

If $u$ and $x_{k;w}$ are log-canonical with respect to $\\{-,-\\}$ for some
$k\in[1,n]$, then their $F$-invariant $(u\mid\mid x_{k;w})_{F}$ is zero;

  * (ii)

If $u$ is a cluster monomial and it is log-canonical with $x_{k;w}$, then the
product $u\cdot x_{k;w}$ is still a cluster monomial;

  * (iii)

If $u$ and $x_{k;w}$ are log-canonical with respect to $\\{-,-\\}$ for any
$k\in[1,n]$, then $u$ is a cluster monomial in ${\bf x}_{w}$;

  * (iv)

Let $k$ be an integer in $[1,n]$ and let $({\bf
x}_{t},\widetilde{B}_{t})=\mu_{k}({\bf x}_{w},\widetilde{B}_{w})$. Suppose
that $u$ and $x_{k;w}$ are not log-canonical but $u$ and $x_{i;w}$ are log-
canonical for any $i\in[1,n]\backslash\\{k\\}$, then $u$ is a cluster monomial
in ${\bf x}_{t}$.

###### Proof.

(i) This follows from Lemma 5.1 and Proposition 4.11.

(ii) By (i), we have $(u\mid\mid x_{k;w})_{F}=0$. Since $u$ is a cluster
monomial and by Proposition 4.17, we obtain that the $u\cdot x_{k;w}$ is still
a cluster monomial.

(iii) and (iv) follow from Lemma 5.1 and Corollary 3.12. ∎

## Acknowledgements

I would like to thank Antoine De Saint Germain, Professor Jiang-Hua Lu, and
Jie Pan for their helpful discussions and thank Professor Bernhard Keller for
his email exchanges. I am grateful to Jiarui Fei for his valuable feedback on
a preliminary version of this manuscript and for telling me that the same
(equivalent) invariant will be used and play an important role in his upcoming
paper.

This project is partially supported by the National Natural Science Foundation
of China Grant No. 12071422, and the Guangdong Basic and Applied Basic
Research Foundation Grant No. 2021A1515012035.

## References