††Project supported by the National Natural Science Foundation of China
(No.12071422 and No.12131015).

#  On Galois theory of cluster algebras: general and that from Riemann
surfaces

Jinlei Dong and Fang Li  Jinlei Dong
School of Mathematical Sciences, Zhejiang University, Yuhangtang Road 866,
Hangzhou, Zhejiang 310058, China P.R<EMAIL_ADDRESS>Fang Li
School of Mathematical Sciences, Zhejiang University, Yuhangtang Road 866,
Hangzhou, Zhejiang 310058, China P.R<EMAIL_ADDRESS>
###### Abstract.

One of the key points in Galois theory via field extensions is to build up a
correspondence between subfields of a field and subgroups of its automorphism
group, so as to study fields via methods of groups. As an analogue of the
Galois theory, we want to discuss the relations between cluster subalgebras of
a cluster algebra and subgroups of its automorphism group and then set up the
Galois-like method.

In the first part, we build up a Galois map from a skew-symmetrizable cluster
algebra $\mathcal{A}$ to its cluster automorphism group, and introduce notions
of Galois-like extensions and Galois extensions. A necessary condition for
Galois extensions of a cluster algebra $\mathcal{A}$ is given, which is also a
sufficient condition if $\mathcal{A}$ has a $\mathcal{D}$-stable basis or
stable monomial basis with unique expression. Some properties for Galois-like
extensions are discussed. It is shown that two subgroups $H_{1}$ and $H_{2}$
of the automorphism group $\text{Aut}\mathcal{A}$ are conjugate to each other
if and only if there exists $f\in\text{Aut}\mathcal{A}$ and two Galois-like
extension subalgebras $\mathcal{A}(\Sigma_{1})$, $\mathcal{A}(\Sigma_{2})$
corresponding to $H_{1}$ and $H_{2}$ such that $f$ is an isomorphism between
$\mathcal{A}(\Sigma_{1})$ and $\mathcal{A}(\Sigma_{2})$.

In the second part, as the answers of two conjectures proposed in the first
part, for a cluster algebra from a feasible surface, we prove that Galois-like
extension subalgebras corresponding to a subgroup of a cluster automorphism
group have the same rank. Moreover, it is shown that there are order-
preserving reverse Galois maps for these cluster algebras. We also give
examples of $\mathcal{D}$-stable bases and some discussions on the Galois
inverse problem in this part.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on cluster algebras and Riemann oriented surfaces
  3. 3 Cluster automorphism groups, extensions and $\mathcal{D}$-stable bases for cluster algebras
    1. 3.1 Galois group, Galois map and cluster automorphism groups embedded into $S_{n}\rtimes\mathbb{Z}_{2}$
    2. 3.2 Galois-like extensions, conjugate of subgroups and some conjectures
    3. 3.3 Galois extensions, stable monomial bases and $\mathcal{D}$-stable bases for cluster algebras
  4. 4 Examples of $\mathcal{D}$-stable bases and Galois theory of cluster algebras from surfaces
    1. 4.1 Bangle, bracelet, and band bases as $\mathcal{D}$-stable bases
    2. 4.2 Reverse Galois maps and the Galois inverse problem for cluster algebras from feasible surfaces

##  1\. Introduction

From Lagrange and Gauss to Abel and Galois, people have gradually come to
understand symmetry as the code used by the creator to design the laws of all
things.

The framework of the Galois theory was established by Galois via solving the
problem of root solutions for higher order equations.

The usefulness of Galois groups provides the invariant under the meaning of
symmetry in the structure of fields, which reflects the understanding of the
“laws” and is also the key to solving algebraic equations of higher order
through the distribution law of roots.

The meaning of the method of Galois lies in its ability to extend to almost
all mathematical objects. This is the significance of the automorphism group
of any mathematical object.

The specific operation method is achieved through the correspondence between
the sub-objects chains of mathematical objects and the subgroup chains of
automorphism groups.

This paper aims to apply the method to cluster algebras so as to consider the
symmetric invariant in the structure of cluster algebras.

Cluster algebras come from many fields of mathematics, such as canonical bases
and total positivity. It is a large class algebras that was axiomatically
defined and introduced by Fomin and Zelevinsky in [FZ02].

The cluster algebras in [FZ02, HL23] were defined recursively: Firstly, give a
seed consisting of exchange matrix and initial cluster variables, then
introduce a formula called mutation to generate new cluster variables, and all
cluster variables obtained by finite steps mutations generate an algebra
called cluster algebras.

In [FZ03] Fomin and Zelevinsky defined a strong cluster isomorphism between
the cluster algebras $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ which requires
any seed in $\mathcal{A}_{1}$ under the isomorphism to be a seed in
$\mathcal{A}_{2}$. And it can be easily proved that it is equivalent to the
condition that the isomorphism can commute with mutations in a sense in
[ASS14], which was called cluster isomorphism in [ASS14].

On the other hand, the author in [HLY18] gave a definition of the morphism
between cluster algebras $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ requiring it
commutes with mutations in a sense. When the morphism is isomorphic, the
definition is also cluster isomorphism mentioned above. Besides, if the
morphism is monomorphic, $\mathcal{A}_{1}$ is called a cluster subalgebra of
the cluster algebra $\mathcal{A}_{2}$.

The author in [HLY18] also gave a structure of all cluster subalgebras by
mixing-type sub-seed. As for the cluster automorphisms of cluster algebras,
the author in [ASS14] obtained a family of cluster automorphism groups using
the method of cluster categories and put forward a conjecture that these
groups of cluster algebras from surfaces are isomorphic to mapping class
groups, which has been proved in [Gu11, BS15, BQ15, BY18].

One of the key points in Galois theory via field extensions is to build up a
correspondence between subfields of a field and subgroups of its automorphism
group, so as to study fields via methods of groups. As an analogue of the
Galois theory, we want to discuss the relations between cluster subalgebras of
a cluster algebra and subgroups of its automorphism group and then set up the
Galois-like method.

In this paper, we try to build up the Galois theory of cluster algebras. We
construct a Galois map from cluster subalgebras of a cluster algebra to
subgroups of its Galois group and introduce the notions of Galois-like
extensions and Galois extensions. In addition, we give a sufficient and
necessary condition for Galois extensions. Finally, we discuss some properties
of Galois-like extensions and confirm some conjectures proposed in Section 3
for the cluster algebras coming from surfaces with certain conditions.
Moreover, we also discuss the Galois inverse problem for these cluster
algebras.

As a supplement, we also introduce the notion of complement sub-seed which
gives a way to embed the Galois group into a cluster automorphism group of a
cluster subalgebra, and offer examples for Galois maps.

The article is organized as follows:

In Section 2, we introduce definitions and some preparation knowledge of
cluster algebras, cluster morphisms, cluster subalgebras, and mapping class
groups of marked surfaces.

In Section 3, we firstly introduce the Galois map from cluster subalgebras of
a cluster algebra to their fixed subgroups which are called Galois group, and
introduce the notion of complement sub-seed which gives a way to embed the
Galois group into a cluster automorphism group of a cluster subalgebra.

Suppose $H_{1},H_{2}\in\mathit{2}^{\text{\em
Aut}\mathcal{A}}\backslash\text{ker}\phi$ for a cluster algebra $\mathcal{A}$
and $\mathcal{M}_{sub}^{H_{i}},i=1,2$ are their corresponding sets of Galois-
like extension subalgebras, we have the following theorem:

######  Theorem A (Theorem 3.15).

$H_{1}$ is conjugate to $H_{2}$ if and only if there exist $f\in\text{\em
Aut}\mathcal{A}$, $\mathcal{A}(\Sigma_{1})\in\mathcal{M}_{sub}^{H_{1}}$,
$\mathcal{A}(\Sigma_{2})\in\mathcal{M}_{sub}^{H_{2}}$ such that the
restriction of $f$ is an isomorphism between $\mathcal{A}(\Sigma_{1})$ and
$\mathcal{A}(\Sigma_{2})$.

Moreover, we introduce the notions of Galois-like extensions and Galois
extensions and give a necessary condition for Galois extensions of cluster
algebras, which is also a sufficient condition if cluster algebras have
$\mathcal{D}$-stable bases or stable monomial bases with unique expression.

######  Theorem B (Theorem 3.24).

Let $\mathcal{A}=\mathcal{A}(\Sigma)$ be a cluster algebra and $H$ be a
subgroup of the cluster automorphism group of $\mathcal{A}(\Sigma)$. Suppose
$\mathcal{A}(\Sigma_{1})$ is maximal in $\mathcal{A}^{H}$ as a cluster
subalgebra of $\mathcal{A}$.

(i) If $\mathcal{A}^{H}=\mathcal{A}(\Sigma_{1})$, then $|Hz|=+\infty$ for any
$z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ which can be represented as
Laurent polynomials of some cluster with positive coefficients.

(ii) Conversely, $\mathcal{A}^{H}=\mathcal{A}(\Sigma_{1})$ holds if either (a)
$\mathcal{A}$ has a stable monomial basis with the unique expression and
$|Hz|=+\infty$ for any cluster variable
$z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ or (b) $\mathcal{A}$ has a
$\mathcal{D}$-stable basis $|Hz|=+\infty$ for any cluster variable or quasi
cluster variable $z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$.

Unfortunately, $\mathcal{A}(\Sigma)^{H}$ is not a cluster subalgebra in many
cases. Thus we discuss Galois-like extensions. That is,
$\mathcal{A}(\Sigma_{1})\subset\mathcal{A}(\Sigma)$ satisfies that
$\mathcal{A}(\Sigma_{1})$ is maximal in $\mathcal{A}(\Sigma)^{H}$ as a cluster
subalgebra of $\mathcal{A}(\Sigma)$ and
$\text{Gal}_{\Sigma_{1}}\mathcal{A}(\Sigma)=H$.

In Section 4, we introduce the relation between mapping class groups and the
corresponding cluster automorphism groups and give some examples of
$\mathcal{D}$-stable bases for cluster algebras from the so-called feasible
surfaces, which is given in Definition 4.2.

Furthermore, we give the Galois map and reverse Galois for cluster algebras
from feasible surfaces as a partial confirmation of Conjecture 3.18.

Suppose $(S,M)$ be a feasible surface and $\mathcal{A}(S,M)$ is the cluster
algebra without coefficients corresponding to $(S,M)$, we have:

######  Theorem C (Theorem 4.10).

Let $(S,M)$ be a feasible surface, $\mathcal{A}(S,M)$ its cluster algebra
without coefficient, and a subgroup $H\leq\text{\em Aut}\mathcal{A}(S,M)$. If
$\mathcal{A}(\Sigma_{i}),i=1,2$ are maximal in $\mathcal{A}(S,M)^{H}$ as
cluster subalgebras of $\mathcal{A}(S,M)$, then the ranks of
$\mathcal{A}(\Sigma_{i}),i=1,2$ are the same.

######  Theorem D (Theorem 4.16).

Let $H\leq\text{\em Aut}\mathcal{A}(S,M)$, $\mathcal{A}(\Sigma_{1})$,
$\mathcal{A}(\Sigma_{2})\in\mathcal{M}^{H}_{sub}$,
$H_{1},H_{2},...,H_{s}\in\mathit{2}^{\text{\em
Aut}\mathcal{A}(S,M)}\backslash\text{ker}\phi$ and $H_{1}\leq
H_{2}\leq\cdots\leq H_{s}$. Then for any
$\mathcal{A}(\Sigma_{i})\in\mathcal{M}^{H_{i}}_{sub},1\leq i\leq s$, there
exist $\mathcal{A}(\Sigma_{j})\in\mathcal{M}^{H_{j}}_{sub}$ for
$j=1,...,i-1,i+1,...,s$ such that

$\mathcal{A}(\Sigma_{1}^{(d)})\geq\cdots\geq\mathcal{A}(\Sigma_{i}^{(d)})\geq\cdots\geq\mathcal{A}(\Sigma_{s}^{(d)}).$

That is, there is an order-preserving reverse Galois map for
$\mathcal{A}(S,M)$.
In particular, if $H_{1}<H_{2}<\cdots<H_{s}$ and any $f\in H_{s}$ is
orientation-preserved, then we have

$\mathcal{A}(\Sigma_{1})>\cdots>\mathcal{A}(\Sigma_{i})>\cdots>\mathcal{A}(\Sigma_{s}).$

Here the reduced cluster subalgebras $\mathcal{A}(\Sigma_{i}^{(d)})$ and the
orientation-preserved cluster automorphism are defined in Section 3.2 and 4.2,
respectively.

Besides, we also discuss the Galois inverse problem for cluster algebras from
feasible surfaces.

######  Proposition E (Proposition 4.14).

Assume $\mathcal{A}(S,M)$ is a cluster algebra without coefficient from a
feasible surface $(S,M)$. Let $C$ be a subsurface of $S$ such that there is no
connected component of $U=\overline{S\backslash C}$ which is a closed annulus,
a closed disk, or a closed once-punctured disk and there is a connected
component of $U$ which i different from a digon, a closed once-punctured
annulus and a closed twice-punctured disk, where $\overline{S\backslash C}$
means the closure of $S\backslash C$. Then we have the following statements:

(i) $\rho^{\prime-1}(\text{\em Mod}_{\bowtie}(C,C\cap M))$ is a Galois group.

(ii) Conversely, for a Galois group $H$ whose any element $f$ is orientation-
preserved, there is a subsurface $C$ such that $H=\rho^{\prime-1}(\text{\em
Mod}_{\bowtie}(C,C\cap M))$.

Here the definition of subsurfaces is given in Section 4.2, $\text{\em
Mod}_{\bowtie}(C,C\cap M)$ is a subgroup of the mapping class group of
$(C,C\cap M)$ which is also given in Section 4.2 and $\rho^{\prime}$ is the
isomorphism between the mapping class group and cluster automorphism group,
see Theorem 4.1.

##  2\. Preliminaries on cluster algebras and Riemann oriented surfaces

In this section, we introduce some background materials of cluster algebras,
the cluster algebras from bordered surfaces with marked points, and their
mapping class groups.

Let $\mathcal{F}$ be the field of rational polynomials over $\mathbb{Q}$ in
$m$ independent variables, the cluster algebra $\mathcal{A}$ is a
$R-$subalgebra of $\mathcal{F}$, where $R=\mathbb{Q}[x_{n+1},...,x_{m}]$, the
ring of polynomials of the frozen variables.

######  Definition 2.1.

A seed in $\mathcal{F}$ is a pair $(\textbf{\em x}_{ex},\textbf{\em
x}_{fr},\widetilde{B})$ where

(1) $\textbf{\em x}_{ex}=(x_{1},...,x_{n}),\textbf{\em
x}_{fr}=(x_{n+1},...,x_{m})$ and $x_{1},...,x_{m}$ are algebraically
independent in $\mathcal{F}$;

(2) $\widetilde{B}$ is an $m\times n$ extended skew-symmetrizable integer
matrix, i.e., there exists a diagonal non-negative integer matrix $D$ such
that $(DB)^{\top}=-DB$, where $B$ is the upper $n\times n$ submatrix of
$\widetilde{B}$, called the exchange matrix; $\widetilde{B}$ is called the
extended exchange matrix and $D$ is called the skew-symmetrizer. Meanwhile,
the lower $n\times n$ submatrix of $\widetilde{B}$, denoted as $C$, is called
the $C$-matrix.

(3) $\textbf{\em x}=\textbf{\em x}_{ex}\cup\textbf{\em x}_{fr}$ is called the
cluster of the seed $(\textbf{\em x}_{ex},\textbf{\em x}_{fr},\widetilde{B})$.
The elements of $\textbf{\em x}_{ex}$ are called exchange variables and the
elements of $\textbf{\em x}_{fr}$ are called frozen variables. The exchange
variables and frozen variables are collectively called cluster variables. The
monomials consisting of cluster variables (or exchange variables) in a cluster
are called extended cluster monomials (or cluster monomials).

We say a cluster algebra to be with coefficients, if the set of frozen
variables in the cluster algebra is not empty, otherwise we say the cluster
algebra to be without coefficient.

######  Definition 2.2.

Let $(\textbf{\em x}_{ex},\textbf{\em x}_{fr},\widetilde{B})$ be a seed in
$\mathcal{F}$, $k\in\\{1,...,n\\}$ The seed mutation $\mu_{k}$ transforms
$(\textbf{\em x}_{ex},\textbf{\em x}_{fr},\widetilde{B})$ to the new seed
$\mu_{k}(\textbf{\em x}_{ex},\textbf{\em x}_{fr},\widetilde{B})=(\textbf{\em
x}_{ex}^{\prime},\textbf{\em x}_{fr}^{\prime},\widetilde{B^{\prime}})$ defined
as follows:

(1) $\widetilde{B^{\prime}}=(b_{ij}^{\prime})_{m\times n}$, where

$b_{ij}^{\prime}=\left\\{\begin{array}[]{ll}-b_{ij}&\text{if }i=k\text{ or
}j=k;\\\
b_{ij}+sgn(b_{ik})max(b_{ik}b_{kj},0)&\text{otherwise}.\end{array}\right.$

(2) $(\textbf{\em x}_{ex}^{\prime},\textbf{\em
x}_{fr}^{\prime})=(x_{1}^{\prime},...,x_{m}^{\prime})$, where

$x_{i}^{\prime}=\left\\{\begin{array}[]{ll}x_{i}&\text{if }i\neq k;\\\
x_{i}^{-1}(\prod_{b_{jk}>0}x^{b_{jk}}_{j}+\prod_{b_{jk}<0}x^{-b_{jk}}_{j})&\text{otherwise}.\end{array}\right.$

In the sequel we also denote by $\mu_{x_{k}}=\mu_{k}$ and
$b_{x_{i}x_{j}}=b_{ij}$, for
$x_{k},x_{i}\in\textbf{x},\,x_{j}\in\textbf{x}_{ex}$.

Let $\mathcal{X}$ be the set of all cluster variables obtained from the
initial seed $\Sigma$ by finite steps of mutations, the cluster algebra
$\mathcal{A}$ is defined as the $\mathbb{Q}-$subalgebra of $\mathcal{F}$
generated by $\mathcal{X}$. The rank of $\mathcal{A}$ is defined as the number
of exchange variables in any seed $\Sigma$.

Note that if $D$ is the skew-symmetrizer of $\widetilde{B}$, it is also the
skew-symmetrizer of $\widetilde{B^{\prime}}$.

In a cluster algebra, any exchange variable can be expressed in terms of any
given cluster as a Laurent polynomial with coefficients in $R$, which is
called the Laurent phenomenon  in the cluster algebra. It is proved in
[GHKK18] that the coefficients in the Laurent polynomial mentioned above are
positive.

Let $\mathbb{T}_{n}$ be the $n$-regular tree whose $n$ edges emanating from
the same vertex are labelled bijectively by $1,...,n$. We assign a seed to
each vertex of $\mathbb{T}_{n}$ such that if two vertices are connected by an
edge labelled $k$, then the seeds assigned to them are obtained from each
other by the mutation at direction $k$. This assignment is called a cluster
pattern.

Let $\mathcal{S}$ be a cluster pattern,
$\mathcal{L}(t)=\mathbb{Q}[x_{1;t}^{\pm 1},...,x_{m;t}^{\pm 1}]$, where
$t\in\mathbb{T}_{n}$ and $x_{1;t},...,x_{m;t}$ are all cluster variables in
the seed of $\mathcal{S}$ corresponding to $t$. The upper cluster algebra
associated with $\mathcal{S}$ is defined as

$\mathcal{U}=\bigcap_{t\in\mathbb{T}_{n}}\mathcal{L}(t).$

Let $\mathcal{A}(\Sigma)$ be a cluster algebra, where initial seed
$\Sigma=(\textbf{x}_{ex},\textbf{x}_{fr},\widetilde{B}_{m\times n})$,
$\textbf{x}_{ex}=\\{x_{1},...,x_{n}\\}$,
$\textbf{x}_{fr}=\\{x_{n+1},...,x_{m}\\}$. We denote by

(2.1) $\widetilde{B}=\left(\begin{array}[]{cc}B_{n\times n}\\\ C_{(m-n)\times
n}\end{array}\right).$

A matrix is called column sign-coherent if any two nonzero entries of this
matrix in the same column have the same sign.

Following the notions given above, $C_{(m-n)\times n}$ is called uniformly
column sign-coherent with respect to the exchange matrix $B_{n\times n}$ if
$C_{(m-n)\times n}$ is column sign-coherent and after any sequence of
mutations, the lower $(m-n)\times n$ submatrix is column sign-coherent.

A cluster algebra with principal coefficients is a cluster algebra
$\mathcal{A}(\Sigma)$ with initial seed
$\Sigma=(\textbf{x}_{ex},\textbf{x}_{fr},\widetilde{B}_{2n\times n})$, where
$\textbf{x}_{ex}=\\{x_{1},x_{2},...,x_{n}\\}$,
$\textbf{x}_{fr}=\\{x_{n+1},x_{n+2},...,x_{2n}\\}$ and the lower $n\times n$
submatrix $B$ of $\widetilde{B}$ is the identity matrix.

It was proved in [GHKK18] that $C$-matrices of a skew-symmetrizable cluster
algebra $\mathcal{A}$ with principal coefficients are always uniformly column
sign-coherent.

For a skew-symmetrizable cluster algebra $\mathcal{A}(\Sigma)$ with principal
coefficients, there is a $\mathbb{Z}^{n}$-grading structure which is defined
by

$deg(x_{i})=\textbf{e}_{i},\;deg(x_{n+j})=-\textbf{b}_{j},\,i,j=1,...,n,$

where $\\{e_{1},...,e_{n}\\}$ is the standard basis for $\mathbb{Z}^{n}$ and
$\textbf{b}_{j}$ is the $j$-th column of the upper $n\times n$ submatrix $B$
of $\widetilde{B}$. It was proved in [FZ07] that Laurent polynomials of
exchange variables in $\mathcal{A}(\Sigma)$ for
$\textbf{x}=\textbf{x}_{ex}\sqcup\textbf{x}_{fr}$ are homogeneous with respect
to this $\mathbb{Z}^{n}$-grading, and the $\mathbb{Z}^{n}$-grading degree of
an exchange variable is called its $g$-vector. For a cluster
$\textbf{x}^{\prime}$, we denote by $G^{\widetilde{B}}_{\textbf{x}^{\prime}}$
the matrix whose columns consisting of $g$-vectors of all exchange variables
in $\textbf{x}^{\prime}$, called the $G$-matrix of the cluster
$\textbf{x}^{\prime}$. Trivially, $G^{\widetilde{B}}_{\textbf{x}^{\prime}}$ is
an $n\times n$ matrix.

The $F$-polynomial of an exchange variable in the principal coefficients
cluster algebra $\mathcal{A}(\Sigma)$ is the polynomial obtained by
specializing all of initial exchange variables to 1 in the Laurent polynomial
of this exchange variable for the initial cluster. If the $F$-polynomials of
two exchange variables are the same, then these two exchange variables are the
same, see [LP].

For an exchange variable $x^{\prime}$ and a cluster
$\textbf{x}=\textbf{x}_{ex}\sqcup\textbf{x}_{fr}$ in an arbitrary cluster
algebra $\mathcal{A}$ with $\textbf{x}_{ex}=\\{x_{1},...,x_{n}\\}$ and
$\textbf{x}_{fr}=\\{x_{n+1},...,x_{m}\\}$, we can represent $x^{\prime}$ as

$\frac{P(\textbf{x})}{x_{1}^{d_{1}}\cdots x_{n}^{d_{n}}},$

where $P(\textbf{x})$ is a polynomial of x which is not divided by any cluster
variable in x. The denominator vector d of $x^{\prime}$ for x is defined as
d$=(d_{1},...,d_{n})^{\top}$. In fact, the component $d_{i}$ of d
corresponding to $x_{i}$ is independent of the cluster x, see [CL20a]. Thus we
define the compatibility degree $d(x_{i},x^{\prime})$ of $x^{\prime}$ with
respect to $x_{i}$ as $d_{i}$.

Let $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ be cluster algebras with
coefficients. A cluster morphism between them is defined as follows:

######  Definition 2.3.

[HLY18] Let $\Sigma_{1},\Sigma_{2}$ are seeds of cluster algebras
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ respectively, where
$\Sigma_{1}=(\textbf{\em x}_{ex,1},\textbf{\em x}_{fr,1},$
$\widetilde{B}_{1,m_{1}\times n_{1}}),$ $\Sigma_{2}=(\textbf{\em
x}_{ex,2},\textbf{\em x}_{fr,2},\widetilde{B}_{2,m_{2}\times n_{2}})$,
$\textbf{\em x}_{ex,i}=\\{x_{i,1},x_{i,2},...,x_{i,n_{i}}\\},i=1,2$ are
exchange variables and $\textbf{\em x}_{fr,i}=\\{x_{i,n+1},x_{i,n+2},...,$
$x_{i,m_{i}}\\},i=1,2$ are frozen variables. Suppose
$f:\mathcal{A}_{1}\rightarrow\mathcal{A}_{2}$ is a morphism of algebras, if it
satisfies the following conditions:

(1) $f(\textbf{\em x}_{ex,1}\sqcup\textbf{\em x}_{fr,1})\subseteq\textbf{\em
x}_{ex,2}\sqcup\textbf{\em x}_{fr,2}\sqcup\mathbb{Z}$;

(2) $f(\textbf{\em x}_{ex,1})\subseteq\textbf{\em x}_{ex,2}\sqcup\mathbb{Z}$;

(3) If $(y_{1},y_{2},...,y_{s})$ is a $(f,\Sigma_{1},\Sigma_{2})$ bi-
admissible sequence, i.e., $y_{i}$ is an exchange variable in
$\mu_{y_{i-1}}...\mu_{y_{1}}(\Sigma_{1})$ and $f(y_{i})$ is an exchange
variable in $\mu_{f(y_{i-1})}...$ $\mu_{f(y_{1})}(f(\Sigma_{1}))$,
$i=1,...,s$, assume $\mu_{y_{0}}(\Sigma_{1})=\Sigma_{1}$ and
$\mu_{f(y_{0})}(f(\Sigma_{1}))=f(\Sigma_{1})$, we have

$f(\mu_{y_{s}}...\mu_{y_{1}}(y))=\mu_{f(y_{s})}...\mu_{f(y_{1})}(f(y)),$

then we call $f$ a cluster morphism bewteen $\mathcal{A}_{1}$ and
$\mathcal{A}_{2}$.

If a cluster morphism $f$ between $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ is
monomorphic (epimorphic, isomorphic) as a morphism of algebras, we say it to
be cluster monomorphic (epimorphic, isomorphic).

Let $\mathcal{A}(\Sigma)$ be a cluster algebra, where initial seed
$\Sigma=(\textbf{x}_{ex},\textbf{x}_{fr},\widetilde{B}_{m\times n})$,
$\textbf{x}_{ex}=\\{x_{1},...,x_{n}\\}$,
$\textbf{x}_{fr}=\\{x_{n+1},...,x_{m}\\}$,
$f\in\text{Aut}\mathcal{A}(\Sigma)$. We denote by

(2.2) $\widetilde{B}=\left(\begin{array}[]{cc}B_{n\times n}\\\ C_{(m-n)\times
n}\end{array}\right),\widetilde{B}_{f}=\left(\begin{array}[]{cc}B_{f,n\times
n}\\\ C_{f,(m-n)\times n}\end{array}\right),$

where $\widetilde{B}_{f}$ is the skew-symmetrizable matrix corresponding to
the cluster $f(\textbf{x}_{ex}\cup\textbf{x}_{fr})$, which can be obtained by
relabelling indexes of the extended exchange matrix corresponding to
$f(\textbf{x}_{ex}\cup\textbf{x}_{fr})$ in a seed in $\mathcal{A}(\Sigma)$,
and the $i$ row (or $j$ column ) corresponds to $f(x_{i})$ (or $f(x_{j})$). As
in [ASS14], we have

$\left(\begin{array}[]{cc}B\\\
C\end{array}\right)=\left(\begin{array}[]{cc}B_{f}\\\
C_{f}\end{array}\right)\text{or}\left(\begin{array}[]{cc}B\\\
C\end{array}\right)=-\left(\begin{array}[]{cc}B_{f}\\\
C_{f}\end{array}\right).$

If the former holds, we call $f$ a direct cluster automorphism and denote by
the subgroup consisting of direct cluster automorphisms ${\rm
Aut}^{+}\mathcal{A}(\Sigma)$, otherwise $f$ is called an inverse cluster
automorphism.

######  Definition 2.4.

Let $\mathcal{A}_{2}$ and $\mathcal{A}_{1}$ be cluster algebras such that
$\mathcal{A}_{2}\subset\mathcal{A}_{1}$. If the embedding from
$\mathcal{A}_{2}$ to $\mathcal{A}_{1}$ is a cluster monomorphism, we say
$\mathcal{A}_{2}$ to be a cluster subalgebra of $\mathcal{A}_{1}$.

In [HLY18] the cluster subalgebra was defined as the preimage of a monomorphic
cluster morphism. But in this article, we require that the cluster subalgebra
be a subalgebra of a cluster algebra which is different from [HLY18], since we
only care about the cluster subalgebras which are fixed under the action of
subgroups of cluster automorphism groups.

In fact, for a cluster algebra $\mathcal{A}$ and its cluster subalgebra
$\mathcal{A}(\Sigma^{\prime})$, the initial seed $\Sigma^{\prime}$ can be
obtained by a sequence of operations on a seed belonging to $\mathcal{A}$, see
[HLY18], which is given as follows.

######  Definition 2.5.

[HLY18] Let $\Sigma=(\textbf{\em x}_{ex},\textbf{\em
x}_{fr},\widetilde{B}_{m\times n})$ be a seed of a cluster algebra
$\mathcal{A}$, Assume $I_{0}$ and $I_{1}$ are subsets of $\textbf{\em
x}_{ex}\sqcup\textbf{\em x}_{fr}$, such that $I_{0}\subseteq\textbf{\em
x}_{ex},I_{1}\subseteq\textbf{\em x}_{ex}\sqcup\textbf{\em x}_{fr}$ and
$I_{0}\cap I_{1}=\emptyset$. Denote by $\textbf{\em
x}_{ex}^{\sharp}=\textbf{\em x}_{ex}\backslash(I_{0}\cup I_{1}),\textbf{\em
x}_{fr}^{\sharp}=(\textbf{\em x}_{fr}\cup I_{0})\backslash I_{1}$, and
$\widetilde{B}^{\sharp}=(b_{ij}^{\sharp})$ is a matrix with
$\sharp(\textbf{\em x}_{ex}^{\sharp}\sqcup\textbf{\em
x}_{fr}^{\sharp})\times\sharp\textbf{\em x}_{ex}^{\sharp}$ order, such that
$b^{\sharp}_{xy}=b_{xy}$, for $x\in\textbf{\em
x}_{ex}^{\sharp}\sqcup\textbf{\em x}_{fr}^{\sharp},y\in\textbf{\em
x}_{ex}^{\sharp}$. Then we can define a new seed
$\Sigma_{I_{0},I_{1}}=(\textbf{\em x}_{ex}^{\sharp},\textbf{\em
x}_{fr}^{\sharp},\widetilde{B}^{\sharp})$, which is called a mixing-type sub-
seed or $(I_{0},I_{1})$-type sub-seed of the seed $\Sigma$.

The initial seed of a cluster subalgebra of a cluster algebra $\mathcal{A}$ is
a $(I_{0},I_{1})$-type sub-seed of some seed of $\mathcal{A}$. Formally, we
have the following theorem.

######  Theorem 2.6.

[HLY18] A cluster algebra $\mathcal{A}^{\prime}$ is a cluster subalgebra of a
cluster algebra $\mathcal{A}$ if and only if there exists a seed
$\Sigma^{\prime}$ belonging to $\mathcal{A}^{\prime}$ such that
$\Sigma^{\prime}=\Sigma_{I_{0},I_{1}}$ for some seed $\Sigma=(\textbf{\em
x}_{ex},\textbf{\em x}_{fr},\widetilde{B})$ of $\mathcal{A}$ satisfying
$b_{xy}=0$, for $x\in I_{1}$ and $y\in\textbf{\em x}_{ex}\backslash(I_{0}\cup
I_{1})$.

Thus from now on we call a cluster algebra $\mathcal{A}(\Sigma^{\prime})$ a
cluster subalgebra of a cluster algebra $\mathcal{A}=\mathcal{A}(\Sigma)$, if
$\Sigma^{\prime}$ is a mixing-type sub-seed of $\Sigma$ satisfying the
condition of Theorem 2.6.

A marked surface is a pair $(S,M)$, where $S$ is a Riemann surface and $M$
consists of finite points of $S$ called marked points of $S$ such that each
boundary component contains at least one marked point. We also call $(S,M)$ a
surface for short. The points in $M\backslash\partial S$ are called punctures,
where $\partial S$ is the boundary of $S$.

It should be noted that all surfaces $(S,M)$ mentioned in this article are
connected oriented Riemann surfaces $S$ with finite boundary components.
Besides, following [FST08], we require all surfaces to be not one of the
following:

A sphere with 1, 2, or 3 marked points, an unpunctured disk with 1,2 or 3
marked points on the boundaries, or the once-punctured disk with 1 marked
point on the boundaries.

Cluster algebras from surfaces are a particular type of cluster algebras whose
exchange matrices are determined by a triangulation of surfaces.

A curve $\gamma$ in $S$ is a continuous map from the unit interval to $S$. We
can assign a natural orientation on $\gamma$ from $\gamma(0)$ to $\gamma(1)$.
The curve with the reverse orientation is denoted by $\gamma^{-1}$.

######  Definition 2.7.

An arc $\gamma$ in $(S,M)$ is a curve (up to isotopy) in $S$ such that

(i) the endpoints of $\gamma$ lie in $M$;

(ii) $\gamma$ does not intersect with boundary of $S$ except endpoints;

(iii) $\gamma$ does not cut out an unpunctured monogon or an unpunctured
digon;

(iv) $\gamma$ does not intersect with itself.

If a curve $\gamma$ satisfies (i), (ii), (iii) of Definition 2.7, then
$\gamma$ is called a generalized arc. If a curve $\alpha$ is isotopic to a
segment of the boundary of $S$ such that endpoints are in $M$, then $\alpha$
is called a boundary arc of $S$.

Two arcs in $(S,M)$ are called compatible if there are curves in their
respective isotopy classes that do not intersect. A maximal collection of
distinct pairwise compatible arcs is called an ideal triangulation.

######  Definition 2.8.

An tagged arc is an arc in which each endpoint has been tagged plain or
notched, so that the following conditions are satisfied:

(i) the arc does not cut out a once-punctured monogon;

(ii) all endpoints lying on the boundaries are tagged plain;

(iii) if its endpoints are the same, they are tagged in the same way.

Two tagged arcs $\alpha$ and $\beta$ are called compatible if the plain arcs
$\bar{\alpha}$, $\bar{\beta}$ obtained by forgetting their taggings are
compatible and satisfy

(a) if $\bar{\alpha}=\bar{\beta}$, then at least one endpoint of
$\bar{\alpha}$ and $\bar{\beta}$ is tagged in the same way;

(b) if $\bar{\alpha}\neq\bar{\beta}$, but they have common endpoints, then
they are tagged in the same way at common endpoints.

Similarly, a maximal collection of distinct pairwise compatible tagged arcs in
$(S,M)$ is called A tagged triangulation of $(S,M)$.

In fact, each ideal triangulation (resp. tagged triangulation) has the same
number of tagged arcs (resp. tagged arcs).

######  Proposition 2.9.

[FST08] Each ideal triangulation (resp. tagged triangulation) of $(S,M)$
consists of $n=6g+3b+3p+c-6$ arcs (resp. tagged arcs), where $g$ is the genus
of $(S,M)$, $b$ is the number of boundary components, $p$ is the number of
punctures, and $c$ is the number of marked points on the boundaries.

Suppose $T$ is a triangulation (resp. tagged triangulation) of $(S,M)$, and
$\gamma\in T$. Then there is another arc (resp. tagged arc) $\gamma^{\prime}$
such that $T^{\prime}=(T\backslash\\{\gamma\\})\cup\\{\gamma^{\prime}\\}$ is
another triangulation (resp. tagged triangulation) of $(S,M)$. And the way to
obtain $T^{\prime}$ from $T$ is called the flip at $\gamma$.

Suppose $\mathcal{A}$ is the cluster algebra whose exchange matrix is
determined by a triangulation of $(S,M)$.

######  Theorem 2.10.

[FST08] (1) If $(S,M)$ is not a closed surface with one puncture, then there
are bijections:

$\\{\text{tagged arcs in }(S,M)\\}\to\\{\text{exchange variables in
}\mathcal{A}\\},$ $\\{\text{tagged triangulations of
}(S,M)\\}to\\{c\text{lusters in }\mathcal{A}\\}.$

(2) If $(S,M)$ is a closed surface with one puncture, then there are
bijections:

$\\{\text{arcs in }(S,M)\\}\to\\{\text{exchange variables }in\mathcal{A}\\},$
$\\{\text{triangulations of }(S,M)\\}\to\\{\text{clusters in }\mathcal{A}\\}.$

And in these bijections, a flip at $\gamma$ coincides with the mutation at the
corresponding exchange variable $x_{\gamma}$.

Note that a loop is a closed curve with no self-intersection, for a loop
$\zeta$ surrounding a puncture $p$ which is a tagged arc, its corresponding
element $x_{\zeta}$ in the corresponding cluster algebra $\mathcal{A}$
satisfies $x_{\zeta}=x_{\gamma}x_{\gamma^{p}}$, where $\gamma$ is the arc
lying in the monogon bounded by $\zeta$ and $\gamma^{p}$ is the tagged arc
obtained from $\gamma$ via changing the tagging at $p$. Then $\zeta$ and
$\gamma$ is said to form a self-folding triangle

######  Definition 2.11.

[BY18] The mapping class group of a bordered surface with marked points
$(S,M)$ is defined by

$\mathcal{MCG}(S,M):=\text{\em Homeo}^{+}(S,M)/\text{\em Homeo}^{+}_{0}(S,M),$

where $\text{\em Homeo}^{+}(S,M)$ is the group of orientation-preserving
homeomorphisms such that $g(M)=M$ for any $g\in\text{\em Homeo}^{+}(S,M)$, and
$\text{\em Homeo}^{+}_{0}(S,M)$ is its subgroup of homeomorphisms isotopic to
the identity. Let $P$ be the set of punctures of $(S,M)$, the tagged mapping
class group of $(S,M)$ is defined to be the semidirect product

$\mathcal{MCG}_{\bowtie}(S,M):=\mathcal{MCG}(S,M)\ltimes\mathbb{Z}_{2}^{P},$

if $(g_{1},p_{1}),(g_{2},p_{2})\in\mathcal{MCG}_{\bowtie}(S,M)$,
$(g_{1},p_{1})(g_{2},p_{2}):=(g_{1}g_{2},p_{1}g_{1}(p_{2}))$.

We can also define an unsigned version of the mapping class group. Denote by
$\text{Homeo}(S,M)$ the group of homeomorphisms of $S$ that takes $M$ to
itself. Define

$\mathcal{MCG}^{\pm}(S,M):=\text{Homeo}(S,M)/\text{Homeo}^{+}_{0}(S,M);$
$\mathcal{MCG}^{\pm}_{\bowtie}(S,M):=\mathcal{MCG}^{\pm}(S,M)\ltimes\mathbb{Z}_{2}^{P}.$

In fact, the tagged mapping class group of a bordered surface with marked
points is isomorphic to the cluster automorphism group of the cluster algebra
corresponding to this surface, which is introduced in Section 4.

##  3\. Cluster automorphism groups, extensions and $\mathcal{D}$-stable bases
for cluster algebras

In this section, we introduce some notions of Galois theory for cluster
algebras and give a sufficient and necessary condition for Galois extensions
of cluster algebras with $\mathcal{D}$-stable bases. And, we discuss some
properties of Galois-like extensions. Moreover, the conjugate of subgroups of
a cluster automorphism group is given.

As a supplement, we also introduce the notion of complement sub-seed which
gives a way to embed the Galois group into a cluster automorphism group of a
cluster subalgebra, and offer an example for cluster automorphism groups
embedded into $S_{n}\rtimes\mathbb{Z}_{2}$.

###  3.1. Galois group, Galois map and cluster automorphism groups embedded
into $S_{n}\rtimes\mathbb{Z}_{2}$

If there is a cluster subalgebra of a cluster algebra $\mathcal{A}$, whose
initial seed is a mixing-type sub-seed of a seed $\Sigma$ of $\mathcal{A}$,
then we can define the complement of this sub-seed, which is also a mixing-
type sub-seed.

######  Definition 3.1.

Suppose $\mathcal{A}(\Sigma^{\prime})$ is a cluster subalgebra of a cluster
algebra $\mathcal{A}=\mathcal{A}(\Sigma)$, where $\Sigma=(\textbf{\em
x}_{ex},\textbf{\em x}_{fr},\widetilde{B}_{m\times n})$ with a mixing-type
sub-seed $\Sigma^{\prime}=(\textbf{\em x}_{ex}^{\prime},\textbf{\em
x}_{fr}^{\prime},\widetilde{B^{\prime}}_{m^{\prime}\times n^{\prime}})$. The
complement sub-seed of $\Sigma^{\prime}$ is defined as

$\Sigma^{\prime\prime}=(\textbf{\em x}_{ex}^{\prime\prime},\textbf{\em
x}_{fr}^{\prime\prime},\widetilde{B^{\prime\prime}}),$

where $\textbf{\em x}_{ex}^{\prime\prime}=\textbf{\em
x}_{ex}\backslash(\textbf{\em x}_{ex}^{\prime}\cup\textbf{\em
x}_{fr}^{\prime})$, $\textbf{\em x}_{fr}^{\prime\prime}=\textbf{\em
x}_{fr}\cup(\textbf{\em x}_{ex}\cap\textbf{\em x}_{fr}^{\prime})$, and
$\widetilde{B^{\prime\prime}}$ is a submatrix of $\widetilde{B}$ with columns
labelled by $\textbf{\em x}_{ex}^{\prime\prime}\subset\textbf{\em x}_{ex}$ and
lows labelled by $\textbf{\em x}_{ex}^{\prime\prime}\sqcup\textbf{\em
x}_{fr}^{\prime\prime}\subset\textbf{\em x}_{ex}\sqcup\textbf{\em x}_{fr}$.

Excluding the frozen variables in $\Sigma^{\prime}$, the exchange variables of
$\Sigma$ is the disjoint union of the exchange variables in $\Sigma^{\prime}$
and the exchange variables in $\Sigma^{\prime\prime}$, i.e.,
$\textbf{x}_{ex}\backslash\textbf{x}_{fr}^{\prime}=\textbf{x}_{ex}^{\prime}\sqcup\textbf{x}_{ex}^{\prime\prime}$.

######  Proposition 3.2.

For a seed $\Sigma$ and its a mixing-type sub-seed $\Sigma^{\prime}$ as given
in Definition 3.1, the complement $\Sigma^{\prime\prime}$ is a mixing-type
sub-seed of $\Sigma$ and $\mathcal{A}(\Sigma^{\prime\prime})$ is a cluster
subalgebra of $\mathcal{A}(\Sigma)$.

###### Proof.

It is obvious that $\Sigma^{\prime\prime}$ is a $(I_{0},I_{1})$-type sub-seed
of $\Sigma$. According to Theorem 2.6, it is sufficient to prove $b_{xy}=0$,
for all $x\in I_{1}$, and $y\in\textbf{x}_{ex}\backslash I_{0}\cup I_{1}$.

We can see
$I_{0}=\textbf{x}_{ex}\cap\textbf{x}_{fr}^{\prime},I_{1}=\textbf{x}_{ex}^{\prime}$.
Because $I_{1}=\textbf{x}_{ex}^{\prime}$, the condition $b_{xy}=0$, for all
$x\in I_{1}$, and $y\in\textbf{x}_{ex}\backslash I_{0}\cup I_{1}$ is
equivalent to $b_{yx}=0$, for all $x\in I_{1}$, and
$y\in\textbf{x}_{ex}\backslash I_{0}\cup I_{1}$.

On the other hand, because $\Sigma^{\prime}$ is
$(I_{0}^{\prime},I_{1}^{\prime})$-type sub-seed of $\Sigma$, we have
$I_{0}^{\prime}=(\textbf{x}_{ex}\backslash\textbf{x}_{ex}^{\prime})\cap\textbf{x}_{fr}^{\prime},I_{1}^{\prime}=\textbf{x}_{ex}\sqcup\textbf{x}_{fr}\backslash\textbf{x}_{ex}^{\prime}\sqcup\textbf{x}_{fr}^{\prime}$.
Since $\mathcal{A}(\Sigma^{\prime})$ is a cluster subalgebra of
$\mathcal{A}(\Sigma_{1})$, we have $b_{xy}=0$, for all $x\in I_{1}^{\prime}$,
and $y\in\textbf{x}_{ex}\backslash I_{0}^{\prime}\cup I_{1}^{\prime}$.

$\textbf{x}_{ex}\backslash I_{0}^{\prime}\cup
I_{1}^{\prime}=(\textbf{x}_{ex}\backslash
I_{0}^{\prime})\cap(\textbf{x}_{ex}\backslash
I_{1}^{\prime})=\textbf{x}_{ex}^{\prime}$ and $\textbf{x}_{ex}\backslash
I_{0}\cup I_{1}=(\textbf{x}_{ex}\backslash
I_{0})\cap(\textbf{x}_{ex}\backslash I_{1})$, where $\textbf{x}_{ex}\backslash
I_{0}=\textbf{x}_{ex}\backslash\textbf{x}_{fr}^{\prime}\subset I_{1}^{\prime}$
and $\textbf{x}_{ex}\backslash
I_{1}=\textbf{x}_{ex}\backslash\textbf{x}_{ex}^{\prime}\subset
I_{1}^{\prime}$, thus $b_{yx}=0$, for all $x\in I_{1}$, and
$y\in\textbf{x}_{ex}\backslash I_{0}\cup I_{1}$. So
$\mathcal{A}(\Sigma^{\prime\prime})$ is a cluster subalgebra of the cluster
algebra $\mathcal{A}(\Sigma)$.

∎

Following this proposition, we call the subalgebra
$\mathcal{A}(\Sigma^{\prime\prime})$ the complement cluster subalgebra of
$\mathcal{A}(\Sigma)$ associated to $\mathcal{A}(\Sigma^{\prime})$.

Let $\mathcal{A}(\Sigma^{\prime})$ be a cluster subalgebra of a cluster
algebra $\mathcal{A}$. Suppose $\text{Aut}\mathcal{A}$ is the cluster
automorphism group of $\mathcal{A}$, then we can define the fixed subgroup of
$\text{Aut}\mathcal{A}$ over $\mathcal{A}(\Sigma^{\prime})$ as follows:

$\text{Gal}_{\Sigma^{\prime}}\mathcal{A}=\\{f\in\text{Aut}\mathcal{A},f|_{\mathcal{A}(\Sigma^{\prime})}=id_{\mathcal{A}(\Sigma^{\prime})}\\},$

which is called the Galois group of $\mathcal{A}$ over
$\mathcal{A}(\Sigma^{\prime})$.

If the cluster subalgebra $\mathcal{A}(\Sigma^{\prime})$ is trivial, i.e.,
$\mathcal{A}(\Sigma^{\prime})=\mathbb{Q}$ whose mixing-type sub-seed
$\Sigma^{\prime}$ is just the empty set $\emptyset$, then
$\text{Aut}\mathcal{A}(\Sigma)=\text{Gal}_{\emptyset}\mathcal{A}(\Sigma)$.

Now we show that the Galois group $\text{
Gal}_{\Sigma^{\prime}}\mathcal{A}(\Sigma)$ can be embedded into
$\text{Aut}\mathcal{A}(\Sigma^{\prime\prime})$. let
$\mathcal{A}(\Sigma^{\prime\prime})$ be the complement cluster subalgebra of
$\mathcal{A}(\Sigma)$ associated to $\mathcal{A}(\Sigma^{\prime})$, we have:

######  Proposition 3.3.

Following the notions given above, we have the following statements.

(i) The map

$\pi:\;\text{\em Gal}_{\Sigma^{\prime}}\mathcal{A}(\Sigma)\rightarrow\text{\em
Gal}_{\textbf{\em
x}_{fr}^{\prime}}\mathcal{A}(\Sigma^{\prime\prime})\;\;\;\text{via}\;\;\;f\mapsto
f|_{\mathcal{A}(\Sigma^{\prime\prime})}$

is a monomorphism, where $\textbf{\em x}_{fr}^{\prime}$ is the set of frozen
variables in $\Sigma^{\prime}$, which is considered as a mixing-type sub-seed
of the cluster subalgebra $\mathbb{Q}[\textbf{\em
x}_{fr}^{\prime}][\emptyset]$ without exchange variables.

(ii) Suppose $f^{\prime}\in\text{\em Gal}_{\textbf{\em
x}_{fr}^{\prime}}\mathcal{A}(\Sigma^{\prime\prime})$. Then

(a) $\bar{\textbf{\em x}}=\textbf{\em x}^{\prime}_{ex}\cup
f^{\prime}(\textbf{\em x}^{\prime\prime}_{ex}\cup\textbf{\em
x}^{\prime\prime}_{fr})$ is a cluster of $\mathcal{A}(\Sigma)$, and

(b) $f^{\prime}\in\text{\em Im}\pi$ if and only if the submatrix $D^{0}_{f}$
of extended exchange matrix of $\bar{\textbf{\em x}}$ corresponding to
$f^{\prime}(\textbf{\em x}^{\prime\prime}_{fr})\times f^{\prime}(\textbf{\em
x}^{\prime}_{fr}\cap\textbf{\em x}_{ex})$ is the same as the submatrix
$D^{0}$of extended exchange matrix of $\textbf{\em x}_{ex}\cup\textbf{\em
x}_{fr}$ corresponding to $\textbf{\em x}^{\prime\prime}_{fr}\times\textbf{\em
x}^{\prime}_{fr}\cap\textbf{\em x}_{ex}$.

In particular, denoting by

$\widetilde{B}^{\prime\prime}=\left(\begin{array}[]{c}B^{\prime\prime}\\\
C^{\prime\prime}\end{array}\right),$

the extended exchange matrix of $\Sigma^{\prime\prime}$, where
$B^{\prime\prime}$ is the exchange matrix of $\Sigma^{\prime\prime}$, if
$C^{\prime\prime}$ is uniformly column sign-coherent with respect to
$B^{\prime\prime}$, then $\text{\em Gal}_{\textbf{\em
x}^{\prime\prime}_{fr}}\mathcal{A}(\Sigma^{\prime\prime})\subset\text{\em
Im}\pi$.

###### Proof.

Suppose $\Sigma=(\textbf{x}_{ex},\textbf{x}_{fr},\widetilde{B})$,
$\Sigma^{\prime}=(\textbf{x}^{\prime}_{ex},\textbf{x}^{\prime}_{fr},\widetilde{B}^{\prime})$
and
$\Sigma^{\prime\prime}=(\textbf{x}^{\prime\prime}_{ex},\textbf{x}^{\prime\prime}_{fr},\widetilde{B}^{\prime\prime})$
with the extended exchange matrices
$\widetilde{B}=(b_{xy})_{x\in\textbf{x}_{ex}\cup\textbf{x}_{fr},y\in\textbf{x}_{ex}}$,
$\widetilde{B}^{\prime}=(b_{xy})_{x\in\textbf{x}^{\prime}_{ex}\cup\textbf{x}^{\prime}_{fr},y\in\textbf{x}^{\prime}_{ex}}$
and
$\widetilde{B}^{\prime\prime}=(b_{xy})_{x\in\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr},y\in\textbf{x}^{\prime\prime}_{ex}}$.
According to Definition 3.1, we have
$\textbf{x}_{ex}=\textbf{x}^{\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{ex}\cup(\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex})$,
$\textbf{x}^{\prime}_{ex}\cap\textbf{x}^{\prime\prime}_{ex}=\emptyset$ and
$\textbf{x}^{\prime\prime}_{fr}=(\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex})\cup\textbf{x}_{fr}$.
We have

$\widetilde{B}=\left(\begin{array}[]{ccc}B_{\textbf{x}^{\prime}_{ex}\times\textbf{x}^{\prime}_{ex}}^{1}&A_{\textbf{x}^{\prime}_{ex}\times\textbf{x}^{\prime\prime}_{ex}}^{2}&B_{\textbf{x}^{\prime}_{ex}\times\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}}^{3}\\\
A_{\textbf{x}^{\prime\prime}_{ex}\times\textbf{x}^{\prime}_{ex}}^{1}&C_{\textbf{x}^{\prime\prime}_{ex}\times\textbf{x}^{\prime\prime}_{ex}}^{1}&C_{\textbf{x}^{\prime\prime}_{ex}\times\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}}^{3}\\\
B_{\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}\times\textbf{x}^{\prime}_{ex}}^{2}&C_{\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}\times\textbf{x}^{\prime\prime}_{ex}}^{2}&D_{\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}\times\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}}^{1}\\\
\hline\cr
B_{\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{fr}\times\textbf{x}^{\prime}_{ex}}^{4}&C_{\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{fr}\times\textbf{x}^{\prime\prime}_{ex}}^{4}&D_{\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{fr}\times\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}}^{2}\\\
A_{\textbf{x}_{fr}\backslash\textbf{x}^{\prime}_{fr}\times\textbf{x}^{\prime}_{ex}}^{3}&C_{\textbf{x}_{fr}\backslash\textbf{x}^{\prime}_{fr}\times\textbf{x}^{\prime\prime}_{ex}}^{5}&D_{\textbf{x}_{fr}\backslash\textbf{x}^{\prime}_{fr}\times\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}}^{3}\end{array}\right).$

Suppose $\textbf{x}_{ex}=\\{x_{1},...,x_{n}\\}$,
$\textbf{x}_{fr}=\\{x_{n+1},...,x_{m}\\}$ and
$D=diag(d_{x_{1}},...,d_{x_{n}})$ is a skew-symmetrizer of $\widetilde{B}$.

(i) Firstly, it is easy to check that the map is a well-defined homomorphism.
Besides, if $g,g^{\prime}\in\text{Gal}_{\Sigma^{\prime}}\mathcal{A}(\Sigma)$
and $g\neq g^{\prime}$, then there exists
$z\in\Sigma\backslash\Sigma^{\prime}\subset\Sigma^{\prime\prime}$ such that
$g(z)\neq g^{\prime}(z)$. Thus $g|_{\mathcal{A}(\Sigma^{\prime\prime})}(z)\neq
g^{\prime}|_{\mathcal{A}(\Sigma^{\prime\prime})}(z)$, which means the map
$\pi$ is a monomorphism.

(ii) (a) Let $f$ be the endomorphism of the rational polynomials field over
$\textbf{x}_{ex}\cup\textbf{x}_{fr}$, which is an extension of $f^{\prime}$
defined by

(3.1)
$f(x)=\left\\{\begin{array}[]{ll}x,&\text{if}\;\;x\in\textbf{x}^{\prime}_{ex},\\\
f^{\prime}(x),&\text{if}\;\;x\in\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr}.\end{array}\right.$

Since
$f^{\prime}\in\text{Gal}_{\textbf{x}_{fr}^{\prime}}\mathcal{A}(\Sigma^{\prime\prime})$,
$f^{\prime}(\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr})$
is a cluster of $\mathcal{A}(\Sigma^{\prime\prime})$, and hence we can make a
sequence of mutations on exchange variables of
$\textbf{x}^{\prime\prime}_{ex}$ and the obtained successors from
$\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr}$ to
$f^{\prime}(\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr})$.
Besides, $\mathcal{A}(\Sigma^{\prime\prime})$ is a cluster subalgebra of
$\mathcal{A}(\Sigma)$, the embedding from $\mathcal{A}(\Sigma^{\prime\prime})$
to $\mathcal{A}(\Sigma)$ is a cluster monomorphism. Thus by (3.1) we have
$\bar{\textbf{x}}=\textbf{x}^{\prime}_{ex}\cup
f^{\prime}(\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr})=f(\textbf{x}^{\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr})=f(\textbf{x}_{ex}\cup\textbf{x}_{fr})$
and $\bar{\textbf{x}}$ can be obtained by doing the sequence of mutations
mentioned above from $\textbf{x}_{ex}\cup\textbf{x}_{fr}$. Hence
$f(\textbf{x}_{ex}\cup\textbf{x}_{fr})$ is a cluster of $\mathcal{A}(\Sigma)$.

(b) The algebraic independence of clusters guarantees $f$ is well-defined.
Thus $f^{\prime}\in\text{Im}\pi$ if and only if $f$ is a cluster automorphism
of $\mathcal{A}(\Sigma)$, which is equivalent to the extended exchange
matrices
$\widetilde{B},\widetilde{B}_{f}=(b_{xy}^{f})_{f(\textbf{x}_{ex}\cup\textbf{x}_{fr})\times
f(\textbf{x}_{ex})}$ associated with $\textbf{x}_{ex}\cup\textbf{x}_{fr}$ and
$f(\textbf{x}_{ex}\cup\textbf{x}_{fr})$ are the same, according to [ASS14].

“only if”: If $f^{\prime}\in\text{Im}\pi$, then $D^{0}_{f}$ of the extended
exchange matrix of $\textbf{x}^{\prime}$ corresponding to
$f^{\prime}(\textbf{x}^{\prime\prime}_{fr})\times
f^{\prime}(\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex})$ is the same as the
submatrix $D^{0}$ of the extended exchange matrix of
$\textbf{x}_{ex}\cup\textbf{x}_{fr}$ corresponding to
$\textbf{x}^{\prime\prime}_{fr}\times(\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex})$.

“if”: Suppose $A^{i}_{f},B^{j}_{f},C^{k}_{f},D^{l}_{f}$ correspond to
$A^{i},B^{j},C^{k},D^{l}$ for $i,l=1,2,3$; $j=1,2,3,4$; $k=1,2,3,4,5$. Since
$\mathcal{A}(\Sigma^{\prime})$ is a cluster subalgebra, The submatrices
$A^{1}$ and $A^{3}$ of $\widetilde{B}$ are equal to zero, according to Theorem
2.6. The skew-symmetrizablity of $\widetilde{B}$ guarantees that $A^{2}=0$.

Since
$f^{\prime}\in\text{Gal}_{\textbf{x}_{fr}^{\prime}}\mathcal{A}(\Sigma^{\prime\prime})$
and $\mathcal{A}(\Sigma^{\prime\prime})$ is a cluster subalgebra of
$\mathcal{A}(\Sigma)$, there is a sequence of mutation from $\Sigma$ at
indexes in $\textbf{x}^{\prime\prime}_{ex}$ to $f(\Sigma)$, which is denoted
by $\mu_{f^{\prime}}$, and $\mu_{f^{\prime}}$ also can be consider as a
sequence of mutation from $\Sigma^{\prime\prime}$ to
$f^{\prime}(\Sigma^{\prime\prime})$. Since $A^{i}=0$, $\mu_{f}$ does not
change the submatrices $A^{i},B^{j}$, $i=1,2,3;j=1,2,4$. Thus we have
$A^{i}=A^{i}_{f},B^{j}=B^{j}_{f}$.

Besides,
$f^{\prime}\in\text{Gal}_{\textbf{x}_{fr}^{\prime}}\mathcal{A}(\Sigma^{\prime\prime})$
means $\widetilde{B}^{\prime\prime}$ and
$\widetilde{B}^{\prime\prime}_{f}=(b_{xy}^{f})_{f^{\prime}(\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr})\times
f^{\prime}(\textbf{x}^{\prime\prime}_{ex})}$ are the same, i.e.,
$b_{xy}=b_{f^{\prime}(x)f^{\prime}(y)},x\in\textbf{x}^{\prime\prime}_{ex}\cup\textbf{x}^{\prime\prime}_{fr},y\in\textbf{x}^{\prime\prime}_{ex}$.
Note that $\widetilde{B}^{\prime\prime}_{f}$ is the submatrix of
$\widetilde{B}_{f}$ corresponding to
$C_{1}^{f},C_{2}^{f},C_{4}^{f},C_{5}^{f}$. Thus we have
$C_{i}=C_{i}^{f},i=1,2,4,5$. Because $D$ is a skew-symmetrizer of
$\widetilde{B}$, $D_{f}=(d_{f(x_{1})},...,d_{f(x_{n})})$ is a skew-symmetrizer
of $\widetilde{B}_{f}$. Thus we have $d_{x}b_{xy}=-d_{y}b_{yx}$ and
$d_{f(x)}b_{f(x)f(y)}=-d_{f(y)}b_{f(y)f(x)}$ for entries $b_{xy}$ belong to
$B^{2},C^{2}$. Since $f(x)=x$ for
$x\in\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}$, we have

$b_{yx}=-\frac{d_{x}}{d_{y}}b_{xy}=-\frac{d_{f(x)}}{d_{f(y)}}b_{f(x)f(y)}=b_{f(y)f(x)},$

for entries $b_{xy}$ belong to $B^{2},C^{2}$. Hence
$C^{3}=C^{3}_{f},B^{3}=B^{3}_{f}$.

The condition $D^{0}=D^{0}_{f}$ means $D^{i}=D^{i}_{f},i=1,2,3$. Thus we have
$\widetilde{B}=\widetilde{B}_{f}$ and hence $f^{\prime}\in\text{Im}\pi$.

According to the definition of mutations of matrices, mutation at
$z\in\textbf{x}^{\prime\prime}_{ex}$,
$\mu_{z}(b_{xy})=b_{xy}+sgn(b_{xz})max(b_{xz}b_{zy},0)$ for
$x\in\textbf{x}^{\prime\prime}_{fr},y\in\textbf{x}^{\prime}_{fr}\cap\textbf{x}_{ex}$.
If $C^{\prime\prime}$, i.e., the submatrix consisting of $C^{2},C^{4},C^{5}$,
is uniformly column sign-coherent, then

$\begin{array}[]{cl}\mu_{z}(b_{xy})&=b_{xy}+sgn(b_{xz})max(b_{xz}b_{zy},0)\\\
&=b_{xy}+sgn(b_{xz})max(-b_{xz}\frac{d_{y}}{d_{z}}b_{yz},0)\\\
&=b_{xy},\end{array}$

for $b_{xy}$ belonging to $D^{i},i=1,2,3$. Thus muatations at
$\textbf{x}^{\prime\prime}_{ex}$ do not change entries belong to
$D^{i},i=1,2,3$ and hence for
$f^{\prime}\in\text{Gal}_{\textbf{x}_{fr}^{\prime\prime}}\mathcal{A}(\Sigma^{\prime\prime})$,
we have $\widetilde{B}=\widetilde{B}_{f}$, which means $f\in\text{Im}\pi$.

∎

According to [CL19], if $C^{2},C^{4},C^{5}$ are nonnegative submatrix, then
$C^{\prime\prime}$ is uniformly column sign-coherent, and hence we have
$\text{Gal}_{\textbf{x}_{fr}^{\prime\prime}}\mathcal{A}(\Sigma^{\prime\prime})\subset\text{Im}\pi$.

Assume $\mathcal{A}(\Sigma_{i})$ is a cluster subalgebra of
$\mathcal{A}(\Sigma_{i-1}),i=1,2,...,s$ where $\mathcal{A}(\Sigma_{s})=0$ and
$\mathcal{A}(\Sigma_{0})=\mathcal{A}(\Sigma)$, then we have

(3.2)
$\mathcal{A}(\Sigma_{0})\geq\mathcal{A}(\Sigma_{1})\geq...\geq\mathcal{A}(\Sigma_{s});$
(3.3)
$\text{Gal}_{\Sigma_{0}}\mathcal{A}(\Sigma)\leq\text{Gal}_{\Sigma_{1}}\mathcal{A}(\Sigma)\leq...\leq\text{Gal}_{\Sigma_{s}}\mathcal{A}(\Sigma).$

The sequence (3.3) is a filtration of subgroups of the cluster automorphism
group $\text{Aut}\mathcal{A}(\Sigma)$ which corresponds to the filtration of
cluster subalgebras of $\mathcal{A}(\Sigma)$ in (3.2). Thus some properties of
the sequence of algebras may be reflected via the sequence (3.3) of subgroups.

We denote by:
(i) $\mathit{2}^{\text{Aut}\mathcal{A}}$ the set of all subgroups of
$\text{Aut}\mathcal{A}$,
(ii) $\mathit{2}^{\mathcal{A}}$ the set of cluster subalgebras of
$\mathcal{A}$,
(iii) $\mathit{2}^{\mathit{2}^{\mathcal{A}}}$ the set of all sets consisting
of some cluster subalgebras of $\mathcal{A}$.

Define a map

(3.4)
$\xi:\;\;\mathit{2}^{\mathcal{A}}\rightarrow\mathit{2}^{\text{Aut}\mathcal{A}}\text{
via}\;\;\;\mathcal{A}(\Sigma^{\prime})\mapsto\text{Gal}_{\Sigma^{\prime}}\mathcal{A}(\Sigma)$

where $\Sigma^{\prime}$ is a mixing-type sub-seed of $\Sigma$. We call $\xi$
the Galois map from cluster subalgebras of $\mathcal{A}(\Sigma)$ to subgroups
of its cluster automorphism group $\text{Aut}\mathcal{A}(\Sigma)$.

By (3.2) and (3.3), the Galois map $\xi$ sends a descending sequence in
$\mathit{2}^{\mathcal{A}}$ to a ascending sequence in
$\mathit{2}^{\text{Aut}\mathcal{A}}$.

However, when cluster subalgebras
$\mathcal{A}(\Sigma_{1})\lneqq\mathcal{A}(\Sigma_{2})$ or even
$\mathcal{A}(\Sigma_{1})$ and $\mathcal{A}(\Sigma_{2})$ do not include each
other, it is still possible to hold that
$\text{Gal}_{\Sigma_{1}}\mathcal{A}(\Sigma)=\text{Gal}_{\Sigma_{2}}\mathcal{A}(\Sigma)$.

For example, the fixed subgroups of any cluster subalgebra of type $A_{n}$
cluster algebras with exchange variables are trivial. And Proposition 3.6 also
gives another example.

Before that, we need the following theorem and lemma.

######  Theorem 3.4.

[NZ12] Let $\mathcal{A}(\Sigma)$ be a cluster algebra with principal
coefficients about the initial seed $\Sigma=(\textbf{\em x}_{ex},\textbf{\em
x}_{fr},\widetilde{B}_{2n\times n})$ with $C=I_{n}$ the lower $n\times n$
submatrix of $\widetilde{B}$, then

$G^{\widetilde{B}}_{\textbf{\em x}^{\prime}}=D((C^{\widetilde{B}}_{\textbf{\em
x}^{\prime}})^{-1})^{\top}D^{-1},$

where $C^{\widetilde{B}}_{\textbf{\em x}^{\prime}}$ is the lower $n\times n$
submatrix of extended exchange matrix $\widetilde{B}_{\textbf{\em
x}^{\prime}}$ of $\textbf{\em x}^{\prime}$ and D is the skew-symmetrizer.

######  Lemma 3.5.

[CL20a] Let $\mathcal{A}(\Sigma)$ be a cluster algebra with principal
coefficients, suppose $x_{1},x_{2}$ are exchange variables of
$\mathcal{A}(\Sigma)$. Then $x_{1}=x_{2}$ if and only if their $g$-vectors
$\textbf{g}(x_{1})$ and $\textbf{g}(x_{2})$ are the same.

For $\sigma\in S_{n}$ is a permutation of $n$, we can consider $\sigma$ as a
permutation matrix. Its action on $n\times n$ matrix $M_{n\times n}$ is the
multiplication of matrices. Then we have the following proposition.

######  Proposition 3.6.

Let $\mathcal{A}(\Sigma)$ be a cluster algebra with principal coefficients,
where $\Sigma=(\textbf{\em x}_{ex},\textbf{\em x}_{fr},\widetilde{B}_{2n\times
n})$. Then $\text{\em Aut}\mathcal{A}(\Sigma)$ is a subgroup of
$S_{n}\rtimes\mathbb{Z}_{2}$ where $S_{n}$ is the group of all permutations on
the initial exchange variables.

###### Proof.

According to [ASS14], cluster automorphism group is semiproduct of direct
cluster automorphism group and $\mathbb{Z}_{2}$, hence we only consider the
direct cluster automorphism group of $\mathcal{A}$. For any
$f\in\text{Aut}^{+}\mathcal{A}(\Sigma)$, suppose $G_{f}$ is the $G$-matrix of
$f(\textbf{x}_{ex})$ and $\widetilde{B}_{f}$ is the skew-symmetrizable matrix
corresponding to $f(\textbf{x}_{ex},\textbf{x}_{fr})$, then we have

$\widetilde{B}=\left(\begin{array}[]{cc}B\\\
C\end{array}\right)=\left(\begin{array}[]{cc}B_{f}\\\
C_{f}\end{array}\right)=\widetilde{B}_{f}.$

According to Theorem 3.4, since $C=I$, after the action of some permutions, we
have

$(G_{f})\sigma_{1}=D((\sigma_{2}(C_{f})\sigma_{1})^{-1})^{\top}D^{-1}=D((\sigma_{2}\sigma_{1})D^{-1}.$

Thus

$(G_{f})\sigma_{2}^{-1}=D((\sigma_{2}\sigma_{1})D^{-1}\sigma_{1}^{-1}\sigma_{2}^{-1},$

The left side of the equality is a matrix with their entries belonging to
$\mathbb{Z}$, so is the right side. And the right side is a multiplication of
diagonal matrices $D$ and
$(\sigma_{2}\sigma_{1})D^{-1}\sigma_{1}^{-1}\sigma_{2}^{-1}$, where the
entries of $(\sigma_{2}\sigma_{1})D^{-1}\sigma_{1}^{-1}\sigma_{2}^{-1}$ are
reciprocal of the entries of $D$, thus the right side of the equality is the
identity matrix and so is the left side.

According to Lemma 3.5, after the action of a permution, the elements of
$f(\textbf{x}_{ex})$ is equal to the elements of $\textbf{x}_{ex}$, thus
$\text{Aut}\mathcal{A}(\Sigma)$ is a subgroup of $S_{n}\rtimes\mathbb{Z}_{2}$.

∎

This proposition says that despite $\mathcal{A}$ has many cluster subalgebras,
their fixed subgroups of $\text{Aut}\mathcal{A}$ could also be the same. In
fact, the cluster automorphism groups of cluster algebras with principal
coefficients are finite, but their cluster subalgebras may be infinite, which
means there exist infinite cluster subalgebras such that their fixed subgroups
are the same.

So we should consider a special class of subalgebras and subgroups.

###  3.2. Galois-like extensions, conjugate of subgroups and some conjectures

For a cluster algebra $\mathcal{A}=\mathcal{A}(\Sigma)$, let $H$ be a subgroup
of $\text{Aut}\mathcal{A}$. Denote by

$\mathcal{A}^{H}=\\{z\in\mathcal{A}\;|\;f(z)=z,\forall f\in H\\}.$

In general, $\mathcal{A}^{H}$ maybe not a cluster subalgebra of $\mathcal{A}$.
However, we have that for a cluster subalgebra $\mathcal{A}(\Sigma_{1})$, if
$H=\text{Gal}_{\Sigma_{1}}\mathcal{A}$ then
$\mathcal{A}(\Sigma_{1})\subseteq\mathcal{A}^{H}\subseteq\mathcal{A}(\Sigma)$.

######  Remark 3.7.

If $\mathcal{A}$ is a cluster algebra of finite type, for any cluster
subalgebra $\mathcal{A}(\Sigma_{1})$ of $\mathcal{A}$,
$\mathcal{A}(\Sigma_{1})\subsetneqq\mathcal{A}^{\text{\em
Gal}_{\Sigma_{1}}\mathcal{A}}$. In fact, let $z$ be the product of all cluster
variables of $\mathcal{A}$, then for any $f\in\text{\em Aut}\mathcal{A}$,
$f(z)=z$, since $f$ is a bijection among cluster variables. Hence $z$ is not
in any proper cluster subalgebra of $\mathcal{A}$.

In general, it is not difficult to find a cluster subalgebra
$\mathcal{A}(\Sigma_{1})$ of a cluster algebras $\mathcal{A}$ satisfying the
condition
$\mathcal{A}(\Sigma_{1})\subsetneqq\mathcal{A}^{\text{Gal}_{\Sigma_{1}}\mathcal{A}}$.

Thus for a given subgroup $H\leq\text{Aut}\mathcal{A}$, we can consider a
maximal cluster subalgebra $\mathcal{A}(\Sigma_{1})$ of $\mathcal{A}$ such
that $\mathcal{A}(\Sigma_{1})\subseteq\mathcal{A}^{H}$.

Motivated by this fact, for $H$ a subgroup of $\text{Aut}\mathcal{A}$, we
define a set of some cluster subalgebras as follows:

$\mathcal{M}_{sub}^{H}:=\left\\{\mathcal{A}(\Sigma^{\prime})\left|\begin{array}[]{ccc}\mathcal{A}(\Sigma^{\prime})\;\text{
is a maximal cluster subalgebra}\;\text{ of }\;\mathcal{A}\;\text{ in
}\mathcal{A}^{H}\;\text{
satisfying}\;\text{Gal}_{\Sigma^{\prime}}\mathcal{A}=H\end{array}\right.\right\\}.$

######  Remark 3.8.

We will show Example 3.17 in the sequel which gives us a maximal cluster
subalgebra $\mathcal{A}(\Sigma^{\prime})$ in $\mathcal{A}^{H}$ such that
$\text{\em Gal}_{\Sigma^{\prime}}\mathcal{A}\neq H\leq\text{\em
Aut}\mathcal{A}$. Hence in the definition of $\mathcal{M}_{sub}^{H}$ above,
the condition $\text{\em Gal}_{\Sigma^{\prime}}\mathcal{A}=H$ is not trivial.

Trivially, $\mathcal{M}_{sub}^{H}\in\mathit{2}^{\mathit{2}^{\mathcal{A}}}$.
Then we can define a map

$\phi:\mathit{2}^{\text{Aut}\mathcal{A}}\rightarrow\mathit{2}^{\mathit{2}^{\mathcal{A}}}\;\text{via}\;H\mapsto\mathcal{M}_{sub}^{H}.$

And, define

$\text{ker}\phi=\\{H\leq\text{Aut}\mathcal{A}\;|\;\phi(H)=\emptyset\\}.$

which means

$H\in\text{ker}\phi\;\;\text{if and only
if}\;\;\mathcal{M}_{sub}^{H}=\varnothing\;\;\text{if and only
if}\;\;\nexists\mathcal{A}(\Sigma^{\prime})\;\text{such
that}\;\text{Gal}_{\Sigma^{\prime}}\mathcal{A}=H.$

Now we give an example of $H\in\text{ker}\phi$ i.e.,
$\mathcal{M}_{sub}^{H}=\varnothing$.

######  Example 3.9.

Let $S$ be a hexagon, $\mathcal{A}$ the cluster algebra from $S$, $H$ the
subgroup of the cluster automorphism group $\text{\em Aut}\mathcal{A}$
generated by the $60^{\circ}$ rotation. Then the only cluster subalgebra in
$\mathcal{A}^{H}$ is trivial. However, if the cluster subalgebra
$\mathcal{A}^{\prime}$ of $\mathcal{A}$ is trivial, then its Galois group is
all cluster automorphism group. Thus $\mathcal{M}_{sub}^{H}=\emptyset$.

Next, we have

######  Proposition 3.10.

The map $\phi$ restricted to $\mathit{2}^{\text{\em
Aut}\mathcal{A}}\backslash\text{ker}\phi$ is injective.

###### Proof.

Suppose
$H_{1},H_{2}\in\mathit{2}^{\text{Aut}\mathcal{A}}\backslash\text{ker}\phi$,
$\phi(H_{1})=\mathcal{M}^{H_{1}}_{sub}=\mathcal{M}^{H_{2}}_{sub}=\phi(H_{2})$.
Thus there exists a cluster subalgebra
$\mathcal{A}(\Sigma^{\prime})\in\mathcal{M}^{H_{1}}_{sub}=\mathcal{M}^{H_{2}}_{sub}$,
which means $H_{1}=\text{Gal}_{\Sigma^{\prime}}\mathcal{A}=H_{2}$. ∎

######  Proposition 3.11.

For $\xi$ defined in (3.4),
Im$\xi=\mathit{2}^{\text{Aut}\mathcal{A}}\backslash\text{ker}\phi$.

###### Proof.

If $H=\xi(\mathcal{A}(\Sigma^{\prime}))$ for some cluster subalgebra
$\mathcal{A}(\Sigma^{\prime})\subset\mathcal{A}$, then there exist a maximal
cluster subalgebra between $\mathcal{A}(\Sigma^{\prime})$ and
$\mathcal{A}^{H}$ denoting by $\mathcal{A}(\Sigma^{\prime\prime})$. It means
$\mathcal{A}(\Sigma^{\prime\prime})\in\mathcal{M}_{sub}^{H}$ and hence
$H\in\mathit{2}^{\text{Aut}\mathcal{A}}\backslash\text{ker}\phi$. Conversely,
if $H\in\mathit{2}^{\text{Aut}\mathcal{A}}\backslash\text{ker}\phi$, then
$\xi(\mathcal{A}(\Sigma^{\prime\prime}))=H$ for
$\mathcal{A}(\Sigma^{\prime\prime})\in\mathcal{M}_{sub}^{H}$.

∎

Refer to the notion of Galois extension of a field, we introduce the following
notion:

######  Definition 3.12.

For a cluster algebra $\mathcal{A}=\mathcal{A}(\Sigma)$ and
$H\in\mathit{2}^{\text{\em Aut}\mathcal{A}}\backslash\text{ker}\phi$,

(i) For a cluster subalgebra
$\mathcal{A}(\Sigma_{1})\in\mathcal{M}_{sub}^{H}$, let $H=\text{\em
Gal}_{\Sigma_{1}}\mathcal{A}$. Then we call
$\mathcal{A}(\Sigma_{1})\subseteq\mathcal{A}^{H}\subseteq\mathcal{A}(\Sigma)$
a Galois-like extension, and $\mathcal{A}(\Sigma_{1})$ is called a Galois-like
extension subalgebra of $\mathcal{A}(\Sigma)$.

(ii) Moreover, following (i), we call
$\mathcal{A}(\Sigma_{1})\subseteq\mathcal{A}(\Sigma)$ a Galois extension, if
it holds that

(3.5) $\mathcal{A}^{H}=\mathcal{A}(\Sigma_{1}).$

In this case $\mathcal{A}(\Sigma_{1})$ is called an Galois extension
subalgebra of $\mathcal{A}(\Sigma)$.

If we consider a strictly descending sequence of Galois-like extension
subalgebra, then the sequence of subgroups obtained similarly as (3.3) is
strictly ascending.

######  Proposition 3.13.

If there is a strictly descending sequence of Galois-like extension
subalgebras of a cluster algebra $\mathcal{A}$

$\mathcal{A}(\Sigma_{1})>\cdots>\mathcal{A}(\Sigma_{i})>\cdots>\mathcal{A}(\Sigma_{s}),$

then there is a strictly ascending sequence of subgroups of $\text{\em
Aut}\mathcal{A}$:

$\text{\em Gal}_{\Sigma_{1}}\mathcal{A}<\cdots<\text{\em
Gal}_{\Sigma_{i}}\mathcal{A}<\cdots<\text{\em Gal}_{\Sigma_{s}}\mathcal{A}.$

###### Proof.

There is a natural ascending sequence of subgroups:

$\text{Gal}_{\Sigma_{1}}\mathcal{A}\leq\cdots\leq\text{Gal}_{\Sigma_{i}}\mathcal{A}\leq\cdots\leq\text{Gal}_{\Sigma_{s}}\mathcal{A}.$

Since $\mathcal{M}_{sub}^{H_{i}}\not=\mathcal{M}_{sub}^{H_{j}}$ induces
$H_{i}\neq H_{j}$, all subgroups in the above sequence are different. So, the
result follows. ∎

######  Proposition 3.14.

Suppose $\mathcal{A}(\Sigma)$ is a cluster algebra and $H\leq\text{\em
Aut}\mathcal{A}(\Sigma)$, if $\mathcal{A}(\Sigma_{1})$ is maximal in
$\mathcal{A}(\Sigma)^{H}$ as a cluster subalgebra of $\mathcal{A}(\Sigma)$.
Denoting $H^{\prime}=\text{\em Gal}_{\Sigma_{1}}\mathcal{A}(\Sigma)$. Then
$\mathcal{A}(\Sigma_{1})\subseteq\mathcal{A}^{H^{\prime}}\subseteq\mathcal{A}(\Sigma)$
is a Galois-like extension.

###### Proof.

Since $\mathcal{A}(\Sigma_{1})$ is maximal in
$\mathcal{A}(\Sigma)^{H^{\prime}}\subset\mathcal{A}(\Sigma)^{H}$ as a cluster
subalgebra of $\mathcal{A}(\Sigma)$, we have
$\mathcal{A}(\Sigma_{1})\in\mathcal{M}_{sub}^{H^{\prime}}$.

∎

Being similar to the Galois theory in fields, we have the following theorem
for cluster subalgebras of a cluster algebra and subgroups of its automorphism
group.

######  Theorem 3.15.

Let $H_{1},H_{2}\in\mathit{2}^{\text{\em
Aut}\mathcal{A}}\backslash\text{ker}\phi$, where $\mathcal{A}$ is a cluster
algebra. Then $H_{1}$ is conjugate to $H_{2}$ if and only if there exist
$f\in\text{\em Aut}\mathcal{A}$,
$\mathcal{A}(\Sigma_{1})\in\mathcal{M}_{sub}^{H_{1}}$,
$\mathcal{A}(\Sigma_{2})\in\mathcal{M}_{sub}^{H_{2}}$ such that the
restriction of $f$ is a cluster isomorphism between $\mathcal{A}(\Sigma_{1})$
and $\mathcal{A}(\Sigma_{2})$.

###### Proof.

Suppose $H_{1}$ is conjugate to $H_{2}$, then there exists
$f\in\text{Aut}\mathcal{A}$ such that $f^{-1}H_{2}f=H_{1}$ or
$fH_{1}f^{-1}=H_{2}$. If $x\in\mathcal{A}(\Sigma_{1})$, let
$\mathcal{A}(\Sigma_{1})\in\mathcal{M}_{sub}^{H_{1}}$, then
$f(\mathcal{A}(\Sigma_{1}))$ is also a cluster subalgebra of $\mathcal{A}$,
which is denoted by $\mathcal{A}(\Sigma_{2})$. Since
$\mathcal{A}(\Sigma_{1}),\mathcal{A}(\Sigma_{2})$ are cluster subalgebras of
$\mathcal{A}$ and $f\in\text{Aut}\mathcal{A}$, $f$ commutes with mutations in
$\mathcal{A}(\Sigma_{1})$ and $\mathcal{A}(\Sigma_{2})$. Thus $f$ is a cluster
isomorphism between $\mathcal{A}(\Sigma_{1})$ and $\mathcal{A}(\Sigma_{2})$.

The cluster subalgebra $\mathcal{A}(\Sigma_{2})$ consist of
$f(x),x\in\mathcal{A}(\Sigma_{1})$, which is invariant under the action of
$H_{2}=fH_{1}f^{-1}$, thus $H_{2}\leq\text{Gal}_{\Sigma_{2}}\mathcal{A}$.
Besides, if $g\in\text{Gal}_{\Sigma_{2}}\mathcal{A}$, then
$f^{-1}(g(f(x)))=f^{-1}(f(x))=x,x\in\mathcal{A}(\Sigma_{1})$, $f^{-1}gf\in
H_{1}=f^{-1}H_{2}f$. Thus $g\in H_{2}$, which means
$H_{2}\geq\text{Gal}_{\Sigma_{2}}\mathcal{A}$. So
$H_{2}=\text{Gal}_{\Sigma_{2}}\mathcal{A}$.

On the other hand, if there exists a cluster subalgebra
$\mathcal{A}(\Sigma_{3})$ such that
$\mathcal{A}(\Sigma_{2})\subsetneq\mathcal{A}(\Sigma_{3})\subset\mathcal{A}^{H_{2}}$,
then $f^{-1}(\mathcal{A}(\Sigma_{3}))$ is the cluster subalgebra and
$\mathcal{A}(\Sigma_{1})\subsetneq
f^{-1}(\mathcal{A}(\Sigma_{3}))\subset\mathcal{A}^{H_{1}}$, which is contrary
to the definition of $\mathcal{A}(\Sigma_{1})$. Thus $\mathcal{A}(\Sigma_{2})$
is a maximal cluster subalgebra in $\mathcal{A}^{H_{2}}$ and
$\mathcal{A}(\Sigma_{2})\in\mathcal{M}_{sub}^{H_{2}}$.

Conversely, if there exist $f\in\text{Aut}\mathcal{A}$,
$\mathcal{A}(\Sigma_{1})\in\mathcal{M}_{sub}^{H_{1}}$,
$\mathcal{A}(\Sigma_{2})\in\mathcal{M}_{sub}^{H_{2}}$ such that $f$ is an
isomorphism from $\mathcal{A}(\Sigma_{1})$ to $\mathcal{A}(\Sigma_{2})$. We
can see that $f^{-1}H_{2}f\leq H_{1}$. Besides, $f^{-1}$ is also an
isomorphism from $\mathcal{A}(\Sigma_{2})$ to $\mathcal{A}(\Sigma_{1})$, thus
we have $f^{H}_{1}f^{-1}\leq H_{2}$, which means $f^{-1}H_{2}f\geq H_{1}$. So
$f^{-1}H_{2}f=H_{1}$.

∎

The maximal cluster subalgebras belonging to $\mathcal{M}_{sub}^{H}$ in
$\mathcal{A}^{H}$ is not unique in general. Now we discuss some common
properties of such maximal cluster algebras in $\mathcal{M}_{sub}^{H}$ and
then connect them with the method of Galois theory. We begin with some
conjectures:

######  Conjecture 3.16.

For a cluster algebra $\mathcal{A}$ and a subgroup $H$ of $\text{\em
Aut}\mathcal{A}$, let $\mathcal{A}(\Sigma_{1})$,
$\mathcal{A}(\Sigma_{2})\in\mathcal{M}^{H}_{sub}$. Then the rank of
$\mathcal{A}(\Sigma_{1})$ is the same as that of $\mathcal{A}(\Sigma_{2})$.

Conversely, assume $\mathcal{A}(\Sigma_{1})$ and $\mathcal{A}(\Sigma_{2})$ are
cluster subalgebras of $\mathcal{A}$ with the same number of exchange
variables, they may not always be in a same set $\mathcal{M}_{sub}^{H}$ for
some $H\leq\text{Aut}\mathcal{A}$.

######  Example 3.17.

In fact, consider a hexagon with its vertices labelled by $1,...,6$ clockwise,
then the cluster subalgebra $\mathcal{A}_{1}$ with a frozen variable
corresponding to the diagonal $\gamma_{14}$ is in $\mathcal{M}_{sub}^{H_{1}}$,
where $H_{1}$ is generated by two reflections and a rotation fixing
$\gamma_{14}$. However, the cluster subalgebra $\mathcal{A}_{2}$ with two
frozen variables corresponding to the diagonals $\gamma_{26},\gamma_{35}$ is
in $\mathcal{M}_{sub}^{H_{2}}$, where $H_{2}$ is generated by one of these
reflections fixing $\gamma_{26},\gamma_{35}$. Furthermore, the numbers of
exchange variables in these subalgebras are all zero.

By the Galois map in (3.4), give a totally ordered sequence of cluster
subalgebras, and their corresponding fixed subgroups also form a totally
ordered sequence. Conversely, give a totally ordered sequence of cluster
subgroups, we wonder to ask if there is a totally ordered sequence of cluster
subalgebras. This is our consideration in the following conjecture.

Let $\mathcal{A}(\Sigma)$ be a cluster algebra with initial extended exchange
matrix $\widetilde{B}=(b_{xy})$. We define $\Sigma^{(d)}$ to be the mixing-
type sub-seed of $\Sigma$ obtained by removing any frozen variable $y$
satisfying $b_{yx}=0$ for any exchange variable $x$ in $\Sigma$. Obviously,
$\mathcal{A}(\Sigma^{(d)})$ is a cluster subalgebra of $\mathcal{A}(\Sigma)$.
We call $\Sigma^{(d)}$ and $\mathcal{A}(\Sigma^{(d)})$ the reduced sub-seed
and reduced cluster subalgebra of $\Sigma$ and $\mathcal{A}(\Sigma)$
respectively.

######  Conjecture 3.18.

Let $H_{1},H_{2},...,H_{s}\in\mathit{2}^{\text{\em
Aut}\mathcal{A}}\backslash\text{ker}\phi$ and $H_{1}\leq H_{2}\leq\cdots\leq
H_{s}$. Then for any
$\mathcal{A}(\Sigma_{i})\in\mathcal{M}^{H_{i}}_{sub},1\leq i\leq s$, there
exist $\mathcal{A}(\Sigma_{j})\in\mathcal{M}^{H_{j}}_{sub}$ for
$j=1,...,i-1,i+1,...,s$ such that

$\mathcal{A}(\Sigma_{1}^{(d)})\geq\cdots\geq\mathcal{A}(\Sigma_{i}^{(d)})\geq\cdots\geq\mathcal{A}(\Sigma_{s}^{(d)}).$

Suppose Conjecture 3.18 holds for a cluster algebra $\mathcal{A}$, then we can
define a map such that

(3.6) $\zeta:\;\;\text{The set of ascending sequences
in}\;\mathit{2}^{\text{Aut}\mathcal{A}}\rightarrow\text{The set of descending
sequences in}\;\mathit{2}^{\mathcal{A}}$ $\text{
via}\;\;\;\zeta(H_{i})=\mathcal{A}(\Sigma_{i}^{(d)}),\;i=1,\cdots,s.$

We call $\zeta$ the reverse Galois map from
$\mathit{2}^{\text{Aut}\mathcal{A}(\Sigma)}$ to
$\mathit{2}^{\mathcal{A}(\Sigma)}$.

Let $A\subset\mathit{2}^{\mathcal{A}(\Sigma)}$ and
$G\subset\mathit{2}^{\text{Aut}\mathcal{A}(\Sigma)}$ be maximal subsets such
that $\zeta$ and $\xi$ are mutual invertible maps between $A$ and $G$. Then we
call $\xi$ a Galois correspondence and $\zeta$ a reverse Galois correspondence
respectively between $A$ and $G$.

In the next section, we confirm Conjecture 3.18 and Conjecture 3.16 to be true
for cluster algebras from surfaces with some conditions.

###  3.3. Galois extensions, stable monomial bases and $\mathcal{D}$-stable
bases for cluster algebras

######  Definition 3.19.

Let $\mathcal{A}$ be a cluster algebra. If $W$ is a $\mathbb{Q}$-basis for
$\mathcal{A}$ such that elements in $W$ are monomials of cluster variables and
for any $f\in\text{Aut}\mathcal{A}$, $W$ is stable for $f$, i.e., $f(W)\subset
W$, then $W$ is called a stable monomial basis for $\mathcal{A}$.

######  Definition 3.20.

(i) Let $\mathcal{A}$ be a cluster algebra and $R$ its polynomial ring of
frozen variables over $\mathbb{Q}$. We choose a set $C_{q}$ such that elements
in $C_{q}$ can be represented as Laurent polynomials of some cluster with
positive coefficients.

Let $\mathcal{D}$ be a set of some subsets of $C_{q}\cup\mathcal{X}$, such
that for any $U\in\mathcal{D}$, $U\cap\mathcal{X}$ is contained in a cluster,
where $\mathcal{X}$ is the set of all cluster variables.

(ii) Besides, for $z\in C_{q}\cup\mathcal{X}$ we set

$\langle k,z\rangle=\left\\{\begin{array}[]{ll}z^{k}&x\in\mathcal{X}\\\
P_{k}(z)&x\notin\mathcal{X}\end{array}\right.$

where $k\in\mathbb{Z}_{+}$ and $P_{k}(z)\in R[z]$.

(iii) Assume that for any $f\in\text{\em Aut}\mathcal{A}$,
(a) it holds $f(\langle k,z\rangle)=\langle k,f(z)\rangle$ for any
$k\in\mathbb{N},z\in U$;
(b) $\mathcal{D}$ is stable for $f$;
(c) $W=\bigcup_{U\in\mathcal{D}}\\{\prod_{x\in U^{\prime}}\langle
k_{x},x\rangle|U^{\prime}\subset U,k_{x}\in\mathbb{Z}_{+}\\}$ is a basis for
$\mathcal{A}$ and if $U_{1}\neq U_{2}$, then $\prod_{x\in U_{1}}\langle
k_{x},x\rangle\neq\prod_{x\in U_{2}}\langle k_{x},x\rangle$.

Then $W$ is called a $\mathcal{D}$-stable basis for $\mathcal{A}$, the
elements in $C_{q}$ are called quasi cluster variables of $\mathcal{A}$ and
the element in $\mathcal{D}$ is called a maximal compatible set of
$\mathcal{A}$.

Suppose $W$ is a $\mathcal{D}$-stable basis for $\mathcal{A}$ and
$f\in\text{Aut}\mathcal{A}$. If $\prod_{x\in U^{\prime}}\langle
k_{x},x\rangle\in W$ for $U^{\prime}\subset U\in\mathcal{D}$, then
$f(U)\in\mathcal{D}$ and $f(U^{\prime})\subset U$. Thus $f(\prod_{x\in
U^{\prime}}\langle k_{x},x\rangle)=\prod_{f(x)\in f(U^{\prime})}\langle
k_{x},f(x)\rangle\in W$ and hence $W$ is also stable for any
$f\in\text{Aut}\mathcal{A}$. It means $f$ is a bijection over $W$.

Suppose $W$ is a stable monomial basis for $\mathcal{A}$. We say that $W$ has
unique expression if for
$v_{i}=x_{i1}^{k_{i1}}x_{i2}^{k_{i2}}...x_{is_{i}}^{k_{is_{i}}}\in W,i=1,2$,
$v_{1}=v_{2}$ means $s_{1}=s_{2}$ and there is a permutation $\sigma\in
S_{s_{1}}$ such that $x_{1j}=x_{2\sigma(j)},$ for $j=[1,s_{1}]$.

######  Lemma 3.21.

Let $\mathcal{A}$ be a cluster algebra with a stable monomial basis $W$. If
the upper cluster algebra $\mathcal{U}$ of $\mathcal{A}$ is factorial, then
$W$ has the unique expression.

###### Proof.

For $v_{1},v_{2}\in W$, write
$v_{i}=x_{i1}^{k_{i1}}x_{i2}^{k_{i2}}...x_{is_{i}}^{k_{is_{i}}},\;i=1,2$.
Since $v_{1}=v_{2}$, we have

(3.7)
$x_{11}^{k_{11}}x_{12}^{k_{12}}...x_{1s_{1}}^{k_{1s_{1}}}=x_{21}^{k_{21}}x_{22}^{k_{22}}...x_{2s_{2}}^{k_{2s_{2}}}.$

According to [GLS13], exchange variables are irreducible in $\mathcal{U}$, and
invertible elements are Laurent monomials of frozen variables. If
$\mathcal{U}$ is factorial,
$x_{11}^{k_{11}}x_{12}^{k_{12}}...x_{1s_{1}}^{k_{1s_{1}}}=x_{21}^{k_{21}}x_{22}^{k_{22}}...x_{2s_{2}}^{k_{2s_{2}}}$
means the numbers of exchange variables in two sides are the same and there is
a permutation $\sigma^{\prime}$ such that
$x_{1j}=p_{j}x_{2\sigma^{\prime}(j)}$ for exchange variables $x_{1j}$ and
$x_{2\sigma^{\prime}(j)}$, where $p_{j}$ are Laurent monomials of frozen
variables.

Let $p_{j}=p_{j1}/p_{j2}$, where $p_{j1}$ and $p_{j2}$ are monomials of frozen
variables. Then we have $p_{j2}x_{1j}=p_{j1}x_{2\sigma^{\prime}(j)}$ in
$\mathcal{A}$. Note that both sides are extended cluster monomials. According
to [CL20b], extended cluster monomials are linearly independent. Thus
$x_{1j}=x_{2\sigma^{\prime}(j)}$. Removing the exchange variables of both
sides in (3.7), the monomials of the remaining frozen variables are equal.
Because of algebraic independence of frozen variables, $s_{1}=s_{2}$ and there
is a permutation $\sigma\in S_{s_{1}}$ such that
$x_{1j}=x_{2\sigma(j)},j\in[1,s_{1}]$.

∎

######  Lemma 3.22.

Let $\mathcal{A}(\Sigma)$ be a cluster algebra with the extended exchange
matrix of full rank, exchange variables
$x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime}\in\mathcal{A}$ and frozen variables
$y_{1},...,y_{s_{1}}$, $y_{1}^{\prime},...,y_{s_{2}}^{\prime}\in\mathcal{A}$.
If $y_{1}^{k_{1}}\cdots y_{s_{1}}^{k_{s_{1}}}x_{1}x_{2}=y_{1}^{\prime
k_{1}}\cdots y_{s_{2}}^{\prime k_{s_{2}}}x_{1}^{\prime}x_{2}^{\prime}$ and the
compatibility degree $d(x_{1},x_{2})\leq 2$ for
$k_{1},...,k_{s_{1}},k_{2}^{\prime},...,k_{s_{2}}^{\prime}\in\mathbb{N}$, then
$x_{1}=x_{1}^{\prime},x_{2}=x_{2}^{\prime}$, or
$x_{1}=x_{2}^{\prime},x_{2}=x_{1}^{\prime}$, and $s_{1}=s_{2}$ and there is a
permutation $\sigma\in S_{s_{1}}$ such that
$y_{i}=y_{\sigma(i)}^{\prime},i\in[1,s_{1}]$.

###### Proof.

Since $x_{1},x_{2},x_{1}^{\prime},x_{2}^{\prime}$ are exchange variables of
$\mathcal{A}$, $x_{2},x_{1}^{\prime},x_{2}^{\prime}$ can be express by a
cluster $\textbf{x}=\\{w_{1},w_{2},...,w_{m}\\}$, where $w_{1}=x_{1}$ as
Laurent polynomials:

$x_{2}=\frac{P_{2}(\textbf{x})}{D_{2}(\textbf{x})},\;\;x_{1}^{\prime}=\frac{P_{1}^{\prime}(\textbf{x})}{D_{1}^{\prime}(\textbf{x})},\;\;x_{2}^{\prime}=\frac{P_{2}^{\prime}(\textbf{x})}{D_{2}^{\prime}(\textbf{x})},$

where $P_{2},P_{1}^{\prime},P_{2}^{\prime}$ are polynomials of all cluster
variables x without monomials as their factors and
$D_{2},D_{1}^{\prime},D_{2}^{\prime}$ are Laurent monomials of x. In this
case, for $i=1,2,...,n$, the compatibility degrees
$d(w_{i},x_{2}),d(w_{i},x_{1}^{\prime})$, $d(w_{i},x_{2}^{\prime})$ are the
degrees of $D_{2},D_{1}^{\prime},D_{2}^{\prime}$ corresponding to $w_{i}$
respectively. According to [CL20a], we have
$d(w_{i},x_{2}),d(w_{i},x_{1}^{\prime}),d(w_{i},x_{2}^{\prime})\geq-1,i=1,2,...,n$,
and

(3.8) $y_{1}^{k_{1}}\cdots
y_{s_{1}}^{k_{s_{1}}}x_{1}\frac{P_{2}(\textbf{x})}{D_{2}(\textbf{x})}=y_{1}^{\prime
k_{1}}\cdots y_{s_{2}}^{\prime
k_{s_{2}}}\frac{P_{1}^{\prime}(\textbf{x})}{D_{1}^{\prime}(\textbf{x})}\frac{P_{2}^{\prime}(\textbf{x})}{D_{2}^{\prime}(\textbf{x})}\;\;\Longrightarrow\;\;y_{1}^{k_{1}}\cdots
y_{s_{1}}^{k_{s_{1}}}x_{1}=y_{1}^{\prime k_{1}}\cdots y_{s_{2}}^{\prime
k_{s_{2}}}\frac{D_{2}(\textbf{x})}{D_{1}^{\prime}(\textbf{x})D_{2}^{\prime}(\textbf{x})}\frac{P_{1}^{\prime}(\textbf{x})P_{2}^{\prime}(\textbf{x})}{P_{2}(\textbf{x})}.$

We now prove $x_{1}^{\prime}=x_{1}$ or $x_{2}^{\prime}=x_{1}$.

Since $P_{2},P_{1}^{\prime},P_{2}^{\prime}$ have not $w_{1}=x_{1}$ as a
factor,
$\frac{P_{1}^{\prime}(\textbf{x})P_{2}^{\prime}(\textbf{x})}{P_{2}(\textbf{x})}$
has not $x_{1}$ as a factor. So by (3.8),
$\frac{D_{2}(\textbf{x})}{D_{1}^{\prime}(\textbf{x})D_{2}^{\prime}(\textbf{x})}$
has $x_{1}$ as a factor.

Since $d(x_{1},x_{2})\leq 2$, it follows that $D_{1}^{\prime}$ or
$D_{2}^{\prime}$ has not $x_{1}$ as a factor.

In the case if $D_{1}^{\prime}$ has not $x_{1}$ as a factor. According to
[CL20a], there exists a cluster containing both $x_{1}$ and $x_{1}^{\prime}$.
We can assume that x also contain $x_{1}^{\prime}$, thus

$y_{1}^{k_{1}}\cdots
y_{s_{1}}^{k_{s_{1}}}x_{1}\frac{P_{2}(\textbf{x})}{D_{2}(\textbf{x})}=y_{1}^{\prime
k_{1}}\cdots y_{s_{2}}^{\prime
k_{s_{2}}}x_{1}^{\prime}\frac{P_{2}^{\prime}(\textbf{x})}{D_{2}^{\prime}(\textbf{x})}.$

Because $P_{2}$ and $P_{1}$ do not have monomials as their factor, we have
$P_{2}=P_{2}^{\prime}$.

Assume $\textbf{x}\in\Sigma$, where
$\Sigma=(\textbf{x}_{ex},\textbf{x}_{fr},\widetilde{B}_{m\times n})$ is a seed
of $\mathcal{A}$, and suppose $\mathcal{A}(\Sigma^{\prime})$ be the cluster
algebras with principle coefficients where initial seed
$\Sigma^{\prime}=(\textbf{x}_{ex},\textbf{x}_{fr}^{\prime},$
$\widetilde{B}^{\prime}_{2n\times n})$,
$\textbf{x}_{ex}=\\{w_{1},w_{2},...,w_{n}\\}$,
$\textbf{x}_{fr}^{\prime}=\\{w_{n+1}^{\prime},w_{n+2}^{\prime},...,w_{2n}^{\prime}\\}$.

Since $\widetilde{B}$ has full rank, we have
$\widetilde{B}^{\prime}=\Psi\widetilde{B}$ where $\Psi=(\psi_{ij})_{2n\times
m}$. Consider two cluster patterns

$(\Sigma(t))_{t\in\mathbb{T}_{n}}=(\textbf{x}_{ex}(t),\textbf{x}_{fr}(t),\widetilde{B}(t))_{t\in\mathbb{T}_{n}},\;\;\;(\Sigma^{\prime}(t))_{t\in\mathbb{T}_{n}}=(\textbf{x}_{ex}^{\prime}(t),\textbf{x}_{fr}^{\prime}(t),\widetilde{B}^{\prime}(t))_{t\in\mathbb{T}_{n}}$

with initial seeds $\Sigma(t_{0})=\Sigma$,
$\Sigma^{\prime}(t_{0})=\Sigma^{\prime}$ respectively. According to [FWZ], if
we define two morphisms $\gamma$ and $\varphi$ of algebras by setting

$\gamma(w_{j})=\prod_{i=1}^{n}w_{i}^{\psi_{ij}}\prod_{i=n+1}^{2n}w_{i}^{\prime{\psi_{ij}}},\;\;\;\varphi(w_{j})=\prod_{i=n+1}^{2n}w_{i}^{\prime\psi_{ij}},\;\;\;j=1,...,m,$

then we have $x_{i;t}^{\prime}=\frac{\gamma(x_{i;t})}{\varphi(x_{i,t})}.$

Thus for $x_{2},x_{2}^{\prime}$, we can set

$z_{2}=\frac{\gamma(P_{2})/\gamma(D_{2})}{\varphi(P_{2})/\varphi(D_{2})},z_{2}^{\prime}=\frac{\gamma(P_{2}^{\prime})/\gamma(D_{2}^{\prime})}{\varphi(P_{2}^{\prime})/\varphi(D_{2}^{\prime})},$

which are exchange variables in $\mathcal{A}(\Sigma^{\prime})$. Since
$\psi_{ij}=0,i\neq j;\,\phi_{ii}=1$, for $i,j\leq n$, then we have
$\gamma(w_{j})=w_{j}\varphi(w_{j}),j=1,...,m$, and hence
$\varphi(D_{2})/\gamma(D_{2})=D_{2}$,
$\varphi(D_{2}^{\prime})/\gamma(D_{2}^{\prime})=D_{2}^{\prime}$. Thus the
$F$-polynomial of $z_{2}$ is equal to

${\frac{\gamma(P_{2})/\gamma(D_{2})}{\varphi(P_{2})/\varphi(D_{2})}}\Big{|}_{w_{1}=\cdots=w_{n}=1}=\frac{\gamma(P_{2})}{\varphi(P_{2})}\frac{\varphi(D_{2})}{\gamma(D_{2})}\Big{|}_{w_{1}=\cdots=w_{n}=1}={\frac{\gamma(P_{2})}{\varphi(P_{2})}}\Big{|}_{w_{1}=\cdots=w_{n}=1},$

and the $F$-polynomial of $z^{\prime}_{2}$ is equal to
$\gamma(P_{2}^{\prime})/\varphi(P_{2}^{\prime})|_{w_{1}=\cdots=w_{n}=1}$. They
are equal since $P_{2}=P^{\prime}_{2}$, which means $z_{2}=z_{2}^{\prime}$. On
the other hand, $\widetilde{B}=\Psi^{\prime}\widetilde{B}^{\prime}$, so
according to [FWZ], we have $x_{2}=x_{2}^{\prime}$ and hence
$x_{1}=x_{1}^{\prime}$.

In the case if $D_{2}^{\prime}$ does not have $x_{1}$ as a factor, we can
similarly prove $x_{2}=x_{1}^{\prime}$ and $x_{1}=x_{2}^{\prime}$.

Considering the algebraically independence of frozen variables, we have
$s_{1}=s_{2}$ and there is a permutation $\sigma\in S_{s_{1}}$ such that
$y_{i}=y_{\sigma(i)}^{\prime},i\in[1,s_{1}]$.

∎

Thus we have the following corollary.

######  Corollary 3.23.

Let $\mathcal{A}$ be a cluster algebra with a stable monomial basis
$W=\\{w_{\lambda}\\}_{\lambda\in\Lambda}$ and let the extended exchange
matrices of $\mathcal{A}$ be of full rank. If the degrees of elements in $W$
except extended cluster monomials are $2$ and the compatibility degree of each
two exchange variables in the expression of $w_{\lambda}$ is not larger than
$2$, then $W$ has the unique expression.

###### Proof.

Due to the condition, any pair $v_{i},i=1,2$ in $W$ except extended cluster
monomials can be expressed as $v_{i}=p_{i}x_{i1}x_{i2}$, where $p_{i}$ are a
monomial of frozen variables and $x_{1},x_{2}$ are cluster variables such that
the compatibility degree $d(x_{i1},x_{i2})\leq 2$. If $v_{1}=v_{2}$, we have
$p_{1}x_{11}x_{12}=p_{2}x_{21}x_{22}$. The corollary follows immediately from
Lemma 3.22.

∎

Note that a standard monomial basis is not necessarily a $\mathcal{D}$-stable
basis and vice versa.

According to [CK08], all extended cluster monomials form a basis if the
cluster algebra is of finite type, Thus the set of all extended cluster
monomials from the cluster algebra of finite type is a stable monomial basis
as well as a $\mathcal{D}$-stable basis.

We have also some examples from cluster algebras of non-finite type. For
example, the bases $\mathcal{B}^{\circ}$, $\mathcal{B}$ and
$\mathcal{B}^{\prime}$ given in [MSW13, Thu14] without coefficient are
$\mathcal{D}$-stable bases, which is introduced and proved in the next
section.

If a cluster algebra $\mathcal{A}$ is acyclic, i.e., there is an acyclic
exchange matrix in a seed of $\mathcal{A}$, a standard monomial basis for
$\mathcal{A}$ was introduced in [BFZ05], which is a stable monomial basis with
the unique expression.

If $\mathcal{A}$ has a $\mathcal{D}$-stable basis, then we give a discussion
on the condition for $\mathcal{A}(\Sigma_{1})\subseteq\mathcal{A}(\Sigma)$ to
be a Galois extension.

######  Theorem 3.24.

Let $\mathcal{A}=\mathcal{A}(\Sigma)$ be a cluster algebra and $H$ be a
subgroup of the cluster automorphism group of $\mathcal{A}(\Sigma)$. Suppose
$\mathcal{A}(\Sigma_{1})$ is maximal in $\mathcal{A}^{H}$ as a cluster
subalgebra of $\mathcal{A}$.

(i) If $\mathcal{A}^{H}=\mathcal{A}(\Sigma_{1})$, then $|Hz|=+\infty$ for any
$z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ which can be represented as
Laurent polynomials of some cluster with positive coefficients.

(ii) Conversely, $\mathcal{A}^{H}=\mathcal{A}(\Sigma_{1})$ holds if either (a)
$\mathcal{A}$ has a stable monomial basis with the unique expression and
$|Hz|=+\infty$ for any cluster variable
$z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ or (b) $\mathcal{A}$ has a
$\mathcal{D}$-stable basis $|Hz|=+\infty$ for any cluster variable or quasi
cluster variable $z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$.

###### Proof.

(i) It is equivalent to prove that if there exists a cluster variable
$x\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ such that $|Hx|<+\infty$,
then $\mathcal{A}^{H}\neq\mathcal{A}(\Sigma_{1})$.

According to Theorem 2.6, we can assume $\Sigma_{1}$ is a mixing-type sub-seed
of $\Sigma$. If $\Sigma_{1}$ contains all cluster variables in $\Sigma$, then
$H$ is the trivial group and the sufficiency is obvious.

Thus we can assume $\textbf{x}^{\prime}\neq\textbf{x}$, where x (resp. x’) is
the cluster of $\Sigma$ (resp. $\Sigma_{1}$). Suppose
$z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ and
$|Hz|=\\{z_{i},i\in[1,t]\\},z_{1}=z$. The Laurent expression of $z_{i}$ with
respect to x are

$z_{i}=\frac{P_{i}(\textbf{x})}{D_{i}(\textbf{x})},i\in[1,t]$

where $P_{i}(\textbf{x})$ are polynomials of x with positive coefficients and
$D_{i}(\textbf{x})$ are extended cluster monomials of x. For any monomial $M$
of $\textbf{x}^{\prime}$, if $z_{1}M\in\mathcal{A}(\Sigma_{1})$, then
$f(z_{1})=z_{1}$, $f\in H$. Thus
$z_{1}\in\mathcal{A}^{H}\backslash\mathcal{A}(\Sigma_{1})$ and this completes
the proof. Otherwise, we can replace $z_{1}$ with $z_{1}M$ for some monomial
$M$ such that $\frac{P_{i}(\textbf{x})}{D_{i}(\textbf{x})},i\in[1,t]$ are
polynomials with respect to $\textbf{x}^{\prime}$.

Let $Z=\sum_{i=1}^{t}z_{i}$, then $Z\in\mathcal{A}^{H}$. Assume
$Z\in\mathcal{A}(\Sigma_{1})$, then $z=L(\textbf{x}^{\prime})$, where
$L(\textbf{x}^{\prime})$ is a Laurent polynomial of $\textbf{x}^{\prime}$. And
hence we have

(3.9)
$L(\textbf{x}^{\prime})=\sum_{i}^{t}\frac{P_{i}(\textbf{x})}{D_{i}(\textbf{x})}.$

Since $z_{1}\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$, there exists
$x\in\textbf{x}\backslash\textbf{x}^{\prime}$ such that $x$ appears in
$\frac{P_{1}(\textbf{x})}{D_{1}(\textbf{x})}$. Since coefficients of
$P_{i}(\textbf{x}),i\in[1,t]$ are positive, $x$ also appears in the right of
(3.9). But $x$ does not appear on left of (3.9), which means the equality does
not hold. Thus $z\notin\mathcal{A}(\Sigma_{1})$ and hence
$\mathcal{A}^{H}\neq\mathcal{A}(\Sigma_{1})$.

(ii) It only needs to prove under the assumption that $\mathcal{A}$ has a
stable monomial basis with the unique expression (resp. $\mathcal{D}$-stable
basis) and $\mathcal{A}^{H}\neq\mathcal{A}(\Sigma_{1})$, it following that
there exists a cluster variable (resp. cluster variable or quasi cluster
variable) $z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ such that
$|Hz|<+\infty$.

Suppose $\mathcal{A}^{H}\neq\mathcal{A}(\Sigma_{1})$, then there exists
$v\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$, such that $f(v)=v$, for
any $f\in H$. Since $\mathcal{A}$ has a stable monomial basis (resp.
$\mathcal{D}$-stable basis) $W$, then $z$ can be represented by
$v=k_{1}v_{1}+k_{2}v_{2}+\cdots+k_{s}v_{s},1,\cdots,s\in I$ and
$k_{1},...,k_{s}\in\mathbb{Q}$. Thus

$f(z)=k_{1}f(v_{1})+k_{2}f(v_{2})+\cdots+k_{s}f(v_{s})=k_{1}v_{1}+k_{2}v_{2}+\cdots+k_{s}v_{s}.$

Because $W$ is stable for $f$, the equality holds if and only if $f$ is
bijective over $\\{v_{1},v_{2},...,v_{s}\\}$.

Since $v\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$, there exists $v_{k}$
such that $v_{k}\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$.

(a) In the case $W$ is a stable monomial basis, denote by $X$ the set of all
cluster variables appearing in expressions of $v_{i},i\in[1,s]$. Then $f$
restricted to $X$ is bijective. Since
$v_{k}\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$, there is $z\in X$ such
that $z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$. Because of the
arbitrariness of $f$, we have $|Hz|<+\infty$.

(b) In the case $W$ is a $\mathcal{D}$-stable basis, write $v_{j}=\prod_{x\in
U_{j}}\langle k_{x},x\rangle,j=[1,s],k_{x}\in\mathbb{Z}_{+}$. Denote by
$X_{1}$ and $X_{2}$ all cluster variables and quasi cluster variables in
$\bigcup_{j=1}^{s}U_{j}$, respectively. Since the set of cluster variables is
stable for $f$ and according to Definition 3.20 (iii), $f$ restricted to
$X_{1}$ and $X_{2}$ are bijective. Since
$v_{k}\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$, there is a cluster
variable $z\in X_{1}$ or quasi cluster variables $z\in X_{2}$ such that
$z\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$. Because of the
arbitrariness of $f$, we have $|Hz|<+\infty$.

∎

According to Lemma 3.21 and Corollary 3.23, we have

######  Corollary 3.25.

Let $\mathcal{A}=\mathcal{A}(\Sigma)$ be a cluster algebra with stable
monomial basis $W=\\{w_{\lambda}\\}_{\lambda\in\Lambda}$ and a subgroup
$H\leq\text{Aut}\mathcal{A}$ such that there is a maximal cluster subalgebra
$\mathcal{A}(\Sigma_{1})$ in $\mathcal{A}^{H}$. Suppose $|Hx|=+\infty$ for any
cluster variable $x\in\mathcal{A}\backslash\mathcal{A}(\Sigma_{1})$ and
either (i) the upper cluster algebra $\mathcal{U}$ of $\mathcal{A}$ is
factorial
or (ii) the extended exchange matrices of $\mathcal{A}$ are of full rank and
the degrees of elements in $W$ except extended cluster monomials are $2$ and
the compatibility degree of each two exchange variables in the expression of
$w_{\lambda}$ is not larger than $2$.

Then $\mathcal{A}^{H}=\mathcal{A}(\Sigma_{1})$.

According to Theorem 3.24, we have the following remark.

######  Remark 3.26.

For any skew-symmetrizable cluster algebra $\mathcal{A}$, if $\text{\em
Aut}\mathcal{A}$ is finite, then $\text{\em Aut}\mathcal{A}x<+\infty$ for any
cluster variable x. According to Theorem 3.24 (i), any cluster subalgebra
$\mathcal{A}(\Sigma^{\prime})\subsetneq\mathcal{A}$ is not Galois extension
subalgebra.
If $\mathcal{A}$ is the cluster algebra with principal coefficients, according
to proposition 3.6, $\text{\em Aut}\mathcal{A}$ is finite and hence in this
case any cluster subalgebra
$\mathcal{A}(\Sigma^{\prime})\subsetneq\mathcal{A}$ is not Galois extension
subalgebra.

##  4\. Examples of $\mathcal{D}$-stable bases and Galois theory of cluster
algebras from surfaces

In this section, we give some examples of $\mathcal{D}$-stable bases for the
cluster algebras from surfaces and discuss the Galois correspondence and
Galois-like extensions for this cluster algebras.

Suppose $\mathcal{A}$ is a cluster algebra with coefficients and
$\mathcal{A}^{\prime}$ is the cluster algebra obtained from $\mathcal{A}$ by
specializing all frozen variables to 1, then $\mathcal{A}^{\prime}$ is a
cluster algebra without coefficients. Note that cluster automorphisms of
$\mathcal{A}$ are also automorphisms of $\mathcal{A}^{\prime}$ by specializing
all frozen variables to 1, but in general, cluster automorphisms of
$\mathcal{A}^{\prime}$ are not necessarily that of $\mathcal{A}$. So, in order
to let the cluster automorphism group be as large as possible, we always
suppose the corresponding cluster algebras of marked surfaces are without
coefficient in this section if there is no additional explanation.

###  4.1. Bangle, bracelet, and band bases as $\mathcal{D}$-stable bases

Let $(S,M)$ be a marked surface, $h$ be a homeomorphism of $(S,M)$ such that
$h(M)=M$ and $R$ is a subset of the set consisting of all punctures in
$(S,M)$, then we can obtain a cluster automorphisms of $\mathcal{A}(S,M)$
corresponding to homeomorphism of $(S,M)$ induced by $h$ and $R$:

$\psi_{h}\in\text{Aut}\mathcal{A}(S,M)\quad\text{defined by
}\psi_{h}(x_{\gamma}):=x_{h(\gamma)},$

where $\gamma\in A_{\bowtie}(S,M)$ the set consisting of all tagged arcs of
$(S,M)$ and $x_{\gamma}$ is its counterpart cluster variable in
$\mathcal{A}(S,M)$. Besides, we set

$\psi_{R}\in\text{Aut}\mathcal{A}(S,M)\quad\text{defined by
}\psi_{R}(x_{\gamma}):=x_{\gamma^{R}},$

where $\gamma^{R}$ denote the tagged arc obtained from $\gamma$ by changing
the taggings of it at those of its endpoints that belong to $R$. For $h$ and
$R$ as above, denote by $\psi_{h,R}:=\psi_{R}\psi_{h}.$

According to [Gu11, BS15, BQ15, BY18], we have the following theorem:

######  Theorem 4.1.

[Gu11, BS15, BQ15, BY18] Let $(S,M)$ be a marked surface different from any
one of (1) the 4-punctured sphere, (2) the once-punctured 4-gon, (3) the
twice-punctured digon. Then the group of cluster automorphisms of the
corresponding cluster algebra $\mathcal{A}(S,M)$ is given as follows:

(a) if $(S,M)$ is a once-punctured closed surface, then

$\mathcal{MCG}^{\pm}(S,M)\cong\text{\em
Aut}\mathcal{A}(S,M),\;\mathcal{MCG}(S,M)\cong\text{\em
Aut}^{+}\mathcal{A}(S,M)\;\;\;via\;\;\;h\mapsto\psi_{h},$

(b) if $(S,M)$ is not a once-punctured closed surface, then

$\mathcal{MCG}^{\pm}_{\bowtie}(S,M)\cong\text{\em
Aut}\mathcal{A}(S,M),\;\mathcal{MCG}_{\bowtie}(S,M)\cong\text{\em
Aut}^{+}\mathcal{A}(S,M)\;\;\;via\;\;\;(h,R)\mapsto\psi_{h,R},$

where $h$ is a representative element of a class in
$\mathcal{MCG}^{\pm}(S,M)$, $R$ is a subset of sets of all punctures in
$(S,M)$.

This theorem builds up a connection between mapping class groups and cluster
automorphism groups for the surfaces satisfying the conditions (1)–(3). Both
the results in the above existing literature and that obtained below show that
the kind of surfaces satisfying these three conditions is very important and
useful. For convenience, we give the definition for such surfaces as follows:

######  Definition 4.2.

A marked surface $(S,M)$ is called a feasible surface if $(S,M)$ is different
from any one of (1) the 4-punctured sphere, (2) the once-punctured 4-gon, and
(3) the twice-punctured digon.

$\frac{1}{3!}\sum_{\sigma\in S_{3}}\sigma$
---
Figure 1. A bangle $\text{Bang}_{3}\zeta$(left), a bracelet
$\text{Brac}_{3}\zeta$(middle) and a band $\text{Band}_{3}\zeta$ (right).

Next, we introduce some material about bases for cluster algebras from
surfaces.

Suppose $\zeta$ is an essential loop i.e., a loop in the interior of a surface
$S$ which is neither a tagged arc nor contractible nor contractible onto a
single puncture. The bangle $\text{Bang}_{k}\zeta$ is defined to be the union
of $k$ copies of $\zeta$. The bracelet $\text{Brac}_{k}\zeta$ is defined to be
the closed loop obtained by concatenating $\zeta$ exactly $k$ times. Suppose
$I\subset\zeta$ is a short interval, the band $\text{Band}_{k}\zeta$ is
defined to be $k$ copies of $\zeta\backslash I$ with the ends connected by
averaging all ways of pairing the endpoints on the two sides under the formal
addition, see Figure 1.

According to [MSW13, Thu14], essential loops, bangles, bracelets, and bands
correspond to elements of the upper cluster algebra of $\mathcal{A}(S,M)$,
respectively.

The Chebyshev polynomials of the first kind are polynomials $T_{k}(z)$ defined
recursively as follows:

$T_{0}(z)=2,\,\,T_{1}(z)=z,\;\;\text{and
}\;\;T_{n+1}(z)=zT_{n}(z)-T_{n-1}(z)\;\;\;\forall n\in\mathbb{N},$

and the Chebyshev polynomials of the second kind are polynomials $U_{k}(z)$
defined recursively as follows:

$U_{0}(z)=1,\,\,U_{1}(z)=z,\;\;\text{and
}\;\;U_{n+1}(z)=zU_{n}(z)-U_{n-1}(z)\;\;\;\forall n\in\mathbb{N}.$

According to [MSW13, Thu14], we have $x_{\text{Bang}_{k}\zeta}=x_{\zeta}^{k}$,
$x_{\text{Brac}_{k}\zeta}=T_{k}(x_{\zeta})$ and
$x_{\text{Band}_{k}\zeta}=U_{k}(x_{\zeta})$, where $x_{\zeta}$ is the element
corresponding to the essential loop $\zeta$.

Suppose $\gamma,\gamma_{1}$ and $\gamma_{2}$ are generalized arcs or closed
loops in $(S,M)$ satisfying:
(1) $\gamma_{1}$ and $\gamma_{2}$ intersect at a point $x$,
(2) $\gamma$ is self-intersecting at a point $x$.
Then we let $C$ be the multicurve $\\{\gamma_{1},\gamma_{2}\\}$ or
$\\{\gamma\\}$ depending on which of the above two cases we suppose. We define
the smoothing of $C$ at the point $x$ to be the pair of multicurves
$C_{+}=\\{\alpha_{1},\alpha_{2}\\}$ (resp. $\\{\alpha\\}$) and
$C_{-}=\\{\beta_{1},\beta_{2}\\}$ (resp. $\\{\beta\\}$).

Here, the multicurve $C_{+}$ (resp. $C_{-}$) is the same as $C$ except for the
local change via replacing the crossing $\times$ with the pair of segments
$~{}_{\cap}^{\cup}$ (resp. $\supset\subset$).

In a local region, we can choose an arc $\alpha$ for any essential loop
$\zeta$ such that the smoothing of $\zeta$ and $\alpha$ is given as in Figure
2. Thus the corresponding element $x_{\zeta}$ can be represented as a Laurent
polynomial of some cluster with positive coefficients.

$\alpha$
---
$\zeta$
---
$=$
---
$+$
---
$\alpha_{1}$
---
$\alpha_{2}$
---
Figure 2. The smoothing of $\zeta$ and $\alpha$

As known well, a multiset means a gathering of some elements allowing the same
elements included more tha once.

######  Definition 4.3.

(i) A finite multiset $C^{\circ}$ of arcs and bangles is said to be
$\mathcal{C}^{\circ}$-compatible if no two elements of $C^{\circ}$ intersect
with each other and there is at most one bangle for each essential loop in
$C^{\circ}$.

(ii) A finite multiset $C$ of arcs and bracelet is said to be
$\mathcal{C}$-compatible if no two elements of $C$ intersect with each other,
except for the self-intersection of a bracelet, and there is at most one
bracelet for each essential loop in $C$.

(iii) A finite multiset $C^{\prime}$ of arcs and band is said to be
$\mathcal{C}^{\prime}$-compatible if no two elements of $C^{\prime}$ intersect
with each other, except for the self-intersection of a band, and there is at
most one band for each essential loop in $C^{\prime}$.

Suppose $\mathcal{C}^{\circ}$ (resp. $\mathcal{C}$ and $\mathcal{C}^{\prime}$)
is the set consisting of all $\mathcal{C}^{\circ}$-compatible (resp.
$\mathcal{C}$-compatible and $\mathcal{C}^{\prime}$-compatible) multisets. For
any surface $(S,M)$ with at least two marked points and without punctures,
suppose $\mathcal{A}(S,M)$ is the cluster algebra from $(S,M)$, then the
Bangle basis $\mathcal{B}^{\circ}$, the Bracelet basis $\mathcal{B}$ and the
Band basis $\mathcal{B}^{\prime}$ of $\mathcal{A}(S,M)$ are defined in [MSW13]
and [Thu14] respectively as follows:

(4.1) $\mathcal{B}^{\circ}=\\{\prod_{c\in
C^{\circ}}x_{c}|\;C^{\circ}\in\mathcal{C}^{\circ}\\};\;\;\;\mathcal{B}=\\{\prod_{c\in
C}x_{c}|\;C\in\mathcal{C}\\};\;\;\;\mathcal{B}^{\prime}=\\{\prod_{c\in
C^{\prime}}x_{c}|\;C^{\prime}\in\mathcal{C}^{\prime}\\}.$

Set the elements corresponding to essential loops as quasi cluster variables
and the maximal sets of $\mathcal{C}^{\circ}$-compatible (resp.
$\mathcal{C}$-compatible, $\mathcal{C}^{\prime}$-compatible) elements as
elements of maximal compatible sets, then the set of maximal compatible sets
$\mathcal{D}$ is stable for any $f\in\text{Aut}\mathcal{A}(S,M)$.

Thus the Bangle basis $\mathcal{B}^{\circ}$, the Bracelet basis $\mathcal{B}$
and the Band basis $\mathcal{B}^{\prime}$ satisfying the Definition 3.20 and
hence they are $\mathcal{D}$-stable bases for $\mathcal{A}(S,M)$.

###  4.2. Reverse Galois maps and the Galois inverse problem for cluster
algebras from feasible surfaces

Now we prove Conjecture 3.18 and 3.16 for cluster algebras from surfaces with
certain conditions.

Recall that a tagged arc $\gamma$ with the natural orientation is also denoted
by $\gamma$ and the tagged arc with the reverse orientation is denoted by
$\gamma^{-1}$. Denote by $\rho^{\prime}$ the isomorphism from the cluster
automorphism group to the mapping class group in Theorem 4.1. Define a map
$\tau$ which assigns each mapping class to a representative in this mapping
class. Let $\rho=\tau\rho^{\prime}$.

######  Lemma 4.4.

Let $(S,M)$ be a feasible surface, $\mathcal{A}(S,M)$ its cluster algebra
without coefficient, and $H\leq\text{\em Aut}\mathcal{A}(S,M)$. If
$x_{\gamma}\in\mathcal{A}(S,M)^{H}$ for some oriented tagged arc $\gamma$ in
$(S,M)$ and $\rho(f)(\gamma)=\gamma$ for any $f\in H$, then
$x_{\gamma^{p}}\in\mathcal{A}(S,M)^{H}$, where $p$ is a puncture as well as an
endpoint of $\gamma$ and $\gamma^{p}$ are the tagged arc obtained from
$\gamma$ by changed the tagging at $p$.

###### Proof.

For each $f\in H$, we also consider $f$ as an element in the mapping class
group. Since $\rho(f)(\gamma)=\gamma$, the endpoints of $\gamma$ is unchanged
under $f$. Let $\psi_{p}\in\mathcal{A}(S,M)^{H}$ such that
$\psi_{p}(x_{\alpha})=x_{\alpha^{p}}$ for any tagged arc $\alpha$. Then $f$
and $\psi_{p}$ commute with each other if we restrict them to $\gamma$. Thus
$f(x_{\gamma^{p}})=f(\psi_{p}(x_{\gamma}))=\psi_{p}(f(x_{\gamma}))=x_{\gamma^{p}}$.
Because of the arbitrariness of $f$, we have
$x_{\gamma^{p}}\in\mathcal{A}(S,M)^{H}$. ∎

######  Remark 4.5.

Suppose $x$ is an exchange variable in $\mathcal{A}(S,M)^{H}$, i.e., there is
a mixing-type sub-seed $\Sigma^{\prime}$ in $\mathcal{A}(S,M)^{H}$ of $\Sigma$
such that $x$ is exchange variable in $\Sigma^{\prime}$. In this case, for
each variable $y\in\Sigma$, if $b_{xy}\neq 0$, then $y\in\Sigma^{\prime}$.

Suppose $\gamma,\gamma_{1}$ correspond to $x$ and $y$ respectively, such that
$b_{xy}\neq 0$. Since $\rho(f)$ maps $\gamma,\gamma_{1}$ to themselves for
each $f\in H$, $b_{f(x)f(y)}=b_{xy}$, thus $f$ is a direct cluster
automorphism. Hence $H\leq\text{\em Aut}^{+}\mathcal{A}(S,M)$.

Besides, let $\gamma$ be an oriented arc corresponding to an exchange variable
$x_{\gamma}$ in $\mathcal{A}(S,M)^{H}$, then for any $f\in H$, $\rho(f)$ maps
$\gamma$ to itself preserving the orientation.

######  Lemma 4.6.

Let $\mathcal{A}(S,M)$ be a cluster algebra without coefficient from a
feasible surface $(S,M)$, and for any $f\in\text{\em
Aut}^{+}\mathcal{A}(S,M)$, let $x_{\gamma}$ belong to a mixing-type sub-seed
$\Sigma^{\prime}$ in $\mathcal{A}(S,M)^{f}$ such that
$\rho(f)(\gamma)=\gamma^{-1}$. Then there is no oriented tagged arc $\delta$
corresponding to cluster variables $x_{\delta}\in\Sigma^{\prime}$ such that
$\rho(f)(\delta)=\delta$.

###### Proof.

If $\rho(f)(\gamma)=\gamma^{-1}$, then the taggings at two endpoints $p,q$ of
$\gamma$ are the same and we also have
$\rho(f)(\gamma^{pq})=(\gamma^{pq})^{-1}$. Combining with Lemma 4.4, This
conclusion holds for tagged arcs corresponding to cluster variables in
$\Sigma^{\prime}$ if and only if it holds for the tagged arcs obtained from
these arcs corresponding to $\Sigma^{\prime}$ by changing the tagging at $p,q$
at the same time or at other punctures. Thus we can assume tagged arcs
corresponding to cluster variables in $\Sigma^{\prime}$ are contained in a
triangulation $T$ satisfying all punctures are tagged plain except for some
punctures tagged in different ways in $T$.

Now we assume there is an oriented tagged arc $\delta$ such that
$\rho(f)(\delta)=\delta$. In this case, $\gamma$ and $\delta$ do not tag
notched at their endpoints, since $\rho(f)(\gamma)=\gamma^{-1}$ and
$\rho(f)(\delta)=\delta$. We can do flip at each tagged arc that tags notched
at its endpoint to obtain a triangulation whose tagged arcs tag plain. Thus we
can assume all arcs in $T$ tagging plain.

Suppose $\gamma$ is adjacent to $\alpha_{1}$, $\alpha_{i}$ is adjacent to
$\alpha_{i+1}$ for $i\in[1,s-1]$ and $\alpha_{s}$ is adjacent to $\delta$ in
$T$, $\rho(f)(\alpha_{i})=\beta_{i}$ for $i\in[1,s]$. Then $\gamma$ is
adjacent to $\beta_{1}$, $\beta_{i}$ is adjacent to $\beta_{i+1}$ for
$i\in[1,s-1]$ and $\beta_{s}$ is adjacent to $\delta$ (see Figure 3).

Since $f\in\text{Aut}^{+}\mathcal{A}(S,M)$ and $\rho(f)(\gamma)=\gamma^{-1}$,
$\rho(f)$ is the $180^{\circ}$ rotation of the local region in Figure 3. Thus
$\rho(f)$ reverses the orientation of $\delta$ in Figure 3, which is contrary
to that the surface $S$ is oriented. ∎

Figure 3. The local region from $\gamma$ to $\delta$

######  Lemma 4.7.

Let $(S,M)$ be a feasible surface, $\mathcal{A}(S,M)$ its cluster algebra
without coefficient, and there is an oriented tagged arc $\delta$ such that
$\rho(f)(\delta)=\delta$ for some $f\in\text{\em Aut}^{+}\mathcal{A}(S,M)$.
Then there is no oriented tagged arc $\gamma$ such that
$\rho(f)(\gamma)=\gamma^{-1}$.

###### Proof.

Similarly, we can assume all arcs tagged plain. Asume there is an oriented arc
$\gamma$ such that $\rho(f)(\gamma)=\gamma^{-1}$. If $\gamma$ intersects with
$\delta$, then $\rho(f)$ reverses the orientation of local region contains
$\gamma$ and $\delta$, which is contrary to
$f\in\text{Aut}^{+}\mathcal{A}(S,M)$. If there is no intersection between
$\gamma$ and $\delta$, then there is a mixing-type sub-seed $\Sigma^{\prime}$
in $\mathcal{A}(S,M)^{f}$ contains the cluster variables corresponding to
$\gamma$ and $\delta$, which is contrary to Lemma 4.6.

∎

$f\in\text{Aut}^{+}\mathcal{A}(S,M)$ is said to be orientation-preserved if
there is an oriented arc $\gamma$ such that $\rho(f)(\gamma)=\gamma$.

A subsurface $C$ of a marked surface $(S,M)$ is a closed subset of $S$ which
is a two-dimensional submanifold and whose boundary components are a loop in
$(S,M)$ which is neither contractible onto an arc, a boundary arc or a single
puncture nor contractible except the case that one of connected components of
$C$ is the digon. Note that here a loop $\zeta$ is defined to be contractible
onto $X\subset S$ (resp. contractible) if there is a isotopy from the identity
of $(S,M)$ that transforms $\zeta$ into $X$ which fixes $X$ (resp. any point
in $S\backslash(\partial S\cup M)$).

We suppose that for a marked surface $(S,M)$, an open disk, open once-
punctured disk or open annulus mentioned in the sequel, which is a connected
component of $S\backslash C$, has no marked point on its boundary.

Suppose $C$ is a subsurface of $(S,M)$, we consider maximal number of
compatible tagged arcs of $(S,M)$ which lie in $(C,C\cap M)$.

We call a set of compatible tagged arcs lying in $(C,C\cap M)$ which is
maximal a maximal tagged arcs set of $(C,C\cap M)$.

######  Lemma 4.8.

Let $(S,M)$ be a marked surface, $C$ a connected subsurface of $(S,M)$. Then
the cardinalities $n$ of maximal tagged arcs sets of $(S,M)$ are constant.

Concretely, we can obtain the valuation of the cardinalities $\mathfrak{N}$
respectively in the following various cases:
(1) the cardinality $\mathfrak{N}=0$ in the case $(C,C\cap M)$ is a subsurface
with no marked point or the once-punctured disk;
(2) the cardinality $\mathfrak{N}=2$ in the case $(C,C\cap M)$ is the once-
punctured disk with 1 marked point on the boundaries;
(3) the cardinality $\mathfrak{N}=n_{b}$ in the case $(C,C\cap M)$ is the
annulus or the digon;
(4) the cardinality $\mathfrak{N}=6g+3a+3p+c-6-t-s-d+n_{b}$ in other cases,
where $s$ is the number of open annuli, $d$ is the number of open once-
punctured disks, among connected components of $S\backslash C$, $t$ is the
number of all boundary components with no marked point on their boundaries in
$(C,C\cap M)$ and $n_{b}$ is the number of boundary arcs of $C$ which are
tagged arc in $(S,M)$.

###### Proof.

If $C$ is either a once-punctured disk or $C\cap M\neq\emptyset$, there is no
tagged arc lying in $C$. We assume $C$ is different from these cases in the
following proof.

Suppose there are $d$ open once-punctured disks $D_{p_{i}},i=1,...,d$ among
connected components of $S\backslash C$ and each puncture point $p_{i}$ is in
$D_{p_{i}}$. For these tagged arcs connected to two of $p_{i}(i=1,...,d)$, we
do flips at these tagged arcs such that a tagged arc connected to two of
$p_{i}(i=1,...,d)$ is replaced with a tagged arc connected to at most one of
$p_{i}(i=1,...,d)$ at each flip.

Since the least number of tagged arcs connected to $p_{i}$ but connected to
none of $\\{p_{j},j\neq i\\}$ in a triangulation of $(S,M)$ is $2$, the
maximal tagged arcs sets of $(C,C\cap M)$ are exactly the sets obtained from
the maximal tagged arcs set of
$(C\cup\bigcup_{i}D_{p_{i}},(C\cup\bigcup_{i}D_{p_{i}})\cap M)$ with exactly
$2d$ tagged arcs connected to one of $p_{i}(i=1,...,d)$ by removing these $2d$
tagged arcs.

In the following, we always assume that no connected component of $S\backslash
C$ is an open once-punctured disk if there is no additional explanation.

In fact, $C$ is an oriented surface with marked points $C\cap M$, but some
boundary components may contain no marked point. Any triangulation $T$ of
$(S,M)$ with a maximal number of tagged arcs lying in $C$ must contain a
tagged arc $\gamma_{b}$ with the same endpoint for each boundary component $b$
with no marked point such that $\gamma_{b}$ lies in $C$ and cuts out an open
region $U_{\gamma_{b}}$ satisfying that $C\backslash U_{\gamma_{b}}$ contains
all tagged arcs in $T$ lying in $C$.

Note that $\gamma_{b_{1}}$ and $\gamma_{b_{2}}$ may be the same for different
boundary components $b_{1}$, $b_{2}$ (up to isotopy). In this case, $C$ is a
once-punctured cylinder or a connected component of $S\backslash C$ is an open
annulus.

Thus removing these $U_{\gamma_{b}}$,
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ is a marked surface, where the union is over all boundary components with
no marked point. And these tagged arcs lying in $C$ still lie in
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$.

Suppose for $(C,C\cap M)$, the genus is $g$, the number of boundary components
is $a$, the number of punctures is $p$ and the number of marked points on the
boundaries is $c$. Then for $U_{\gamma_{b}}$, if the endpoint of $\gamma_{b}$
is a puncture, then the genus and number of boundary components, punctures,
marked points on the boundaries in $(C\backslash U_{\gamma_{b}},(C\backslash
U_{\gamma_{b}})\cap M)$ are $g$, $a-1+1$, $p-1$ and $c+1$, respectively; if
the endpoint of $\gamma_{b}$ is a marked point on the boundaries, then the
genus and number of boundary components, punctures, marked points in
$(C\backslash U_{\gamma_{b}},(C\backslash U_{\gamma_{b}})\cap M)$ are $g$,
$a-1$, $p$ and $c+1$, respectively.

Let $n_{b}$ be the number of boundary arcs of $C$ which are in $T$ and there
are $s$ open annuli among connected components of $S\backslash C$.

Note that if
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ is a disk with 1, 2 or 3 marked points on the boundaries, or the once-
punctured disk with 1 marked point on the boundaries,
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ does not admit a triangulation.

Now we give a classification of $C$ as follows.

In the case
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ is a disk with $2$ marked points on the boundaries, $C$ is (I) either (i)
the digon, or (ii) the annulus with 1 marked point on the boundaries or (iii)
the once-punctured annulus.

In the case
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ is a disk with $3$ marked points on the boundaries, $C$ is (II) either (i)
the triangle, or (ii) the annulus with 2 marked points on the same boundary
component or (iii) a pair of pants with 1 puncture.

In the case
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ is the once-punctured disk with 1 marked point on the boundaries, $C$ is
(III) either (i) the once-punctured disk with 1 marked point on the boundaries
or (ii) the twice-punctured disk.

Let $\mathfrak{N}$ denote a cardinality of a maximal tagged arcs set. Then
respectively,
$\mathfrak{N}=n_{b}$ for $C$ in the cases (I)(i), (I)(ii) and (II)(i);
$\mathfrak{N}=1-d$ for $C$ in the cases (I)(iii) and (II)(ii);
$\mathfrak{N}=3-d-s$ for $C$ in the case (II)(iii);
$\mathfrak{N}=2$ for $C$ in the case (III)(i)-(ii),
where $d$ is the number of open once-punctured disks among connected
components of $S\backslash C$.

According to Proposition 2.9, the number of tagged arcs in $T$ lying in
$(C\backslash\bigcup_{b}U_{\gamma_{b}},(C\backslash\bigcup_{b}U_{\gamma_{b}})\cap
M)$ except those $\gamma_{b}$ is equal to $n^{\prime}=6g+3a+3p+c-6-2t+n_{b}$,
where $t$ is the number of all boundary components with no marked point in
$(C,C\cap M)$.

Note that $\gamma_{b}$ is counted twice for each pair of boundary components
of an open annulus among connected components of $S\backslash C$. Thus the
cardinality of a maximal tagged arcs set is equal to

$\mathfrak{N}=n^{\prime}+t-s=6g+3a+3p+c-6-2t+n_{b}+t-s=6g+3a+3p+c-6-t-s+n_{b}.$

Suppose $(C,C\cap M)$ is different from a connected subsurface in the case
(I)-(IV) and there are $d$ open once-punctured disks among connected
components of $S\backslash C$, the cardinality of a maximal tagged arcs set of
$(C,C\cap M)$ is equal to $\mathfrak{N}=6g+3a+3p+c-6-t-s-d+n_{b}$.

Note that this formula is also valid for those $C$ in the cases (I)(iii),
(II)(i)-(iii) and (III)(ii).

∎

######  Lemma 4.9.

Let $(S,M)$ be a marked surface, $C_{1},C_{2}$ subsurfaces of $(S,M)$. If the
set of tagged arcs lying in $C_{1}$ is a proper subset of that of $C_{2}$ and
boundary components of $C_{1}$ which contain exactly one marked point are
contained in $\partial S$ and no connected component of $S\backslash C_{1}$ is
a triangle with two edges in $\partial S$, then the cardinality of a maximal
tagged arcs set of $(C_{1},C_{1}\cap M)$ is less than that of
$(C_{2},C_{2}\cap M)$.

###### Proof.

According to the condition, tagged arcs lying in a connected component of
$C_{1}$ also lies in a connected component of $C_{2}$, and there are two
connected components $C_{1}^{\prime},C_{2}^{\prime},$ of $C_{1}$ and $C_{2}$,
respectively, such that the set of tagged arcs lying in $C_{1}^{\prime}$ is a
proper subset of that of $C_{2}^{\prime}$. Thus we can assume that
$C_{1},C_{2}$ is connected.

Denote by $\mathfrak{N}_{1},\mathfrak{N}_{2}$ the cardinalities of a maximal
tagged arcs set of $(C_{1},C_{1}\cap M)$ and $(C_{2},C_{2}\cap M)$,
respectively. If $C_{1}$ satisfies one of the cases (1)-(3) in Lemma 4.8, it
is easy to check that this lemma holds. In the following, we always assume
that $C_{1}$ is different from that in cases (1)-(3).

Since the set of tagged arcs lying in $C_{1}$ is a proper subset of that of
$C_{2}$, at least one of genuses, numbers of boundary components, punctures
and marked points on the boundaries of $(C_{1},C_{1}\cap M)$ and
$(C_{2},C_{2}\cap M)$ must be different each other. Denote by
$g_{i},B_{i},P_{i},A_{i}$ the genus, sets of boundary components, punctures
and marked points on the boundaries of $(C_{i},C_{i}\cap M)$, respectively for
$i=1,2$. In fact, we have $g_{1}\leq g_{2}$, $P_{1}\subset P_{2}$, otherwise
we can find a proper tagged arc lies in $C_{1}$ but does not lie in $C_{2}$.

Since $C_{1}$ is different from $C_{2}$, there must be $B_{1}\backslash
B_{2}\neq\emptyset$. In these cases, some boundary component $b$ in $C_{1}$
vanishes via gluing some subset of the connected component of $S\backslash
C_{1}$ bounded by boundaries with non-empty intersection with $b$ and some
marked points in $A_{1}$ become punctures in $P_{2}$. According to Lemma 4.8,
$\mathfrak{N}_{2}$ is greater than $\mathfrak{N}_{1}$ except the following
cases:

(1) $C_{2}$ is obtained from $C_{1}$ by gluing an annulus with one marked
point on the boundaries along the boundary component with one marked point for
some boundary component $b$ of $C_{1}$ with one marked point, see Figure 5.

(2) $C_{2}$ is obtained from $C_{1}$ by gluing some triangle whose two edges
are boundary arcs along another edge, see Figure 5.

Note that in cases (1) and (2) due to Lemma 4.8,
$\mathfrak{N}_{1}=\mathfrak{N}_{2}$.

Since boundary components of $C_{1}$ which contain exactly one marked point
are contained in $\partial S$ and no connected component of $S\backslash
C_{1}$ is a triangle with two edges in $\partial S$, $n_{2}$ is always greater
than $n_{1}$.

∎

$\Longrightarrow$
---
$C_{1}$
---
$C_{2}$
---
Figure 4. Case (1)

$\Longrightarrow$
---
$C_{1}$
---
$C_{2}$
---
Figure 5. Case (2)

######  Theorem 4.10.

Let $(S,M)$ be a feasible surface, $\mathcal{A}(S,M)$ its cluster algebra
without coefficient, and a subgroup $H\leq\text{\em Aut}\mathcal{A}(S,M)$. If
$\mathcal{A}(\Sigma_{i}),i=1,2$ are maximal in $\mathcal{A}(S,M)^{H}$ as
cluster subalgebras of $\mathcal{A}(S,M)$, then the ranks of
$\mathcal{A}(\Sigma_{i}),i=1,2$ are the same.

###### Proof.

If there is no exchange variable in mixing-type sub-seed in
$\mathcal{A}(S,M)^{H}$, then the ranks of $\mathcal{A}(\Sigma_{i}),i=1,2$ are
equal to zero.

In the next proof, we assume there is an exchange variable in mixing-type sub-
seed in $\mathcal{A}(S,M)^{H}$. Thus according to Remark 4.5,
$H\leq\text{Aut}^{+}\mathcal{A}(S,M)$.

Suppose the oriented tagged arcs whose orientation are preserved under
$\rho(H)$ and correspond to cluster variables in $\Sigma_{1}$ belong to a
triangulation $T$ of $(S,M)$. According to Lemma 4.4, we can assume all
punctures are tagged plain except for some punctures tagged in different ways
in $T$.

Due to Remark 4.5, all oriented tagged arcs corresponding to exchange
variables in $\Sigma_{1}$ are preserved under $\rho(H)$ and hence elements in
$H$ is orientation-preserved. According to Lemma 4.7, oriented tagged arcs
corresponding to cluster variables in $\mathcal{A}(S,M)^{H}$ is invariant
under $\rho(H)$. Thus $T$ contains all tagged arcs corresponding to cluster
variables in $\Sigma_{1}$.

According to Lemma 4.4, we can assume that each tagged arc is tagging plain in
$T$ except some arcs tagging in different ways at some punctures which are
endpoints of tagged arcs corresponding to cluster variables in $\Sigma_{1}$.

We choose some regions determined by cluster variables in $\Sigma_{1}$ as
follows.

(1) Suppose $\gamma,\gamma^{p}$ correspond to cluster variables in
$\Sigma_{1}$. We choose the loop $\zeta$ such that $\gamma$ and $\zeta$ form a
self-folding and denote by $U_{\zeta}$ the region bounded by $\zeta$;

(2) Suppose $\gamma_{i},i=1,2,3$ are ones of the following cases: a loop of a
self-folding triangle such that there is a tagged arc in $T$ tagging notched
at the interior puncture of this self-folding triangle, boundar arc of $S$ and
a tagged arc corresponding to a cluster variables in $\Sigma_{1}$. If
$\gamma_{1},\gamma_{2},\gamma_{3}$ form a triangle, we denote by
$U_{\gamma_{1},\gamma_{2},\gamma_{3}}$ this simply-connected triangle.

(3) If $x_{\gamma}\in\Sigma_{1}$ and $\gamma$ does not satisfy the two
condition mentioned above, we denote by $U_{\gamma}$ the digon in which
$\gamma$ lies.

Let $U_{0}$ be the union of $U_{X}$ over the regions we chosen, then $U_{0}$
contains all tagged arcs corresponding to cluster variables in $\Sigma_{1}$.
Then each $U_{X}$ is invariant under $\rho(H)$, i.e., for each
$\varphi\in\rho(H)$, there is $\varphi^{\prime}$ isotopic to $\varphi$ such
that $\varphi^{\prime}(u)=u,u\in U_{X}$. Hence $U_{0}$ is also invariant under
$\rho(H)$. Suppose either (i) $\gamma_{1},\gamma_{2}$ corresponding to cluster
variables in $\Sigma_{1}$ or (ii) one of them corresponding to cluster
variables in $\Sigma_{1}$ and the other is boundary arc of $S$ such that
$\gamma_{1}$ is adjacent to $\gamma_{2}$. Since each $f\in H$ is orientation-
preserved, for some $\gamma_{3}$ in $T$ that is adjacent to $\gamma_{1}$ and
$\gamma_{2}$, $\gamma_{3}$ is invariant under $\rho(H)$. It means that
$b\backslash M$ is connected for each boundary component $b$.

Let $\\{b_{1},...,b_{h}\\}$ be the set consisting of boundary components of
$U_{0}$ which have one marked point and are not contained in $\partial S$.
Suppose $U$ is obtained by gluing an annulus with one marked point on the
boundaries along the boundary component with one marked point for each
$b_{i}$. Thus $U$ is a subsurface of $(S,M)$. Since $\rho(f),f\in H$ maps
these annuli to themselves and its restriction to each annulus is a power of
the Dehn twist about the boundary, $U$ is invariant under $\rho(H)$.

In the following proof, we show that $x_{\gamma}\in\mathcal{A}(S,M)^{H}$ if
and only if the corresponding tagged arc $\gamma$ lies in $U$ (up to isotopy).

Otherwise, suppose there is an oriented tagged arc $\eta$ invariant under
$\rho(H)$ not lying in $U$. According to Lemma 4.4 we can assume $\eta$ is an
arc tagged plain. Similarly, we can choose a simply-connected digon $U_{\eta}$
containing $\eta$ which is invariant under $\rho(H)$.

Denote by $U^{\prime}$ the union of $U_{\eta}$ and $U$. Suppose $\eta$
intersects the boundary components of $U$ with minimal number of intersection
points. Since $U$ is invariant under $\rho(H)$, the image of each intersection
point lies in the same segment of boundary component (tagged arc) of $U$ and
the order of intersection points is preserved under $\rho(H)$. Since the
connected components of the closure of $U_{\eta}\backslash U$ are simply-
connected regions and $U_{\eta}$ is invariant under $\rho(H)$, the connected
components of the closure of $U_{\eta}\backslash U$ are also invariant under
$\rho(H)$. Thus $U^{\prime}$ is invariant under $\rho(H)$.

Note that if $U^{\prime}$ is not a subsurface of $(S,M)$, we can similarly
gluing annuli to obtain a subsurface, which is invariant under $\rho(H)$. Thus
we assume $U^{\prime}$ is a subsurface for convenience.

It follows from Lemma 4.8 that the cardinality of a maximal tagged arcs set of
$U$ (resp. $U^{\prime}$) is constant, which is denoted by $\mathfrak{N}$
(resp. $\mathfrak{N}^{\prime}$). And it follows from Lemma 4.9 we have
$\mathfrak{N}^{\prime}>\mathfrak{N}$. Then we can find a maximal tagged arc
set of $U^{\prime}$ which contains tagged arcs whose corresponding cluster
variables in $\Sigma_{1}$. It means we can find a mixing-type sub-seed
$\Sigma^{\prime}\subset\mathcal{A}(S,M)^{H}$ containing $\Sigma_{1}$, which is
contrary to that $\mathcal{A}(\Sigma_{1})$ is a maximal cluster subalgebra in
$\mathcal{A}(S,M)^{H}$.

Thus the tagged arcs corresponding cluster variables in $\Sigma_{2}$ lie in
$U$ and hence the rank of $\mathcal{A}(\Sigma_{2})$ is less than or equal to
the rank of $\mathcal{A}(\Sigma_{1})$.

Similarly, we can show the rank of $\mathcal{A}(\Sigma_{1})$ is less than or
equal to the rank of $\mathcal{A}(\Sigma_{2})$. Hence the rank of
$\mathcal{A}(\Sigma_{2})$ is the same as $\mathcal{A}(\Sigma_{1})$’s.

∎

######  Definition 4.11.

Let $(S,M)$ be a feasible surface, $\mathcal{A}(S,M)$ its cluster algebra
without coefficient, $H\leq\text{\em Aut}\mathcal{A}(S,M)$ and any $f\in H$ is
orientation-preserved. A subsurface $U$ of $(S,M)$ is called a maximal
invariant subsurface of $H$ in case $x_{\gamma}\in\mathcal{A}(S,M)^{H}$ if and
only if the corresponding tagged arc $\gamma$ lies in $U$ (up to isotopy).

######  Remark 4.12.

Suppose $U_{1},U_{2}$ are maximal invariant subsurfaces of $H$, their boundary
components including exactly one marked point are contained in $\partial S$
and all tagged arcs are contained in their connected components. Then the
genuses, the sets of boundary components, the punctures and marked points on
the boundaries of $U_{1}$ and $U_{2}$ are the same respectively. Otherwise we
can find a proper tagged arc which lies in one of $U_{1}$ and $U_{2}$ but does
not lie in the other, which is contrary to Definition 4.11. Thus in the
sequel, we always assume maximal invariant subsurfaces satisfying the above
conditions.

Note that the proof of Theorem 4.10 provides a way to obtain a maximal
invariant subsurface $U$ of $H$ for the case that any $f\in H$ is orientation-
preserved. Suppose $\mathcal{A}(\Sigma_{1})$ is a maximal cluster subalgebra
in $\mathcal{A}(S,M)^{H}$. Similar to the proof of Theorem 4.10, we can prove
that for an orientation-preserved cluster automorphism $f$, $\Sigma_{1}$ is
invariant under $f$ if and only if $U$ is invariant under $\rho(f)$. And the
set of tagged arcs corresponding to cluster variables in $\Sigma^{\prime}$ is
a maximal tagged arcs set of $(U,U\cap M)$ if and only if
$\mathcal{A}(\Sigma^{\prime})$ is a maximal cluster subalgebra in
$\mathcal{A}(S,M)^{H}$.

For cluster algebras from feasible surfaces, Conjecture 3.16 also holds for
maximal cluster subalgebras $\mathcal{A}(\Sigma_{i}),i=1,2$ in
$\mathcal{A}(S,M)^{H}$, even though they are not Galois-like extension
subalgebra. In fact, we have the following corollary.

######  Corollary 4.13.

Let $(S,M)$ be a feasible surface, $\mathcal{A}(S,M)$ its cluster algebra
without coefficient, $H\in\mathit{2}^{\text{\em
Aut}\mathcal{A}}\backslash\text{ker}\phi$ whose any element $f$ is
orientation-preserved. Then $\mathcal{A}(S,M)$ is a Galois-like extension over
$\mathcal{A}(\Sigma_{1})$ for any maximal cluster subalgebra
$\mathcal{A}(\Sigma_{1})$ in $\mathcal{A}(S,M)^{H}$.

###### Proof.

Let $U$ be the maximal invariant subsurface of $H$ and
$\mathcal{A}(\Sigma^{\prime})\in\mathcal{M}_{sub}^{H}$.

If the connected components of $U$ are digons, once-punctured annuli or twice-
punctured disks, then the maximal tagged arcs set of $U$ uniquely (neglecting
the taggings) consists of two edges of each digon, the loop lying in each
annulus and two tagged arcs tagged in different ways. Hence
$\text{Gal}_{\Sigma_{1}}\mathcal{A}(S,M)=\text{Gal}_{\Sigma^{\prime}}\mathcal{A}(S,M)$
by Lemma 4.4.

Otherwise, in other cases, for any maximal tagged arcs $A$ set of $U$, after
adding boundary arcs of $S$ and replacing each pair tagged arcs in $A$ which
is the form as $\gamma,\gamma^{p}$ with the self-folding triangle containing
$\gamma$, there exists a triple of tagged arcs in the new set forming a
triangle. According to Remark 4.12, the set of taged arcs corresponding to
$\Sigma_{1}$ is a maximal tagged arcs set $A_{1}$ of $U$. Thus after
modifying, there exists a triple of tagged arcs in the new set obtained from
$A_{1}$ such that this triple forms a triangle. Besides, cluster variables
corresponding to elements in this new set are invariant under
$\text{Gal}_{\Sigma_{1}}\mathcal{A}(S,M)$ and the existence of a triangle
guarantee that elements in $\text{Gal}_{\Sigma_{1}}\mathcal{A}(S,M)$ are
orientation-preserved.

Since elements in $H$ are orientation-preserved, according to Remark 4.12,
$f\in\text{Gal}_{\Sigma_{1}}\mathcal{A}(S,M)$ if and only if $U$ is invariant
under $\rho(f)$, which is equivalent to
$f\in\text{Gal}_{\Sigma^{\prime}}\mathcal{A}(S,M)=H$.

∎

Now we consider the Galois inverse problem for a feasible surface $(S,M)$,
that is, under what condition is a subgroup $H$ of
$\text{Aut}\mathcal{A}(S,M)$ a Galois group?

Let $(C,C\cap M)$ be a subsurface of the marked surface $(S,M)$. Denote by
$\text{Mod}(C,C\cap M)$ (resp. $\text{Mod}_{\bowtie}(C,C\cap M)$ the group
consiting of elements of $\mathcal{MCG}(C,C\cap M)$ (resp.
$\mathcal{MCG}_{\bowtie}(C,C\cap M)$) fixing boundary components of $C$ not
contained in $\partial S$ if $S$ is a once-punctured closed surface $S$ (resp.
if $S$ is not a once-punctured closed surface).

Note that for a feasible surface $(S,M)$ and a subsurface $C$ of $S$, there is
a natural homomorphism $\iota:\text{Mod}_{\bowtie}(C,C\cap
M)\to\mathcal{MCG}_{\bowtie}(S,M)$ induced by the embedding. According to
[FM12], if no connected component of $S\backslash C$ is an open annulus, an
open disk, or an open once-punctured disk, then $\iota$ is a monomorphism. In
this case, $\text{Mod}(C,C\cap M)$ (resp. $\text{Mod}_{\bowtie}(C,C\cap M)$)
is a subgroup of $\mathcal{MCG}(S,M)$ (resp. $\mathcal{MCG}_{\bowtie}(S,M)$)
under the monomorphism.

######  Proposition 4.14.

Assume $\mathcal{A}(S,M)$ is a cluster algebra without coefficient from a
feasible surface $(S,M)$. Let $C$ be a subsurface of $S$ such that there is no
connected component of $U=\overline{S\backslash C}$ which is a closed annulus
or a closed once-punctured disk and there is a connected component of $U$
which i different from a digon, a closed once-punctured annulus and a closed
twice-punctured disk, where $\overline{S\backslash C}$ means the closure of
$S\backslash C$. Then we have the following statements:

(a) If $(S,M)$ is a once-punctured closed surface, then

(i) $\rho^{\prime-1}(\text{\em Mod}(C,C\cap M))$ is a Galois group.

(ii) Conversely, for a Galois group $H$ whose any element $f$ is orientation-
preserved, there is a subsurface $C$ such that $H=\rho^{\prime-1}(\text{\em
Mod}(C,C\cap M))$.

(b) If $(S,M)$ is not a once-punctured closed surface, then

(i) $\rho^{\prime-1}(\text{\em Mod}_{\bowtie}(C,C\cap M))$ is a Galois group.

(ii) Conversely, for a Galois group $H$ whose any element $f$ is orientation-
preserved, there is a subsurface $C$ such that $H=\rho^{\prime-1}(\text{\em
Mod}_{\bowtie}(C,C\cap M))$.

###### Proof.

We only prove (b). The proof of (a) is similar.

Since no connected component of $U$ is a closed annulus, a closed disk, or a
closed once-punctured disk, the homomorphism

$\iota:\;\text{Mod}_{\bowtie}(C,C\cap M)\to\mathcal{MCG}_{\bowtie}(S,M)$

induced by the embedding is monomorphic. Hence we have
$G=\rho^{\prime-1}(\text{Mod}_{\bowtie}(C,C\cap
M))\leq\text{Aut}\mathcal{A}(S,M)$.