Synchronicity phenomenon in cluster patterns

TL;DR

This paper explains the synchronicity phenomenon in cluster patterns, showing how various objects share periodicity under mutations, and extends these properties to generalized cluster algebras.

Contribution

It provides a comprehensive explanation of the synchronicity mechanism in cluster patterns based on fundamental cluster algebra results and extends these properties to generalized cluster algebras.

Findings

Objects in cluster patterns share periodicity under mutations.

The synchronicity mechanism is explained using key cluster algebra results.

Properties are extended to generalized cluster algebras, pending Laurent positivity.

Abstract

It has been known that several objects such as cluster variables, coefficients, seeds, and $Y$-seeds in different cluster patterns with common exchange matrices share the same periodicity under mutations. We call it synchronicity phenomenon in cluster patterns. In this expository note we explain the mechanism of synchronicity based on several fundamental results on cluster algebra theory such as separation formulas, sign-coherence, Laurent positivity, duality, and detropicalization obtained by several authors. We also show that all synchronicity properties studied in this paper are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture.

Equations

$$x_i^{\prime }$$

$$b_{ij}^{\prime }$$

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

$${\hat{y}}_i:=\prod _{j=1}^n{x}_j^{b_{ji}}.$$

$$y_i^{\prime }$$

$$(\mathbf{x},B){\to }^{k_1}({\mathbf{x}}^{\prime },B^{\prime }){\to }^{k_2}\cdots {\to }^{k_L}(\tilde{\mathbf{x}},\tilde{B}),$$

$$(\mathbf{y},B){\to }^{k_1}({\mathbf{y}}^{\prime },B^{\prime }){\to }^{k_2}\cdots {\to }^{k_L}(\tilde{\mathbf{y}},\tilde{B}).$$

$$\tilde{\mathbf{x}}=\mathbf{x}⟺\tilde{\mathbf{y}}=\mathbf{y}.$$

$${\hat{y}}_i^{\prime }$$

$$\prod _{i=1}^m{u}_i^{a_i}\oplus \prod _{i=1}^m{u}_i^{b_i}:=\prod _{i=1}^m{u}_i^{\min (a_i,b_i)}.$$

$$x_i^{\prime }$$

$$y_i^{\prime }$$

$$b_{ij}^{\prime }$$

$${\hat{y}}_i:=y_i\prod _{j=1}^n{x}_j^{b_{ji}}\in \mathcal{F}.$$

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

$${\mathbf{x}}_{t_0}=\mathbf{x}=(x_1,\dots ,x_n),{\mathbf{y}}_{t_0}=\mathbf{y}=(y_1,\dots ,y_n),B_{t_0}=B=(b_{ij}{)}_{i,j=1}^n.$$

$$F_{i;t_0}(\mathbf{u})=1(i=1,\dots ,n),G_{t_0}=C_{t_0}=I,$$

$$F_{i;t^{\prime }}(\mathbf{u})$$

$$g_{ij}^{t^{\prime }}$$

$$c_{ij}^{t^{\prime }}$$

$$\begin{array}{rl}{M}_{k;t}(\mathbf{u})& =\prod _{j=1}^n{u}_j^{[c_{jk}^t{]}_{+}}{F}_{j;t}(\mathbf{u}{)}^{[b_{jk}^t{]}_{+}}+\prod _{j=1}^n{u}_j^{[-c_{jk}^t{]}_{+}}{F}_{j;t}(\mathbf{u}{)}^{[-b_{jk}^t{]}_{+}}\\ & =\left(\prod _{j=1}^n{u}_j^{[-c_{jk}^t{]}_{+}}{F}_{j;t}(\mathbf{u}{)}^{[-b_{jk}^t{]}_{+}}\right)\left(1+\prod _{j=1}^n{u}_j^{c_{jk}^t}{F}_{j;t}(\mathbf{u}{)}^{b_{jk}^t}\right).\end{array}$$

$${\tilde{y}}_{i;t}$$

$$x_{i;t}$$

$$y_{i;t}$$

$$\sigma (x_{1;t},\dots ,x_{n;t})$$

$$\sigma (y_{1;t},\dots ,x_{n;t})$$

$$\sigma B_t$$

$${\mu }_{\sigma (k)}(\sigma {\Sigma }_t)$$

$$\sigma (F_{1;t}(\mathbf{u}),\dots ,F_{n;t}(\mathbf{u}))$$

$$\sigma G_t$$

$$\sigma C_t$$

$${\mu }_{\sigma (k)}(\sigma {\Gamma }_t)$$

$$P_{\sigma }=(p_{ij}),p_{ij}={\delta }_{i,{\sigma }^{-1}(j)}.$$

$$\sigma C_t=C_t{P}_{\sigma },\sigma G_t=G_t{P}_{\sigma },\sigma B_t=P_{\sigma }^T{B}_t{P}_{\sigma },$$

$$\text{(Duality)}{D}^{-1}{G}_t^{T}D{C}_t$$

$$D{B}_t$$

$$|\det G_t|$$

Full text

1. Introduction: Synchronicity problem

Consider a pair $(\mathbf{x},B)$, where $\mathbf{x}=(x_{1},\dots,x_{n})$ is an $n$-tuples of commuting formal variables, and $B=(b_{ij})_{i,j=1}^{n}$ is a skew-symmetrizable integer matrix. For any $k\in\{1,\dots,n\}$, we obtain a new pair $(\mathbf{x}^{\prime},B^{\prime})$ by the following transformation:

[TABLE]

where

[TABLE]

This is called the mutation of a seed without coefficients at direction $k$ [FZ02].

Similarly, consider a pair $(\mathbf{y},B)$, where $\mathbf{y}=(y_{1},\dots,y_{n})$ is another $n$-tuples of commuting formal variables, and $B=(b_{ij})_{i,j=1}^{n}$ is a skew-symmetrizable integer matrix. For any $k\in\{1,\dots,n\}$, we obtain a new pair $(\mathbf{y}^{\prime},B^{\prime})$, where $B^{\prime}$ is given by (1.2) while $\mathbf{y}^{\prime}$ is given by

[TABLE]

This is called the mutation of a $Y$-seed with coefficients in a universal semifield at direction $k$ [FZ02, FZ07].

Now, starting with a common skew-symmetrizable matrix $B$, let us apply the same sequence of mutations at given directions $k_{1}$, …, $k_{L}$ on $(\mathbf{x},B)$ and $(\mathbf{y},B)$ as follows:

[TABLE]

Then, though it is rare, it may happen that $\tilde{\mathbf{x}}=\mathbf{x}$ (resp., $\tilde{\mathbf{y}}=\mathbf{y}$). Let us call such a sequence of mutations as a period of $x$-variables (resp. $y$-variables), respectively.

The following question is frequently asked:

Question/Conjecture 1.1** (e.g., [FG09a, Footnote 3 in Section 1.2]).**

Do the periodicities of $x$- and $y$-variables always coincide? In other words, does the following “synchronicity” of $x$- and $y$-variables hold?

[TABLE]

This question, or the conjecture that the synchronicity (1.8) holds, naturally arose from the observation that it indeed holds in several examples, such as the rank 2 cluster algebras [FZ02], the finite type cluster algebras [FZ03a, FZ03b], the $T$-systems and the $Y$-systems of several types [IIK*+*10], etc.

Two transformations (1.1) and (1.5) certainly look closely related. To be more specific, it has been well known (e.g., [FZ07, Proposition 3.9]) that under the mutation (1.1) the variables in (1.4) ($\hat{y}$-variables) also mutate just like $y$-variables, namely,

[TABLE]

If $B$ is nondegenerate, the initial $\hat{y}$-variables are algebraically independent; therefore, the implication $\tilde{\mathbf{x}}=\mathbf{x}\ \Longrightarrow\ \tilde{\mathbf{y}}=\mathbf{y}$ holds by (1.9). Furthermore, if $\det B=\pm 1$, one can invert the relation (1.4), and the opposite implication also holds. Therefore, the synchronicity (1.8) holds if $\det B=\pm 1$. (We may prove the opposite implication for a more general nondegenerate $B$ by properly extending the field where $y$-variables live, but we do not touch this issue here.)

In many important and interesting cases, the matrix $B$ is degenerate. Then, in general, we do not yet know how to prove the synchronicity (1.8) by directly working on (1.1) and (1.5). However, thanks to the recent development of cluster algebra theory, one can prove it as a consequence of a collection of known fundamental results in cluster algebra theory by several authors. In other words, the cluster algebra structure (to be more precisely, the cluster pattern structure) behind it is essential in our derivation.

Our proof of (1.8) based on the known results is relatively elementary, in the sense that one can work in the framework of [FZ07] without relying on some other machinery. Therefore, we believe that the fact (1.8) and its proof have been already known for some time among experts including the author. In fact, a proof of the implication from left to right in (1.8) was already presented in [CL18, Proposition 6.1]. However, the fact (1.8) seems not to be well known, and it is still a frequently asked question by colleagues to the author. So in this note we present a proof of (1.8), together with other related synchronicity properties in cluster patterns scattering around in the literature as theorems and conjectures. We note that our proof of the implication from left to right in (1.8) is different from the one in [CL18].

The paper consists of two parts: Part I (Sections 2–6) is for the results on cluster patterns in (ordinary) cluster algebras, and Part II (Sections 7–9) is for the results on cluster patterns in generalized cluster algebras introduced by [CS14]. In Sections 2–4 we give a concise review of fundamental properties on cluster patterns that we need. Section 5 is the main section of Part I, where we derive and present several synchronicity properties based on the results in the previous sections. Overall we pay special attention on presenting how these results are logically related. See the introduction of Section 4 for more about this point. We conclude Part I by addressing an important open problem in Section 6. In Part II (Sections 7–9) we show that all synchronicity properties studied in Part I are naturally extended to cluster patterns of generalized cluster algebras, up to the Laurent positivity conjecture. Moreover, we show that any cluster pattern of GCA synchronizes with its “companion cluster patterns” of CA.

Part I is regarded as an extended note of the talks given at the workshops “Cluster Algebras: Twenty Years On” at CIRM, Luminy, 2018, March and “School on Cluster Algebras” at ICTS, Bangalore, 2018, December.

We thank Sergey Fomin for encouragement to write up this note and also for useful comments and suggestions on the manuscript. This work was partially supported by JSPS KAKENHI Grant Number 16H03922.

Part I Synchronicity in cluster patterns of cluster algebras

2. Cluster patterns

Let us recall some basic notions in cluster algebra theory, mostly following [FZ07]. Consult [FZ07] for further details. Throughout the paper we fix a positive integer $n$.

A square integer matrix $B=(b_{ij})_{i,j=1}^{n}$ is said to be skew-symmetrizable if there is a diagonal matrix $D=\mathrm{diag}(d_{1},\dots,d_{n})$ with positive integer diagonal entries $d_{1},\dots,d_{n}$ such that $DB$ is skew-symmetric, i.e., $d_{i}b_{ij}=-d_{j}b_{ji}$ holds. Such a matrix $D$ is called a (left) skew-symmetrizer of $B$.

A semifield $\mathbb{P}$ is a multiplicative abelian group equipped with an “addition” $\oplus$ which is commutative, associative, and distributive with respect to the multiplication. Then, the group ring $\mathbb{Z}\mathbb{P}$ of $\mathbb{P}$ is a domain. Let $\mathbb{Q}\mathbb{P}$ be the fraction field of $\mathbb{Z}\mathbb{P}$.

The following three examples of semifields are especially important in cluster algebra theory.

Example 2.1**.**

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

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

[TABLE]

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

(3). (Trivial semifield $\mathbf{1}$.) Let $\mathbf{1}=\{1\}$ be the trivial multiplicative group. We define the addition by $1\oplus 1=1$. Then, $\mathbf{1}$ becomes a semifield.

Definition 2.2** (Seeds).**

Let $\mathbb{P}$ be any semifield, and let $\mathcal{F}$ be a field which is isomorphic to the rational function field of $n$-variables with coefficients in the field $\mathbb{Q}\mathbb{P}$. A (labeled) seed with coefficients in $\mathbb{P}$ is a triplet $\Sigma=(\mathbf{x},\mathbf{y},B)$, where $\mathbf{x}=(x_{1},\dots,x_{n})$ is an $n$-tuple of any algebraically independent elements (called cluster variables) in $\mathcal{F}$, $\mathbf{y}=(y_{1},\dots,y_{n})$ is an $n$-tuple of any elements (called coefficients) in $\mathbb{P}$, and $B=(b_{ij})_{i,j=1}^{n}$ is a skew-symmetrizable integer matrix (called an exchange matrix).

In this note let us call $x_{i}$’s and $y_{i}$’s in the above as $x$-variables and $y$-variables, respectively, in view of their equal roles in the periodicity phenomenon. They are also called cluster $\mathcal{A}$- and $\mathcal{X}$-coordinates in other fundamental references on cluster algebras by Fock and Goncharov [FG09a, FG09b].

Definition 2.3** (Seed mutations).**

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

[TABLE]

where $\hat{y}_{k}$ in (2.2) is defined by

[TABLE]

The seed $\Sigma^{\prime}$ is called the mutation of $\Sigma$ at direction $k$, and denoted by $\mu_{k}(\Sigma)$. Note that a skew-symmetrizer $D$ of $B$ is also a skew-symmetrizer of $B^{\prime}$.

The mutations are involutive, i.e., $\mu_{k}(\Sigma^{\prime})=\Sigma$ in the above.

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

Definition 2.4** (Cluster patterns).**

A family of seeds $\mathbf{\Sigma}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with coefficients in a semifield $\mathbb{P}$ indexed by $\mathbb{T}_{n}$ is called a cluster pattern with coefficients in $\mathbb{P}$ if, for any vertices $t,t^{\prime}\in\mathbb{T}_{n}$ connected by an edge labeled by $k$, the equality $\Sigma_{t^{\prime}}=\mu_{k}(\Sigma_{t})$ holds.

Following [FZ07], for a seed $\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})$ at $t$ in a cluster pattern, we use the notation

[TABLE]

In parallel to Example 2.1 we introduce three families of cluster patterns.

Example 2.5**.**

(1). (Cluster pattern with universal coefficients) This is a cluster pattern $\mathbf{\Sigma}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with coefficients in $\mathbb{P}$ where there is some $t_{0}\in\mathbb{T}_{n}$ such that, for the universal semifield $\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$ of a given $n$-tuple of variables $\mathbf{u}=(u_{1},\dots,u_{n})$, the semifield homomorphism $\varphi_{t_{0}}:\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})\rightarrow\mathbb{P}$ defined by $u_{i}\mapsto y_{i;t_{0}}$ is injective. In this case one can safely set $\mathbb{P}=\mathbb{Q}_{\mathrm{sf}}(\mathbf{y}_{t_{0}})$ from the beginning. Moreover, one can replace it with $\mathbb{P}=\mathbb{Q}_{\mathrm{sf}}(\mathbf{y}_{t})$ for any other $t\in\mathbb{T}_{n}$ if necessary, thanks to the involution property of the mutations. Sometimes we call $\mathbf{y}_{t}$’s the universal $y$-variables.

We remark that no special name is given for such a cluster pattern in [FZ07], and the terminology “universal coefficients” is used for a different concept [FZ07, Definition 12.3] therein.

(2). (Cluster pattern with principal coefficients at $t_{0}$.) This is a cluster pattern $\tilde{\mathbf{\Sigma}}[{t}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with coefficients in $\mathbb{P}$ where there is some ${t}_{0}\in\mathbb{T}_{n}$ such that, for the tropical semifield $\mathrm{Trop}(\mathbf{u})$ of a given $n$-tuple of variables $\mathbf{u}=(u_{1},\dots,u_{n})$, there is an injective semifield homomorphism $\varphi_{{t}_{0}}:\mathrm{Trop}(\mathbf{u})\rightarrow\mathbb{P}$ such that $u_{i}\mapsto\tilde{y}_{i;{t}_{0}}$. In this case one can safely set $\mathbb{P}=\mathrm{Trop}(\tilde{\mathbf{y}}_{{t}_{0}})$ from the beginning. In contrast to the previous case one cannot replace it with $\mathrm{Trop}(\tilde{\mathbf{y}}_{t})$ with other $t\in\mathbb{T}_{n}$, in general, because the mutation (2.3) is incompatible with the tropical sums of $\mathrm{Trop}(\tilde{\mathbf{y}}_{t})$ and $\mathrm{Trop}(\tilde{\mathbf{y}}_{t^{\prime}})$. Sometimes we call $\tilde{\mathbf{y}}_{t}$’s the tropical $y$-variables with the tropicalization point ${t}_{0}$.

(3). (Cluster pattern without coefficients.) This is a cluster pattern $\underline{\mathbf{\Sigma}}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},\underline{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with coefficients in a trivial semifield $\mathbf{1}$. Since all $y$-variables are $1$, one can safely omit them from seeds; namely, we set $\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})$.

3. Separation formulas

We recall the most fundamental theorem by [FZ07] on the structure of seeds in cluster patterns.

Let $\mathbf{\Sigma}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. We arbitrary choose a distinguished point (the initial point) $t_{0}$ in $\mathbb{T}_{n}$. The seed $(\mathbf{x}_{t_{0}},\mathbf{y}_{t_{0}},B_{t_{0}})$ at $t_{0}$ is called the initial seed of $\mathbf{\Sigma}$. We use the following simplified notation for them:

[TABLE]

Let us extract the family of the exchange matrices $\mathbf{B}=\{B_{t}\mid t\in\mathbb{T}_{n}\}$ from the cluster pattern $\mathbf{\Sigma}$, and call it a $B$-pattern. Following [FZ07], we introduce a family of quadruplets $\mathbf{\Gamma}(\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ uniquely determined from $\mathbf{B}$ and any initial point $t_{0}$ as follows. For each $t\in\mathbb{T}_{n}$, $\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})$ is a quadruplet (let us call it an $FGC$-seed), where

• •

$\mathbf{F}_{t}(\mathbf{u})=(F_{1;t}(\mathbf{u}),\dots,F_{n;t}(\mathbf{u}))$ is an $n$-tuple of rational functions (called $F$-polynomials) in $n$-tuple of variables $\mathbf{u}=(u_{1},\dots,u_{n})$ with coefficients in $\mathbb{Q}$ having a subtraction-free expression, i.e., $F_{i;t}(\mathbf{u})\in\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$,

• •

and $G_{t}$ and $C_{t}$ are $n\times n$ integer matrices (called a $G$-matrix and a $C$-matrix).

At the initial point $t_{0}$, they are given by

[TABLE]

where $I$ is the identity matrix. For $t,t^{\prime}\in\mathbb{T}_{n}$ which are connected by an edge labeled by $k$, $\Gamma_{t}$ and $\Gamma_{t^{\prime}}$ are related by the following mutation at $k$:

[TABLE]

where

[TABLE]

Note that the transformation (3.4) involves the matrix $B=B_{t_{0}}$ which does not depend on $t$. Again, these mutations are involutive. We call $\mathbf{\Gamma}(\mathbf{B},t_{0})$ the $FGC$-pattern of $\mathbf{B}$ with the initial point $t_{0}$.

As the name suggests, a seemingly rational function $F_{i;t}(\mathbf{u})\in\mathbb{Q}_{\mathrm{sf}}(\mathbf{u})$ is indeed a polynomial in $\mathbf{u}$. (This fact follows from the celebrated Laurent phenomenon [FZ02, FZ03a].)

Theorem 3.1** ([FZ07, Proposition 3.6]).**

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

We emphasize that, in our formulation, the initial point $t_{0}$ for a $FGC$-pattern can be chosen independently to the tropicalization point $t_{0}$ of any cluster pattern with principal coefficients in Example 2.5 (2) that we will consider in this paper. However, when they coincide, the $C$-matrices are naturally identified as the (exponents of) tropical $y$-variables as follows:

Proposition 3.2** ([FZ07, Remark 3.2]).**

Let $\tilde{\mathbf{\Sigma}}[t_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with principal coefficients at any point $t_{0}$, and let $\mathbf{\Gamma}(\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}$ with the initial point $t_{0}$. Then, the following formula holds for any $t\in\mathbb{T}_{n}$:

[TABLE]

Now we present formulas expressing arbitrary $x$- and $y$-variables $\mathbf{x}_{t}$, $\mathbf{y}_{t}$ in terms of the initial $x$- and $y$-variables $\mathbf{x}$, $\mathbf{y}$ together with $G$- and $C$-matrices and $F$-polynomials with any initial point $t_{0}$. This is one of the most fundamental properties of cluster patterns, which makes the cluster pattern (or cluster algebra) structure so useful in various applications.

Theorem 3.3** (Separation Formulas [FZ07, Proposition 3.13. Corollary 6.13]).**

For a cluster pattern $\mathbf{\Sigma}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with coefficients in any semifield $\mathbb{P}$, the following formulas hold for any $t\in\mathbb{T}_{n}$:

[TABLE]

where $F_{i;t}|_{\mathbb{P}}(\mathbf{y})$ is the evaluation of $F_{i,t}(\mathbf{u})$ in $\mathbb{P}$ at $\mathbf{y}$.

Remark 3.4**.**

Originally in [FZ07], $FGC$-patterns are defined through cluster patterns with principal coefficients, and the proof of Theorem 3.3 also relied on it. However, it is straightforward to prove Theorem 3.3 only from (3.2)–(3.6) by induction on $t$ (since we already know this result!). Moreover, the original definition of $FGC$-patterns in [FZ07] can be easily recovered from Theorem 3.3. We leave it as an exercise to the readers.

Remark 3.5**.**

In [FZ07] only the first formula (3.8) is called the separation formula, meaning that it separates the additions of $\mathcal{F}$ and $\mathbb{P}$ in the numerator and the denominator therein. In view of the parallelism of $x$- and $y$-variables, we also call the second one (3.9) the separation formula. Here the “separation” has an additional meaning in view of Proposition 3.2 that the tropical part (involving $C$ and $G$ matrices) and nontropical part (involving $F$-polynomials) are separated therein.

Let $S_{n}$ be the symmetric group of degree $n$. For a seed $\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})$ at $t$ and a permutation $\sigma\in S_{n}$, we define the left action of $\sigma$ as $\sigma\Sigma_{t}=(\sigma\mathbf{x}_{t},\sigma\mathbf{y}_{t},\sigma B_{t})$, where

[TABLE]

This action is compatible with mutations; namely, we have

[TABLE]

Similarly, for a $FGC$-seed $\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})$ at $t$ and a permutation $\sigma\in S_{n}$, we define the action of $\sigma$ as $\sigma\Gamma_{t}=(\sigma\mathbf{F}_{t}(\mathbf{u}),\sigma G_{t},\sigma C_{t},\sigma B_{t})$, where

[TABLE]

Again, we have

[TABLE]

(Caution: For the mutation in the left hand side of (3.17) we keep $B$ in the last term of the first line of (3.4) as it is, not replacing it with $\sigma B$. Otherwise, the equality (3.17) fails.) Observe that (3.14)–(3.16) and (3.10)–(3.12) are compatible in view of the separation formulas (3.8)–(3.9).

Let us introduce the permutation matrix $P_{\sigma}$ associated with $\sigma\in S_{n}$ as

[TABLE]

Then, (3.12), (3.15), and (3.16) are written as

[TABLE]

where $M^{T}$ is the transposition of a matrix $M$.

4. Further fundamental results

To extract the power of the separation formulas in our periodicity problem, we need some additional properties on $G$- and $C$-matrices and $F$-polynomials.

It turns out that all necessary results stem from two fundamental and very deep results in cluster algebra theory, namely, the sign-coherence and the Laurent positivity, both of which were proved partially by several authors and proved in full generality by the recent seminal paper [GHKK18]. Unfortunately, no elementary or direct proof is yet known in the framework of [FZ07], and the known proofs (except for [LS15] notably) involve some construction or realization of cluster variables with relatively big machinery such as categorification or scattering diagram method.

Our strategy is to accept this two fundamental results temporarily as inputs from outside, and then work in the framework of [FZ07] in the rest, hoping that we will have some elementary proofs in the framework of [FZ07] in some future.

Throughout this section we fix the $FGC$-pattern $\mathbf{\Gamma}(\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),\allowbreak G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of any $B$-pattern $\mathbf{B}$ with any initial point $t_{0}$. Also, any cluster patterns in this section share the common $B$-pattern $\mathbf{B}$ with $\mathbf{\Gamma}(\mathbf{B},t_{0})$.

4.1. Sign-coherence

The first fundamental result is the following one.

Theorem 4.1** **(Sign-coherence, [GHKK18, Corollary 5.5],

[DWZ10, Theorem 1.7], [Pla11, Theorem 3.7 (1) and Remark 3.11], [Nag13, Theorem 8.8]).

Each $C$-matrix $C_{t}$ is column-sign coherent. Namely, each column vector ($c$-vector) of $C_{t}$ is nonzero vector, and its components are either all nonnegative or all nonpositive.

This was conjectured by [FZ07, Conjecture 5.5], and proved for the skew-symmetric case by [DWZ10, Pla11, Nag13] and in general by [GHKK18].

It was noticed by [FZ07, Proposition 5.6] that Theorem 4.1 is equivalent to the following simple statement for $F$-polynomials. (The equivalence is easily proved from (3.3) by induction on $t$.)

Theorem 4.2**.**

[GHKK18, DWZ10, Pla11, Nag13*]**

The constant term of each $F$-polynomial $F_{i;t}(\mathbf{u})$ is 1.*

In fact, Theorem 4.1 was proved in this form in [DWZ10, Nag13].

Let $D$ be a common skew-symmetrizer of (the matrices in) $\mathbf{B}$.

We have the following consequence of Theorem 4.1. (It was proved earlier than Theorem 4.1 under the assumption of Theorem 4.1.)

Theorem 4.3** ([NZ12, Eqs. (1,11), (1.15), (2.7), (2.9)]).**

The following equalities hold for any $t\in\mathbb{T}_{n}$:

[TABLE]

We say that a permutation $\sigma\in S_{n}$ is compatible with $D=\mathrm{diag}(d_{1},\dots,d_{n})$ if

[TABLE]

With the permutation matrix $P_{\sigma}$ in (3.18), the condition (4.4) is equivalently expressed as

[TABLE]

Proposition 4.4**.**

The following properties hold.

(1). If $C_{t_{1}}=\sigma C_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$.

(2). If $G_{t_{1}}=\sigma G_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$.

Proof.

(1). By (4.1), we have $DC_{t_{1}}D^{-1}=(G^{T}_{t_{1}})^{-1}$ and $DC_{t_{2}}^{-1}D^{-1}=G^{T}_{t_{2}}$. By assumption, $C_{t_{2}}^{-1}C_{t_{1}}=P_{\sigma}$. Therefore, $DP_{\sigma}D^{-1}=G^{T}_{t_{2}}(G^{T}_{t_{1}})^{-1}$. By (4.3), the righthand side is an integer matrix. Therefore, $d_{i}d_{\sigma(i)}^{-1}$ is an integer for any $i=1,\dots,n$. In particular, $d_{\sigma(i)}\leq d_{i}$ for any $i$. This is possible only if $d_{\sigma(i)}=d_{i}$ for any $i$. Therefore, $\sigma$ is compatible with $D$. The proof of (2) is similar. ∎

We have the following corollary of Theorem 4.3 and Proposition 4.4.

Corollary 4.5**.**

The following statements hold for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, where $P_{\sigma}$ is the permutation matrix in (3.18).

[TABLE]

Proof.

They follow from (3.19), (4.1), (4.2), and (4.5), thanks to the compatibility of $\sigma$ with $D$ in Proposition 4.4. ∎

4.2. Laurent positivity

The second fundamental result is the following one.

Theorem 4.6** **(Laurent positivity, [GHKK18, Corollary 0.3],

[MSW11, Theorem 1.1], [KQ14, Corollary 3.39], [LS15, Theorem 1.1], [Dav18, Theorem 2.4]).

Each $F$-polynomial $F_{i;t}(\mathbf{u})$ has only positive coefficients.

This was conjectured by [FZ07, Section 3], and proved for surface type by [MSW11], for acyclic case by [KQ14], for the skew-symmetric case by [LS15, Dav18], and in general by [GHKK18].

Let us give a simple and useful criterion of the triviality of an $F$-polynomial, which quickly follows from Theorems 4.2 and 4.6.

Lemma 4.7**.**

Let $\underline{\mathbf{\Sigma}}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern without coefficients, and let $\underline{\hat{y}}_{i;t}$ be the one (2.5) for $\underline{\mathbf{\Sigma}}$, namely,

[TABLE]

Then, under the specialization $\underline{x}_{i:t_{1}}=1$ ($i=1,\dots,n$) at any point $t_{1}$, we have the following inequality for any $t,t^{\prime}\in\mathbb{T}_{n}$ and $i=1,\dots,n$:

[TABLE]

Moreover, the equality holds if and only if $F_{i;t}({\mathbf{u}})=1$.

Proof.

By $\eqref{eq:xmut1}$, we have $\underline{x}_{i;t}>0$ for any $t$ and $i$ under the specialization. Therefore, by (4.8), we have $\underline{\hat{y}}_{i;t}>0$ for any $t$ and $i$. Then, the claim is an immediate consequence of Theorems 4.2 and 4.6. ∎

The following theorem is a very important consequence of Theorems 4.1, 4.2, and 4.6 together, where only the case $\sigma=\mathrm{id}$ was treated in [CHL18]. It means that the tropical part ($G$-matrix) uniquely determines the nontropical part ($F$-polynomials).

Theorem 4.8** (Detropicalization, [CHL18, Lemma 2.4 & Theorem 2.5]).**

The following statements hold for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, where $P_{\sigma}$ is the permutation matrix in (3.18):

[TABLE]

For completeness, we present a proof for general $\sigma$, slightly (and carefully) modifying the proof of [CHL18].

Proof.

Proof of (4.10). We prove it by following the proof of Lemma 2.4 of [CHL18].

Step 1. Suppose that $G_{t_{1}}=P_{\sigma}$ for some $t_{1}$. Then, by Corollary 4.5, we have $C_{t_{1}}=P_{\sigma}$. Let $\tilde{\mathbf{\Sigma}}[t_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with principal coefficients at $t_{0}$. Then, by Proposition 3.2, we have

[TABLE]

This implies that the cluster pattern $\tilde{\mathbf{\Sigma}}$ is also viewed as the one with principal coefficients at $t_{1}$. Let $\mathbf{\Gamma}^{\prime}(\mathbf{B},t_{1})=\{\Gamma^{\prime}_{t}=(\mathbf{F}^{\prime}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}$ with the initial point $t_{1}$. Then, by reversing the relation (4.12) and again by Proposition 3.2, we have $C^{\prime}_{t_{0}}=P_{\sigma^{-1}}$. Therefore, we have $G^{\prime}_{t_{0}}=P_{\sigma^{-1}}$ by Corollary 4.5.

Step 2. Next, let $\underline{\mathbf{\Sigma}}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern without coefficients. By applying the separation formula (3.8) with both initial points $t_{0}$ and $t_{1}$, we have, for any $i=1,\dots,n$,

[TABLE]

where we used $G_{t_{1}}=P_{\sigma}$ and $G^{\prime}_{t_{0}}=P_{\sigma^{-1}}$. Then, by replacing $i$ in (4.14) with $\sigma^{-1}(i)$ and multiplying it with (4.13), we obtain

[TABLE]

It is important that the left hand side is $1$. Now we do the specialization $\underline{x}_{1;t_{0}}=\dots,=\underline{x}_{n;t_{0}}=1$ in (4.15). Then, by Lemma 4.7, we conclude that $F_{i;t_{1}}(\mathbf{u})=F^{\prime}_{\sigma^{-1}(i);t_{0}}(\mathbf{u})=1$, which is the desired result. (Simplification of the proof using Lemma 4.7 is due to ourselves.)

Proof of (4.11). Again, we prove it by following the proof of Theorem 2.5 of [CHL18]. Suppose that $G_{t_{1}}=\sigma G_{t_{2}}$ for some $t_{1}$ and $t_{2}$. Then, by Corollary 4.5, we have $C_{t_{1}}=\sigma C_{t_{2}}$ and $B_{t_{1}}=\sigma B_{t_{2}}$. Suppose that $t_{2}$ is connected to the initial point $t_{0}$ in $\mathbb{T}_{n}$ as

[TABLE]

Let $t_{3}\in\mathbb{T}_{n}$ be the one such that

[TABLE]

Then, by (3.17) and ignoring $F$-polynomials therein, we have the following commutative diagram:

[TABLE]

It follows that

[TABLE]

Then, by (4.10), we have $F_{i,t_{3}}(\mathbf{u})=1$ for any $i=1,\dots,n$. Therefore, we have

[TABLE]

Then, again by (3.17), we have the commutative diagram

[TABLE]

Therefore, we obtain

[TABLE]

which proves the claim (4.11). ∎

5. Synchronicity phenomenon

In this section we present several synchronicity properties in cluster patterns based on the results in the previous sections. Many of them have already appeared partially and/or in special cases such as skew-symmetric case, $\sigma=1$ case, finite type case, geometric coefficients case, surface type, etc., in various literature with several methods. The references below are not complete by any means. Also, many of the results may be derived in a different manner via the scattering diagram method in [GHKK18]. However, we believe that our approach is still useful in view of cluster algebra theory as explained in the introduction of Section 4.

5.1. $\sigma$-periodicity

Let us formulate the notion of periodicity for cluster patterns, including the partial periodicity up to permutations $\sigma\in S_{n}$.

Let $\mathbf{\Sigma}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. Suppose that $t_{1},t_{2}\in\mathbb{T}_{n}$ are connected in $\mathbb{T}_{n}$ as

[TABLE]

so that

[TABLE]

Definition 5.1** ($\sigma$-periodicity).**

We call a sequence of mutations of seeds (5.8) a $\sigma$-period if

[TABLE]

In this case we also say that seeds are $\sigma$-periodic under the sequence of mutations (5.8). Similarly, we say that $x$-variables (resp., $y$-variables) are $\sigma$-periodic under the sequence of mutations (5.8) if $\mathbf{x}_{t_{1}}=\sigma\mathbf{x}_{t_{2}}$ (resp., $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$) holds.

5.2. $xy/GC$ synchronicity

In this subsection we present some basic synchronicity property in cluster patterns, which we call the $xy/GC$* synchronicity*. Namely, both periodicities of $x$- and $y$-variables coincide with the common periodicity of their tropical counterparts, $G$- and $C$-matrices at any initial point $t_{0}$, where some condition is assumed for $y$-variables. It has been proved partially and/or in special cases, for example, in [FZ07, Pla11, Nag13, IIK*+*13, CIKLFP13, IN14, GHKK18, CHL18, CL18]. We regard it as the most basic synchronicity property, because all other synchronicity properties studied in this paper are obtained from it.

Let us divide the statement of the $xy/GC$ synchronicity into two theorems. The first half of the statement is as follows:

Theorem 5.2** ($xy/GC$ synchronicity).**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$, and let $\mathbf{\Gamma}(\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}$ at any initial point $t_{0}$. Then, for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, the following three conditions are equivalent:

(a). $G_{t_{1}}=\sigma G_{t_{2}}$.

(b). $C_{t_{1}}=\sigma C_{t_{2}}$.

(c). $\mathbf{x}_{t_{1}}=\sigma\mathbf{x}_{t_{2}}$.

Moreover, one of Conditions (a)–(c) implies the following condition:

(d). $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$.

Proof.

((a) $\Longleftrightarrow$ (b)). This was already stated in (4.7).

((a) $\Longrightarrow$ (c), (d)). Assume that $G_{t_{1}}=\sigma G_{t_{2}}$. Then, by (4.7) and (4.11), we have $C_{t_{1}}=\sigma C_{t_{2}}$, $B_{t_{1}}=\sigma B_{t_{2}}$ and $F_{i;t_{1}}(u)=F_{\sigma^{-1}(i);t_{2}}(u)$. Then, applying the separation formulas (3.8) and (3.9) for $\mathbf{x}_{t_{1}}$ and $\mathbf{y}_{t_{1}}$, respectively, we obtain Conditions (c) and (d).

((c) $\Longrightarrow$ (b)). Assume that ${\mathbf{x}}_{t_{1}}=\sigma{\mathbf{x}}_{t_{2}}$. Now let $\underline{\mathbf{\Sigma}}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern without coefficients. First apply a semifield homomorphism (the trivialization map) in the coefficients of $\mathbf{x}_{t}$

[TABLE]

then apply a homomorphism ${x}_{i;t}\mapsto\underline{x}_{i;t}$ from the subfield of the ambient field $\mathcal{F}$ of $\mathbf{\Sigma}$ generated by $\mathbf{x}_{t}$’s to the ambient field $\underline{\mathcal{F}}$ of $\underline{\mathbf{\Sigma}}$. Then, we obtain $\underline{\mathbf{x}}_{t_{1}}=\sigma\underline{\mathbf{x}}_{t_{2}}$.

Let us take the $FGC$-pattern $\mathbf{\Gamma}(\mathbf{B},t_{2})=\{\Gamma^{\prime}_{t}=(\mathbf{F^{\prime}}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with the initial point $t_{2}$. Applying the separation formula (3.8) for $\underline{\bf{x}}_{t_{1}}$ with the initial point $t_{2}$, we have, for any $i=1,\dots,n$,

[TABLE]

Since $\underline{\mathbf{x}}_{t_{1}}=\sigma\underline{\mathbf{x}}_{t_{2}}$, we have

[TABLE]

Let us evaluate the equality (5.12) under the specialization $\underline{x}_{1;t_{2}}=\cdots=\underline{x}_{n;t_{2}}=1$. Then, we see that $F^{\prime}_{i;t_{1}}(\underline{\hat{\mathbf{y}}}_{t_{2}})$ is 1 under the specialization. Therefore, by Lemma 4.7, we have $F^{\prime}_{i;t_{1}}(\mathbf{u})=1$. Then, again by (5.12), we obtain $G^{\prime}_{t_{1}}=P_{\sigma}$. Therefore, by (4.6), we have $C^{\prime}_{t_{1}}=P_{\sigma}$.

Finally, let $\tilde{\mathbf{\Sigma}}[{t}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with principal coefficients at $t_{0}$. We apply the separation formula (3.9) for $\tilde{\mathbf{y}}_{t_{1}}$ with the initial point $t_{2}$. Since $C^{\prime}_{t_{1}}=P_{\sigma}$ and $F^{\prime}_{i;t_{1}}(\mathbf{u})=1$ as shown above, we obtain $\tilde{\mathbf{y}}_{t_{1}}=\sigma\tilde{\mathbf{y}}_{t_{2}}$. This means $C_{t_{1}}=\sigma C_{t_{2}}$ by (3.7). ∎

In contrast to $x$-variables, the implication (d) $\Longrightarrow$ (a), (b) in Theorem 5.2 does not holds in general. In other words, $y$-variables may admit a finer periodicity than $x$-variables. For example, in the extreme case $\mathbb{P}=\mathbf{1}$, we have $\mathbf{y}_{t}=(1,\dots,1)$ for any $t\in\mathbb{T}_{n}$. Then, $\mathbf{y}_{t}$ is periodic by any single mutation for any $B$-pattern $\mathbf{B}$. To provide a sufficient condition that (d) $\Longrightarrow$ (a), (b) holds, we introduce some notion.

Definition 5.3**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$, and $\tilde{\mathbf{\Sigma}}[t^{\prime}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with principal coefficients at $t^{\prime}_{0}$, such that they share a common $B$-pattern $\mathbf{B}$. Let $\langle\mathbf{y}_{t^{\prime}_{0}}\rangle$ be the subsemifield in $\mathbb{P}$ generated by $\mathbf{y}_{t^{\prime}_{0}}$. Then, we say that $\mathbf{\Sigma}$* covers $\tilde{\mathbf{\Sigma}}[t^{\prime}_{0}]$* if there is a semifield homomorphism such that

[TABLE]

(Here we use the symbol $t^{\prime}_{0}$ so that it cannot be confused with the initial point $t_{0}$ in Theorem 5.2.)

Note that for such $\varphi$ in (5.13),

[TABLE]

holds for any $t\in\mathbb{T}_{n}$ and $i=1,\dots,n$ by homomorphism property.

Example 5.4**.**

A cluster pattern with universal coefficients $\mathbf{\Sigma}$ covers a cluster pattern with principal coefficients $\tilde{\mathbf{\Sigma}}[t^{\prime}_{0}]$ at any point $t^{\prime}_{0}$, where $\varphi:\mathbb{Q}_{\mathrm{sf}}(\mathbf{y}_{t^{\prime}_{0}})\rightarrow\mathrm{Trop}(\tilde{\mathbf{y}}_{t^{\prime}_{0}})$ is the tropicalization homomorphism.

Now the second half of the statement of the $xy/GC$ synchronicity is as follows:

Theorem 5.5** ($xy/GC$ synchronicity).**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\allowbreak B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$ which covers a cluster pattern $\tilde{\mathbf{\Sigma}}[t^{\prime}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with principal coefficients at some $t^{\prime}_{0}$. Then, Condition (d) implies Conditions (a)–(c) in Theorem 5.2.

Proof.

We show (d) $\Longrightarrow$ (b) in Theorem 5.2. Assume that $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$. By assumption, there is a semifield homomorphism $\varphi$ in (5.13). Then, by (5.14), we obtain $\tilde{\mathbf{y}}_{t_{1}}=\sigma\tilde{\mathbf{y}}_{t_{2}}$. Let us consider the $FGC$-pattern $\mathbf{\Gamma}(\mathbf{B},t^{\prime}_{0})=\{\Gamma^{\prime}_{t}=(\mathbf{F^{\prime}}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of $\mathbf{B}$ with the initial point $t^{\prime}_{0}$. Then, by (3.7), we have $C^{\prime}_{t_{1}}=\sigma C^{\prime}_{t_{2}}$. Let $\tilde{\mathbf{\Sigma}}[{t}_{0}]=\{\tilde{\Sigma}^{\prime}_{t}=(\tilde{\mathbf{x}}^{\prime}_{t},\tilde{\mathbf{y}}^{\prime}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with principal coefficients at ${t}_{0}$. Then, applying Theorem 5.2 for $\tilde{\mathbf{\Sigma}}[{t}_{0}]$ and $\mathbf{\Gamma}(\mathbf{B},t^{\prime}_{0})$, we obtain $\tilde{\mathbf{y}}^{\prime}_{t_{1}}=\sigma\tilde{\mathbf{y}}^{\prime}_{t_{2}}$. Then, applying (3.7) for $\tilde{\mathbf{y}}^{\prime}_{t}$, we obtain $C_{t_{1}}=\sigma C_{t_{2}}$. ∎

We obtain the following property of the periodicities of $x$- and $y$-variables.

Corollary 5.6**.**

Let $D$ be a common skew-symmetrizer of $\mathbf{B}$.

(1). Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. If $\mathbf{x}_{t_{1}}=\sigma\mathbf{x}_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$.

(2). Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\allowbreak B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$ which covers a cluster pattern with principal coefficients at some $t^{\prime}_{0}$. If $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$.

Proof.

This follows from Proposition 4.4 and Theorems 5.2 and 5.5. ∎

5.3. More synchronicity results

Let us present some consequences of Theorems 5.2 and 5.5. Below “periodicity” is used in the sense of $\sigma$-periodicity.

The following theorem states that the periodicity of $x$-variables is independent of the choice of $\mathbb{P}$, $\mathbf{x}_{t}$’s, and $\mathbf{y}_{t}$’s. It was proved by [CL18] using $d$-vectors. Here we give an alternative proof via Theorem 5.2.

Theorem 5.7** ([CL18, Proposition 6.1]).**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. Then, the periodicity of $x$-variables $\mathbf{x}_{t}$ depends only on the $B$-pattern $\mathbf{B}$ therein.

Proof.

By Theorem 5.2, the periodicity of $x$-variables for any cluster pattern coincides with the periodicity of $C$-matrices (with any initial point), which are uniquely determined from $\mathbf{B}$. ∎

Similarly, by Theorem 5.5, we obtain the counterpart for $y$-variables.

Theorem 5.8**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$ such that $\mathbf{\Sigma}$ covers a cluster pattern with principal coefficients at some $t^{\prime}_{0}$. Then, the periodicity of $y$-variables $\mathbf{y}_{t}$ depends only on the $B$-pattern $\mathbf{B}$ therein.

Combining Theorems 5.7 and 5.8 (and their proofs), we have the following synchronicity of $x$- and $y$-variables.

Theorem 5.9**.**

The $x$-variables in Theorem 5.7 and the $y$-variables in Theorem 5.8 with a common $B$-pattern $\mathbf{B}$ share the same periodicity.

Example 5.10**.**

As a special case of Theorem 5.9, take a cluster pattern $\underline{\mathbf{\Sigma}}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ without coefficients for Theorem 5.7 and a cluster pattern $\overline{\mathbf{\Sigma}}=\{\overline{\Sigma}_{t}=(\overline{\mathbf{x}}_{t},\overline{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ with universal coefficients for Theorem 5.8. Then, we have

[TABLE]

This proves the synchronicity (1.8) in Question 1.1 as the special case $\sigma=\mathrm{id}$.

So far, we only consider the synchronicity of $x$-variables and $y$-variables. Let us extend the result for seeds and $Y$-seeds.

The following theorem tells that the periodicities of $x$-variables alone and seeds including them coincide for any cluster pattern. It was proved by [CL18] using $d$-vectors. Here we give an alternative proof via Theorem 5.2. See also Conjecture 5.15 and Theorem 5.16 for related results.

Theorem 5.11** ([CL18, Proposition 6.1]).**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. Then, the $x$-variables $\mathbf{x}_{t}$ and the seeds $\Sigma_{t}$ share the same periodicity. Furthermore, it depends only on the $B$-pattern $\mathbf{B}$ therein.

Proof.

It is enough to show the following claim:

[TABLE]

This is a consequence of Theorem 5.2 and (4.7). ∎

A pair $(\mathbf{y}_{t},B_{t})$ is called a $Y$-seed in [FZ07]. We have a counterpart of Theorem 5.11 for $Y$-seeds.

Theorem 5.12**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$ such that $\mathbf{\Sigma}$ covers a cluster pattern with principal coefficients at some $t^{\prime}_{0}$. Then, the $y$-variables $\mathbf{y}_{t}$ and the $Y$-seeds $(\mathbf{y}_{t},B_{t})$ share the same periodicity. Furthermore, it depends only on the $B$-pattern $\mathbf{B}$ therein.

Proof.

It is enough to show the following claim:

[TABLE]

This is a consequence of Theorem 5.5 and (4.7). ∎

By combining Theorems 5.9, 5.11, and 5.12, we also have the following synchronicity of seeds and $Y$-seeds.

Theorem 5.13**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$ such that $\mathbf{\Sigma}$ covers a cluster pattern with principal coefficients at some $t^{\prime}_{0}$. Then, the seeds $\Sigma_{t}$ and the $Y$-seeds $(\mathbf{y}_{t},B_{t})$ share the same periodicity. Furthermore, it depends only on the $B$-pattern $\mathbf{B}$ therein.

5.4. Conjectures by Fomin and Zelevinsky

In [FZ03a, FZ07] Fomin and Zelevinsky proposed several important conjectures on periodicity properties of cluster patterns.

Let us recall some definitions from [FZ03a, FZ07].

Definition 5.14**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. An unlabeled cluster of $\mathbf{\Sigma}$ is a set $\{x_{1;t},\dots,x_{n;t}\}$ ($t\in\mathbb{T}_{n}$) of $x$-variables belonging to $\mathbf{x}_{t}$. An unlabeled seed $[\Sigma_{t}]$ of $\mathbf{\Sigma}$ ($t\in\mathbb{T}_{n}$) is an equivalence class of all labeled seeds obtained from $\Sigma_{t}$ by the action of permutations given by (3.10)–(3.12). The exchange graph (of unlabeled seeds) of $\mathbf{\Sigma}$ is a quotient graph of $\mathbb{T}_{n}$ modulo the equivalence relation $t\sim t^{\prime}$ defined by the condition $[\Sigma_{t}]=[\Sigma_{t^{\prime}}]$.

The following conjecture was proposed by [FZ03a].

Conjecture 5.15** ([FZ03a, Section 1.5]).**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with coefficients in any semifield $\mathbb{P}$. Then, the following properties hold:

(a). Each unlabeled cluster $\{x_{1;t},\dots,x_{n;t}\}$ uniquely determines an unlabeled seed containing it.

(b). The exchange graph of $\mathbf{\Sigma}$ depends only on the $B$-pattern $\mathbf{B}$ therein.

(c). Two unlabeled clusters are adjacent in the exchange graph if and only if they have exactly $n-1$ common $x$-variables.

Claim (a) was proved in several cases in [FZ03a, BMRT07, FST08, GSV08, CIKLFP13] and in full generality in [CL18]. Claim (b) was also proved in several cases in [FZ03a, FST08, GSV08, CIKLFP13] and in full generality in [CL18]. It was shown in [GSV08] that Claim (c) follows from Claim (a).

Theorem 5.16** ([CL18, Proposition 6.1]).**

Conjecture 5.15 is true.

Proof.

Claim (a) is equivalent to the first half of Theorem 5.11. Claim (b) follows from the second half of Theorem 5.11. Claim (c) follows from Claim (a) due to [GSV08, Theorem 5] as already mentioned. ∎

Remark 5.17**.**

Conjecture 5.15 (b) was enhanced by [FZ07, Conjeture 4.3] to includes the statement that the canonical connection on the exchange graph (of unlabeled seeds) of $\mathbf{\Sigma}$ depends only on $\mathbf{B}$. This is also true by the following reason. The exchange graph of labeled seeds has the canonical connection induced from the labeling of $\mathbb{T}_{n}$. The canonical connection on the exchange graph of unlabeled seed, is further induced from the above one modulo $\sigma$-periodicity. (This is well defined by (3.13).) Since the $\sigma$-periodicity depends only on $\mathbf{B}$, the induced canonical connection also depends only on $\mathbf{B}$.

There is another conjecture proposed by [FZ07].

Conjecture 5.18** ([FZ07, Conjetures 4.7 & 4.8]).**

Let $\tilde{\mathbf{\Sigma}}[{t}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},\allowbreak B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern with principal coefficients at any ${t}_{0}$. Then, the following properties hold:

(a). The seeds $\tilde{\Sigma}_{t}$ and the $Y$-seeds $(\tilde{\mathbf{y}}_{t},B_{t})$ share the same periodicity.

(b). There are finitely many distinct seeds $\Sigma_{t}$ if and only if there are finitely many distinct $Y$-seeds $(\tilde{\mathbf{y}}_{t},B_{t})$.

Theorem 5.19**.**

Conjecture 5.18 is true.

Proof.

Claim (a) is a special case of Theorem 5.13. Claim (b) is a consequence of Claim (a) as already noted in [FZ07]. ∎

5.5. Synchronicity under dualities

As another application of Theorems 5.2 and 5.11, let us show that the dualities of $B$-patterns considered by Fock and Goncharov in [FG09a, Section 1] preserve the periodicity of cluster patterns.

Let $\mathbf{B}=\{B_{t}\mid t\in\mathbb{T}_{n}\}$ be a $B$-pattern with a common skew-symmetrizer $D$. Then, it is easy to see from (2.4) that the following three families of matrices are also $B$-patterns.

(a). (Transposition duality) $\mathbf{B}^{T}:=\{B^{T}_{t}\mid t\in\mathbb{T}_{n}\}$. A common skew-symmetrizer is given by $D^{\prime}=\tilde{d}D^{-1}$, where $\tilde{d}$ is the least common multiple of $d_{1},\dots,d_{n}$.

(b). (Chiral duality) $-\mathbf{B}:=\{-B_{t}\mid t\in\mathbb{T}_{n}\}$. A common skew-symmetrizer is given by $D$.

(c). (Langlands duality) $-\mathbf{B}^{T}:=\{-B^{T}_{t}\mid t\in\mathbb{T}_{n}\}$. A common skew-symmetrizer is given by $D^{\prime}$ in (a).

Note that the compatibility of a permutation $\sigma\in S_{n}$ with $D$ is equivalent to the compatibility with $D^{\prime}$.

Let us start with the transposition duality. The following theorem was suggested by Sergey Fomin to the author.

Theorem 5.20**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},B^{T}_{t})\mid t\in\mathbb{T}_{n}\}$ be cluster patterns with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. Then, the seeds in ${\mathbf{\Sigma}}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}$ share the same periodicity.

Proof.

Thanks to Theorem 5.11, it is enough to prove

[TABLE]

where $\sigma$ is compatible with $D$ and $D^{\prime}$. Let us consider the pair of $FGC$-patterns $\mathbf{\Gamma}(\mathbf{B},t_{2})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and $\mathbf{\Gamma}(\mathbf{B}^{T},{t}_{1})=\{\Gamma^{\prime}_{t}=(\mathbf{F^{\prime}}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},B^{T}_{t})\mid t\in\mathbb{T}_{n}\}$, where $t_{1}$ and $t_{2}$ are the ones in (5.18). We recall another duality of $G$- and $C$-matrices from [NZ12, Eq. (1.13)], namely,

[TABLE]

(This is also a consequence of Theorem 4.1.) Let us assume that ${\mathbf{x}}_{t_{1}}=\sigma{\mathbf{x}}_{t_{2}}$. Then, by Theorem 5.2, we have $C_{t_{1}}=P_{\sigma}$. Therefore, by (5.19), we have $G^{\prime}_{t_{2}}=P_{\sigma^{-1}}$. Then, again by Theorem 5.2, we have ${\mathbf{x}}^{\prime}_{t_{2}}=\sigma^{-1}{\mathbf{x}}^{\prime}_{t_{1}}$, that is, ${\mathbf{x}}^{\prime}_{t_{1}}=\sigma{\mathbf{x}}^{\prime}_{t_{2}}$. The other direction also follows by symmetry. ∎

Next we consider the chiral duality.

Theorem 5.21**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},-B_{t})\mid t\in\mathbb{T}_{n}\}$ be cluster patterns with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. Then, the seeds in ${\mathbf{\Sigma}}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}$ share the same periodicity.

Proof.

One can prove the claim by Theorem 5.13 and the following involution between $Y$-patterns for $\mathbf{B}$ and $-\mathbf{B}$ with universal $y$-variables [FZ07, Proof of Proposition 5.3]:

[TABLE]

Alternatively, one can also prove it using the following equality [FG19, Theorem 2.8] for the pair of $FGC$-patterns $\mathbf{\Gamma}(\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and $\mathbf{\Gamma}(-\mathbf{B},{t}_{0})=\{\Gamma^{\prime}_{t}=(\mathbf{F^{\prime}}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},-B_{t})\mid t\in\mathbb{T}_{n}\}$ with any initial point $t_{0}$:

[TABLE]

where $F_{t}$ (the $F$-matrix at $t$) is the degree matrix of the $F$-polynomials $\mathbf{F}_{t}(\mathbf{u})$ defined in [FG19]. (This is not a consequence of Theorem 4.1.) By (5.21), (4.7) and (4.11), we obtain

[TABLE]

Then, the claim follows from Theorems 5.2 and 5.11. ∎

Finally we consider the Langlands duality.

Theorem 5.22**.**

Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},-B^{T}_{t})\mid t\in\mathbb{T}_{n}\}$ be cluster patterns with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. Then, the seeds in ${\mathbf{\Sigma}}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}$ share the same periodicity.

Proof.

This is proved by combining Theorems 5.20 and 5.21. Alternatively, one can also prove it using the following yet another duality [NZ12, Eq. (1.11)] for the pair of $FGC$-patterns $\mathbf{\Gamma}(\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and $\mathbf{\Gamma}(-\mathbf{B}^{T},{t}_{0})=\{\Gamma^{\prime}_{t}=(\mathbf{F^{\prime}}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},-B^{T}_{t})\mid t\in\mathbb{T}_{n}\}$ with any initial point $t_{0}$:

[TABLE]

(This is also a consequence of Theorem 4.1. Also, this is actually equivalent to (4.1) due to the relation $C^{\prime}_{t}=DC_{t}D^{-1}$.) By (5.23), we obtain

[TABLE]

Then, the claim follows from Theorems 5.2 and 5.11. ∎

5.6. Synchronicity for conjugate pair

Let $R=\mathrm{diag}(r_{1},\dots,r_{n})$ be a diagonal matrix with positive integer diagonal entries $r_{1},\dots,r_{n}$. Let us consider a pair of $B$-patterns of the form

[TABLE]

where $RB_{t}$ and $B_{t}R$ are (skew-symmetrizable) integer matrices, but $B_{t}$ is not necessarily an integer matrix. They are related by the conjugation

[TABLE]

It is easy to see from (5.27) and (2.4) that if one is a $B$-pattern, then so is the other. We call such $B$-patterns a conjugate pair. Such a pair appears in the context of generalized cluster algebras [Nak15, NR16]. See Definition 7.9.

Lemma 5.23**.**

Let $R\mathbf{B}$ and $\mathbf{B}R$ be a conjugate pair. If $D$ is a common skew-symmetrizer of $R\mathbf{B}$, then $DR^{2}$ is a common skew-symmetrizer of $\mathbf{B}R$.

Proof.

Suppose that $(DRB_{t})^{T}=-DRB_{t}$. Then, $R(DRB_{t})^{T}R=-DR^{2}B_{t}R$. That is, $(DR^{2}B_{t}R)^{T}=-DR^{2}B_{t}R$. ∎

Proposition 5.24**.**

Let $R\mathbf{B}$ and $\mathbf{B}R$ be a conjugate pair. Let $\mathbf{\Gamma}(R\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},RB_{t})\mid t\in\mathbb{T}_{n}\}$ and $\mathbf{\Gamma}(\mathbf{B}R,t_{0})=\{\Gamma^{\prime}_{t}=(\mathbf{F}^{\prime}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},\allowbreak B_{t}R)\mid t\in\mathbb{T}_{n}\}$ be their $FGC$-patterns with any initial point $t_{0}$.

(1). The following equalities hold:

[TABLE]

(2). If $G_{t_{1}}=\sigma G_{t_{2}}$ or $G^{\prime}_{t_{1}}=\sigma G^{\prime}_{t_{2}}$ for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$ and $R$.

Proof.

(1). They are true at $t_{0}$. Then, they are shown from (3.4), (3.5), and (5.27) by induction on $t$.

(2). Suppose that $G_{t_{1}}=\sigma G_{t_{2}}$. Then, $\sigma$ is compatible with $D$ by Proposition 4.4. Meanwhile, by (5.28), $R^{-1}G_{t_{1}}R=G^{\prime}_{t_{1}}$ and $R^{-1}G_{t_{2}}^{-1}R=(G^{\prime}_{t_{2}})^{-1}$. Therefore, $R^{-1}P_{\sigma}R=R^{-1}G_{t_{2}}^{-1}G_{t_{1}}R=(G^{\prime}_{t_{2}})^{-1}G^{\prime}_{t_{1}}$, where the righthand side is an integer matrix. Then, by repeating the argument in the proof of Proposition 4.4, we conclude that $\sigma$ is compatible with $R$. The other case $G^{\prime}_{t_{1}}=\sigma G^{\prime}_{t_{2}}$ is similar. ∎

Theorem 5.25**.**

Let $R\mathbf{B}$ and $\mathbf{B}R$ be a conjugate pair with common skew-symmetrizers $D$ and $DR^{2}$, respectively. Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},RB_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},B_{t}R)\mid t\in\mathbb{T}_{n}\}$ be cluster patterns with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively.

(1). If $\Sigma_{t_{1}}=\sigma\Sigma_{t_{2}}$ or $\Sigma^{\prime}_{t_{1}}=\sigma\Sigma^{\prime}_{t_{2}}$ for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$ and $R$.

(2). The seeds in ${\mathbf{\Sigma}}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}$ share the same periodicity.

Proof.

(1). This follows from Theorems 5.2 and 5.11, and Proposition 5.24.

(2). Let $\mathbf{\Gamma}(R\mathbf{B},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u}),G_{t},C_{t},RB_{t})\mid t\in\mathbb{T}_{n}\}$ and $\mathbf{\Gamma}(\mathbf{B}R,t_{0})=\{\Gamma^{\prime}_{t}=(\mathbf{F}^{\prime}_{t}(\mathbf{u}),G^{\prime}_{t},C^{\prime}_{t},B_{t}R)\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-patterns of $R\mathbf{B}$ and $\mathbf{B}R$ with any initial point $t_{0}$, respectively. For a permutation $\sigma$ which is compatible with $R$, $G_{t_{1}}=\sigma G_{t_{2}}$ if and only if $G^{\prime}_{t_{1}}=\sigma G^{\prime}_{t_{2}}$ thanks to (5.28). Then, the claim follows from Theorems 5.2, 5.11, and Claim (1). ∎

See the forthcoming Corollary 9.14 for a stronger version of Theorem 5.25 (1) when $B$ is an integer matrix.

6. Open problem

We have seen that various objects such as seeds, $Y$-seeds, $x$-variables, $y$-variables, $C$-matrices, $G$-matrices, etc, with a common $B$-pattern $\mathbf{B}$ share the same periodicity. In other words, they share the same exchange graph. Also, a categorification of a cluster pattern share the same exchange graph with the underlying cluster pattern by definition. In addition, in Part II we show that cluster patterns of generalized cluster algebras also share the exchange graphs with ordinary cluster algebras, up to the Laurent positivity conjecture (Theorem 9.13). These results suggest that the underlying exchange graph is the essence of what we vaguely call cluster structure.

In this aspect it may be more convenient to consider the exchange graphs of labeled seeds, rather than unlabeled seeds. Let us introduce the labeled exchange graph of a cluster pattern $\mathbf{\Sigma}$, which is the quotient graph of $\mathbb{T}_{n}$ modulo the equivalence relation $t\sim t^{\prime}$ defined by the condition $\Sigma_{t}=\Sigma_{t^{\prime}}$, endowed with arrows $t_{2}\xrightarrow{\sigma}t_{1}$ if $\Sigma_{t_{1}}=\sigma\Sigma_{t_{2}}$ for a permutation $\sigma\in S_{n}$. Moreover, these labeled exchange graphs are better regarded as groupoids as proposed earlier in [FG09b, Section 2.1], where objects are seeds, $Y$-seeds, etc., and morphisms are generated by their mutations and permutations. Then, the synchronicity means that we have different realizations (representations) of a common groupoid with various objects. It is natural to call this common underlying groupoid $\mathcal{G}(\mathbf{B})$ the cluster groupoid of $\mathbf{B}$. (It is called the cluster modular groupoid in [FG09a].)

In analogy with reflection groups and Coxeter groups, the usual construction of $\mathcal{G}(\mathbf{B})$ by seeds and their mutations may correspond to an explicit construction of reflection groups starting from the simple roots. On the other hand, like Coxeter groups, cluster groupoids should be also defined as abstract groupoids by generators and relations. Unfortunately we do not yet know much about the fundamental relations of morphisms in cluster groupoids except for rank 2 case [FZ02, FG09a] and certain surface type [FST08, Theorem 9.1, Remark 9.19]. Therefore, we believe that the following problem has fundamental importance in cluster algebra theory:

Problem 6.1**.**

Describe the fundamental relations of morphisms in a cluster groupoid $\mathcal{G}(\mathbf{B})$ in terms of the exchange matrices in $\mathbf{B}$.

Part II Synchronicity in cluster patterns of generalized cluster algebras

Generalized cluster algebras (GCAs) were introduced by Chekhov and Shapiro [CS14] as a generalization of (ordinary) cluster algebras (CAs). It turned out that many (possibly all) important properties of CAs are naturally extended to GCAs [CS14, Nak15]. In this part we show that all main results in Part I are naturally extended to cluster patterns of GCA as expected, up to the Laurent positivity which is still a conjecture for GCA. Moreover, we show that any cluster pattern of GCA synchronizes with its “companion cluster patterns” of CA.

7. Cluster patterns and separation formulas of GCA

7.1. Cluster patterns of GCA

We recall the definitions of seeds and cluster patterns of (normalized) GCA [CS14] following the formulation of [Nak15, NR16].

Definition 7.1** (Seeds of GCA).**

Let $\mathbb{P}$ be any semifield. We fix a diagonal matrix $R=\mathrm{diag}(r_{1},\dots,r_{n})$ whose diagonal entries $r_{1},\dots,r_{n}$ are positive integers. (Here we use the notation $r_{i}$ instead of $d_{i}$ and vice versa in [Nak15, NR16].) We call it the degree of GCA. A (labeled) seed of degree $R$ with coefficients in $\mathbb{P}$ is a quadruplet $\Sigma=(\mathbf{x},\mathbf{y},\mathbf{z},B)$, where $(\mathbf{x},\mathbf{y},B)$ is a seed of CA in Definition 2.2, and

[TABLE]

are collection of arbitrary elements in $\mathbb{P}$. For $\mathbf{z}$, we associate an $n$-tuple $\mathbf{Z}(\eta)=(Z_{1}(\eta),\dots,Z_{n}(\eta))$ of polynomials (exchange polynomials) in a single variable $\eta$ with coefficients in $\mathbb{P}$ such that

[TABLE]

with $z_{i,0}=z_{i,r_{i}}=1$.

When $r_{i}=1$, $\mathbf{z}_{i}$ is empty, and $Z_{i}(\eta)=1+\eta$. In particular, when $R=I$, $\mathbf{z}$ is empty, and seeds of GCA reduce to seeds of CA. For $R\neq I$, the seed mutations in Definition 2.3 are generalized as follows.

Definition 7.2** (Seed mutations of GCA).**

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

[TABLE]

where $\hat{y}_{k}$ in (7.4) is defined by

[TABLE]

The seed $\Sigma^{\prime}$ is called the mutation of $\Sigma$ at direction $k$, and denoted by $\mu_{k}(\Sigma)$. Again, these mutations are involutive.

It is easy to see that $RB^{\prime}$ (resp. $B^{\prime}R$) is obtained from $RB$ (resp. $BR$) by the ordinary matrix mutation (2.4). In other words, the (ordinary) $B$-patterns $R\mathbf{B}$ and $\mathbf{B}R$ form a conjugate pair in the sense of Section 5.6.

In terms of exchange polynomials, the mutation of $\mathbf{z}$ in (7.6) is rephrased as

[TABLE]

Accordingly, a *cluster pattern $\mathbf{\Sigma}_{R}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with coefficients in a semifield $\mathbb{P}$ * is defined in the same way as the ordinary one in Definition 2.4 by replacing seeds of CA therein with seeds of degree $R$ of GCA.

Example 7.3**.**

(1). A cluster pattern ${\mathbf{\Sigma}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with coefficients in $\mathbb{P}=\mathbb{Q}_{\mathrm{sf}}({\mathbf{y}}_{{t}_{0}},{\mathbf{z}}_{t_{0}})$ for some $t_{0}\in\mathbb{T}_{n}$ is called a cluster pattern of degree $R$ with universal coefficients.

(2). A cluster pattern $\tilde{\mathbf{\Sigma}}_{R}[{t}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with coefficients in $\mathbb{P}=\mathrm{Trop}(\tilde{\mathbf{y}}_{{t}_{0}},\tilde{\mathbf{z}}_{t_{0}})$ for some $t_{0}\in\mathbb{T}_{n}$ is called a cluster pattern of degree $R$ with principal coefficients at $t_{0}$.

(3). A cluster pattern $\check{\mathbf{\Sigma}}_{R}[{t}_{0}]=\{\check{\Sigma}_{t}=(\check{\mathbf{x}}_{t},\check{\mathbf{y}}_{t},\check{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with coefficients in $\mathbb{P}=\mathrm{Trop}(\check{\mathbf{y}}_{{t}_{0}})$ for some $t_{0}\in\mathbb{T}_{n}$ such that $\check{z}_{i,s;t}=1$ for any $i,s$, and $t$ is called a cluster pattern of degree $R$ with $Y$-principal coefficients at $t_{0}$. In this case one can set $\check{{\Sigma}}_{t}=(\check{\mathbf{x}}_{t},\check{\mathbf{y}}_{t},B_{t})$.

(4). A cluster pattern $\underline{\mathbf{\Sigma}}_{R}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},\underline{\mathbf{y}}_{t},\underline{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with coefficients in the trivial semifield $\mathbf{1}$ is called a cluster pattern of degree $R$ without coefficients. In this case one can set $\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})$.

7.2. $FGC$-patterns of GCA and separation formulas

Next we introduce $FGC$-patterns of GCA following [Nak15].

Let $\mathbf{\Sigma}_{R}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$. We arbitrary choose a distinguished point (the initial point) $t_{0}$ in $\mathbb{T}_{n}$. The seed $(\mathbf{x}_{t_{0}},\mathbf{y}_{t_{0}},\mathbf{z}_{t_{0}},B_{t_{0}})$ at $t_{0}$ is called the initial seed of $\mathbf{\Sigma}_{R}$. We use the following simplified notation

[TABLE]

together with the notation (3.1).

Let us extract a family $\mathbf{B}_{R}=\{B_{t})\mid t\in\mathbb{T}_{n}\}$ from $\mathbf{\Sigma}_{R}$, which we call a $B$-pattern of degree $R$. For any $B$-pattern $\mathbf{B}_{R}=\{B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ and any $t_{0}\in\mathbb{T}_{n}$, we introduce a family of quintuplet $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\mathbf{v}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ uniquely determined from $\mathbf{B}_{R}$ and $t_{0}$ as follows. For each $t\in\mathbb{T}_{n}$, $\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\mathbf{v}_{t},B_{t})$ is a quintuplet (let us call it an $FGC$-seed of degree $R$), where

• •

$\mathbf{F}_{t}(\mathbf{u},\mathbf{v})=(F_{1;t}(\mathbf{u},\mathbf{v}),\dots,F_{n;t}(\mathbf{u},\mathbf{v}))$ is an $n$-tuple of rational functions (called $F$-polynomials) in variables $\mathbf{u}=(u_{1},\dots,u_{n})$ and $\mathbf{v}=(v_{i,s})_{1\leq i\leq n,1\leq s\leq r_{i}-1}$ with coefficients in $\mathbb{Q}$ having a subtraction-free expression, i.e., $F_{i;t}(\mathbf{u},\mathbf{v})\in\mathbb{Q}_{\mathrm{sf}}(\mathbf{u},\mathbf{v})$,

• •

$G_{t}$ and $C_{t}$ are $n\times n$ integer matrices (called a $G$-matrix and a $C$-matrix),

• •

and $\mathbf{v}_{t}=(v_{i,s;t})_{1\leq i\leq n,1\leq s\leq r_{i}-1}$ is obtained from $\mathbf{v}$ by some permutations.

At the initial point $t_{0}$, they are given by

[TABLE]

For $t,t^{\prime}\in\mathbb{T}_{n}$ which are connected by an edge labeled by $k$, $\Gamma_{t}$ and $\Gamma_{t^{\prime}}$ are related by the following mutation at $k$:

[TABLE]

where

[TABLE]

with $v_{k,0;t}=v_{k,r_{k};t}=1$. Note that $v$-variables mutate as $z$-variables in (7.6). Again, these mutations are involutive. We call $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})$ the $FGC$-pattern of $\mathbf{B}_{R}$ with the initial point $t_{0}$.

Here is the Laurent property of GCA.

Theorem 7.4** ([CS14, Theorem 2.5],[Nak15, Proposition 3.3]).**

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

We also note an analogue of a well known formula for CA (Cf. [FZ07, Proposition 5.2]).

Proposition 7.5**.**

The following formula holds for any $i$ and $t\in\mathbb{T}_{n}$:

[TABLE]

Proof.

This can be easily deduced from (7.12) and (7.16) by induction on $t\in\mathbb{T}_{n}$. ∎

There are two kinds of analogues of Proposition 7.6, both of which are useful. The first one is for cluster patterns with principal coefficients.

Proposition 7.6** ([Nak15, Definition 3.7 & Proposition 3.8]).**

Let $\tilde{\mathbf{\Sigma}}_{R}[t_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with principal coefficients at any point $t_{0}$, and let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\allowbreak{\mathbf{v}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with the initial point $t_{0}$. Then, the following formula holds for any $t\in\mathbb{T}_{n}$:

[TABLE]

Proof.

This is proved by comparing the mutations (7.5) and (7.14). ∎

The second one is for cluster patterns with $Y$-principal coefficients.

Proposition 7.7**.**

Let $\check{\mathbf{\Sigma}}_{R}[t_{0}]=\{\check{\Sigma}_{t}=(\check{\mathbf{x}}_{t},\check{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with $Y$-principal coefficients at any point $t_{0}$, and let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\allowbreak{\mathbf{v}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with the initial point $t_{0}$. Then, the following formula holds for any $t\in\mathbb{T}_{n}$:

[TABLE]

Proof.

Again, this is proved by comparing the mutations (7.5) and (7.14). ∎

The separation formulas for GCA are just in the same form as the ones for CA.

Theorem 7.8** (Separation Formulas [Nak15, Theorems 3.22 & 3.23]).**

For a cluster pattern $\mathbf{\Sigma}_{R}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with coefficients in any semifield $\mathbb{P}$, the following formulas hold for any $t\in\mathbb{T}_{n}$:

[TABLE]

7.3. Companion patterns

For any $FGC$-pattern of GCA, we associate a pair of “companion” $FGC$-patterns of CA based on the results in [Nak15, NR16].

As remarked after Definition 7.2, for any $B$-pattern $\mathbf{B}_{R}=\{B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$, we can associate a pair of $B$-patterns (of CA),

[TABLE]

which form a conjugate pair in the sense of Section 5.6. (Here, $B_{t}$ is an integer matrix.)

Definition 7.9** (Companion patterns).**

Let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),\allowbreak G_{t},C_{t},\mathbf{v}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with any initial point $t_{0}$. The $FGC$-patterns (of CA),

[TABLE]

are called the left- and right-companions of $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})$, respectively.

They are companions in the following sense.

Theorem 7.10**.**

The following equalities hold.

(1) [Nak15, Propositions 3.9 & 3.16].

[TABLE]

(2) [NR16, Propositions 4.3 & 4.6].

[TABLE]

Proof.

They are obtained by comparing the mutations (7.12)–(7.16) for GCA and the mutations (3.3)–(3.6) for CA. ∎

8. Further fundamental results in GCA

8.1. Sign-coherence

Let us present analogous results related to the sign-coherence in Section 4.1. Let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),\allowbreak G_{t},C_{t},\mathbf{v}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with any initial point $t_{0}$.

The sign-coherence of $C$-matrices of GCA immediately follows from (7.26) and Theorem 4.1.

Theorem 8.1** (Sign-coherence, [Nak15, Theorem 3.20]).**

Each $C$-matrix $C_{t}$ is column-sign coherent. Namely, each column vector ($c$-vector) of $C_{t}$ is nonzero vector, and its components are either all nonnegative or all nonpositive.

Other related results in Section 4.1 also hold via Theorem 7.10 and/or 8.1.

Theorem 8.2**.**

[Nak15*, Theorem 3.19]**

The constant term of each $F$-polynomial $F_{i;t}(\mathbf{u},\mathbf{v})$ is 1. Moreover, $F_{i;t}(\mathbf{0},\mathbf{v})=1$.*

Proof.

Observe that if all $F_{i;t}(\mathbf{u},\mathbf{v})$ ($i=1,\dots,n$) satisfy this property for some $t$, then $M_{k;t}(\mathbf{u},\mathbf{v})$ in (7.16) also satisfy the same property thanks to the sign-coherence in Theorem 8.1. Then, the claim is proved by induction on $t\in\mathbb{T}_{n}$ using (7.12). ∎

Let $D$ be a common skew-symmetrizer of $R\mathbf{B}$.

Theorem 8.3** ([Nak15, Proposition 3.21]).**

The following equalities hold for any $t\in\mathbb{T}_{n}$:

[TABLE]

Proof.

This is obtained from Theorems 4.3 and 7.10. ∎

Define $\sigma G_{t}$, $\sigma C_{t}$, $\sigma B_{t}$ as before in (3.15), (3.16), (3.12), respectively.

Proposition 8.4**.**

The following properties hold.

(1). If $C_{t_{1}}=\sigma C_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$ and $R$.

(2). If $G_{t_{1}}=\sigma G_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$ and $R$.

Proof.

(1). The compatibility with $D$ is a consequence of Proposition 4.4 and (7.26). Also, by applying the argument in the proof of Proposition 4.4 to (8.1) intead of (4.1), the compatibility with $RD$ is shown. Therefore, $\sigma$ is also compatible with $R$. The proof of (2) is similar. ∎

Corollary 8.5**.**

The following statements hold for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, where $P_{\sigma}$ is the permutation matrix in (3.18).

[TABLE]

Proof.

This follows from Theorem 8.3 and Proposition 8.4. ∎

8.2. Laurent positivity of GCA

Next let us consider analogous results related to the Laurent positivity in Section 4.2. Again, let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),\allowbreak G_{t},C_{t},\mathbf{v}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with any initial point $t_{0}$.

Unfortunately, the Laurent positivity of GCA itself is still an open problem.

Conjecture 8.6** (Laurent positivity).**

Each $F$-polynomial $F_{i;t}(\mathbf{u},\mathbf{v})$ has only positive coefficients.

We note that, though limited, all examples of $F$-polynomials in [Nak15, NR16] have this property. To obtain the synchronicity results, we need to assume the validity of the conjecture in some part. We will make it clear in each proposition, whenever the conjecture is assumed.

Here is an analogous result to Lemma 4.7, which we will use later.

Let $F_{i;t}(\mathbf{u},\mathbf{1})$ denote the polynomial in $\mathbf{u}$ obtained from $F_{i;t}(\mathbf{u},\mathbf{v})$ by setting all $v_{i,s}$ to be $1$.

Lemma 8.7**.**

Assume that Conjecture 8.6 is true. Let $\underline{\mathbf{\Sigma}}_{R}=\{\underline{\Sigma}_{t}=(\underline{\mathbf{x}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ without coefficients, and let $\underline{\hat{y}}_{i;t}$ be the one (7.8) for $\underline{\mathbf{\Sigma}}$, namely,

[TABLE]

Then, under the specialization $\underline{x}_{i:t_{1}}=1$ ($i=1,\dots,n$) at any point $t_{1}$, we have the following inequality for any $t,t^{\prime}\in\mathbb{T}_{n}$ and $i=1,\dots,n$:

[TABLE]

Moreover, the equality holds if and only if $F_{i;t}({\mathbf{u},\mathbf{v}})=1$.

Proof.

Under the assumption of Conjecture 8.6, the polynomial $F_{i;t}(\mathbf{u},\mathbf{1})$ in $\mathbf{u}$ has only positive coefficients. Furthermore, by Theorem 8.2, the constant term of $F_{i;t}(\mathbf{u},\mathbf{1})$ is 1. Then, the rest of the the proof is the same as the proof of Lemma 4.7. ∎

An analogue of Theorem 4.8, which is essential in our study, can be obtained by the same argument as before using Lemma 8.7, where Conjecture 8.6 is assumed. However, fortunately one can also deduce it through Theorem 4.8 without assuming Conjecture 8.6. Here, we present the latter proof.

We use the following property of $F$-polynomials.

Proposition 8.8** (Cf. [FZ07, Proposition 5.3, Conjecture 5.5]).**

(1). Each $F$-polynomial $F_{i;t}(\mathbf{u},\mathbf{v})$ has a unique monomial of maximal degree in $\mathbf{u}$. Furthermore, the coefficient of this monomial is 1.

(2). An $F$-polynomial $F_{i;t}(\mathbf{u},\mathbf{v})$ is 1 if $F_{i;t}(\mathbf{u},\mathbf{0})$ is 1, or equivalently, its right companion ${}^{R}F_{i;t}(\mathbf{u})$ is $1$.

Proof.

(1). This is proved in the same way as Theorem 8.2.

(2). This is an immediate consequence of (1) and (7.29). ∎

Here is an analogue of Theorem 4.8 with a proof which does not rely on Conjecture 8.6.

Theorem 8.9** (Detropicalization).**

The following statements hold for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, where $P_{\sigma}$ is the permutation matrix in (3.18):

[TABLE]

Here, if $t_{2}$ is connected to the initial point $t_{0}$ in $\mathbb{T}_{n}$ as

[TABLE]

then $t_{3}\in\mathbb{T}_{n}$ is the one such that

[TABLE]

Also, $v_{j,s}\leftarrow v_{\sigma(j),s;t_{3}}$ means the replacement of $v_{j,s}$ by $v_{\sigma(j),s;t_{3}}$ for all $j$ in a polynomial in $\mathbf{v}$.

Proof.

Proof of (8.8). Let ${}^{R}\mathbf{\Gamma}$ be the right companion of $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})$ in Definition 7.9. Then, by (7.27), we have ${}^{R}G_{t_{1}}=G_{t_{1}}=P_{\sigma}$. Then, by Theorem 4.8, we have ${}^{R}F_{i;t_{1}}(\mathbf{u})=1$. Therefore, by Proposition 8.8 (2), we obtain ${F}_{i;t_{1}}(\mathbf{u},\mathbf{v})=1$.

Proof of (8.9). The proof is similar to the one for Theorem 4.8. Let $t_{3}$ be the one in (8.23). Repeating the same argument as before, we have

[TABLE]

Thus, by (8.8), we have $F_{i,t_{3}}(\mathbf{u},\mathbf{v})=1$ for any $i=1,\dots,n$. Then, by comparing the mutations of $F$-polynomials (7.12) for the sequences (8.16) and (8.23), one can prove the claim (8.9) by induction on mutations. See (8.56) and (8.57) for the formulas in the first step. ∎

8.3. Strong compatibility

The second result (8.9) in Theorem 8.9 is not simply a permutation of $F$-polynomials. unlike the corresponding one (4.11) in the CA case, Let us study its implication closely. All results in this subsection do not rely on Conjecture 8.6.

Let $\sigma\mathbf{x}$ and $\sigma\mathbf{y}$ be as before in (3.10) and (3.11).

Let us start with an “obvious” result of Corollary 8.5 and Theorem 8.9.

Proposition 8.10**.**

Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$, and let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\mathbf{v}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with any initial point $t_{0}$. Let $\mathbf{x}$, $\mathbf{y}$, $\mathbf{z}$ be the initial variables at $t_{0}$. Then, for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, the following hold.

(1). If $G_{t_{1}}=P_{\sigma}$, then we have

[TABLE]

(2). Let $x_{i;t}(\mathbf{x},\mathbf{y},\mathbf{z})$ and $y_{i;t}(\mathbf{x},\mathbf{y},\mathbf{z})$ be $x_{i;t}$ and $y_{i;t}$ as functions of $\mathbf{x}$, $\mathbf{y}$, $\mathbf{z}$ through the separation formulas (7.20) and (7.21), respectively. If $G_{t_{1}}=\sigma G_{t_{2}}$, then we have

[TABLE]

where $t_{3}$ is the one in (8.23).

Proof.

They are straightforward consequences of the separation formulas in Theorem 7.8 with Corollary 8.5 and Theorem 8.9. ∎

What is less obvious is that Proposition 8.10 further implies the following result.

Proposition 8.11**.**

In the same situation as Proposition 8.10, if $G_{t_{1}}=\sigma G_{t_{2}}$, then we have

[TABLE]

Proof.

Let us apply (8.27) for a cluster pattern $\check{\mathbf{\Sigma}}_{R}[{t}_{2}]=\{\check{\Sigma}_{t}=(\check{\mathbf{x}}_{t},\check{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with $Y$-principal coefficients at $t_{2}$. Since all $z$-variables are trivial, we obtain $\check{\mathbf{y}}_{t_{1}}=\sigma\check{\mathbf{y}}_{t_{2}}$ by (8.27). Now, let $\mathbf{\Gamma}^{\prime}(\mathbf{B}_{R},t_{2})=\{\Gamma^{\prime}_{t}=(\mathbf{F}^{\prime}_{t}(\mathbf{u},\mathbf{v}),G^{\prime}_{t},C^{\prime}_{t},\allowbreak\mathbf{v}^{\prime}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of the above $\mathbf{B}_{R}$ with the initial point $t_{2}$. Then, by Proposition 7.7, we have $C^{\prime}_{t_{1}}=P_{\sigma}$. Therefore, $G^{\prime}_{t_{1}}=P_{\sigma}$ by (8.4). Then, by applying (8.25) for $\Sigma_{R}$, we obtain $\mathbf{x}_{t_{1}}=\sigma\mathbf{x}_{t_{2}}$ and $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$. ∎

The apparent discrepancy between Propositions 8.10 and 8.11 leads to a remarkable condition for $\sigma$ therein, which is stronger than the compatibility with $R$.

Definition 8.12**.**

For a permutation $\sigma\in S_{n}$ and a diagonal matrix $R=\mathrm{diag}(r_{1},\dots,r_{n})$ with positive integer diagonal entries, we say that $\sigma$ is strongly compatible with $R$ if

[TABLE]

Proposition 8.13**.**

Let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\mathbf{v}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with any initial point $t_{0}$. Suppose that $\sigma\in S_{n}$ and $t_{1},t_{2}\in\mathbb{T}_{n}$ satisfy the condition

[TABLE]

Let $t_{3}\in\mathbb{T}_{n}$ be the one in Theorem 8.9. Then, the following properties hold.

(1). The following equality holds for any $i$ with $r_{i}\geq 2$ and $s=1,\dots,r_{i}-1$:

[TABLE]

(2). $\sigma$ is strongly compatible with $R$.

(3). The following equality holds for any $i$:

[TABLE]

(4). Suppose that $t_{0}$ and $t_{3}$ are connected in $\mathbb{T}_{n}$ as

[TABLE]

Then, for each $i$ with $r_{i}\geq 3$, the number of the appearance of $i$ in the sequence $(\ell_{1},\dots,\ell_{m})$ is even.

(5). Suppose that $t_{1}$ and $t_{2}$ are connected in $\mathbb{T}_{n}$ as

[TABLE]

Then, for each $i$ with $r_{i}\geq 3$, the number of the appearance of $i$ in the sequence $(\ell^{\prime}_{1},\dots,\ell^{\prime}_{s})$ is even.

(6). The following equality holds:

[TABLE]

Proof.

(1). Choose any $k=1,\dots,n$, and define $t^{\prime}_{0},t^{\prime}_{3}\in\mathbb{T}_{n}$ as

[TABLE]

Recall that $G_{t_{0}}=C_{t_{0}}=I$, $F_{i;t_{0}}(\mathbf{u},\mathbf{v})=1$, and, as we see in the proof of (8.9), $G_{t_{3}}=C_{t_{3}}=P_{\sigma}$, and $F_{i;t_{3}}(\mathbf{u},\mathbf{v})=1$. Therefore, by the mutation (7.12), we obtain

[TABLE]

(This confirms (8.9) in the simplest case.) We also note that, by (3.17), we have

[TABLE]

Let $\tilde{\mathbf{\Sigma}}_{R}[{t}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with principal coefficients at $t_{0}$. Then, by the separation formula (7.20), Proposition 7.5, and (8.58), we have

[TABLE]

Meanwhile, by (8.58) and Theorem 9.1, we have $\tilde{x}_{k;t^{\prime}_{0}}=\tilde{x}_{\sigma(k);t^{\prime}_{3}}.$ Therefore, we obtain the identity

[TABLE]

Setting $\tilde{x}_{i}=1$ ($i=1,\dots,n$) therein, we have the identity

[TABLE]

Since the principal coefficients $\tilde{\mathbf{y}}$ and $\tilde{\mathbf{z}}$ at $t_{0}$ are independent variables, this is equivalent to the identity

[TABLE]

Then, comparing (8.56) and (8.57), we conclude that (8.31) holds.

(2). Under the mutation rule (7.15), we have $v_{i,s;t}=v_{i,s}$ or $v_{i,r_{i}-s}$ for any $t$. Therefore, (8.31) occurs only when $\sigma$ is strongly compatible with $R$.

(3). This is a consequence of (8.9) and (8.31).

(4). Let $i$ be the one with $r_{i}\geq 3$. During the sequence of mutations (8.39), for fixed $i$ and $s$, the variables $v_{i,s;t}$ alternate between $v_{i,s}$ and $v_{i;r_{i}-s}$ whenever $\mu_{i}$ occurs. Therefore, (8.31) implies that $\mu_{i}$ occurs in even times.

(5). As a path between $t_{1}$ and $t_{2}$ in $\mathbb{T}_{n}$, it is enough to consider the concatenation of (8.16), (8.23), and (8.39). For $i$ with $r_{i}\geq 2$, the numbers of the appearance of $\mu_{i}$ in (8.16) and (8.23) coincide thanks to the strong compatibility $\sigma(i)=i$ proved in (2). Then, combining this fact with (4), we obtain the claim.

(6). We show $v_{i,s;t_{1}}=v_{i,s;t_{2}}$ for $i$ with $r_{i}\geq 2$. For $i$ with $r_{i}\geq 3$, this is a corollary of (5). For $i$ with $r_{i}=2$, this is always true. ∎

Let us formally define the action of a permutation $\sigma\in S_{n}$ on $z$-variables for a seed $(\mathbf{x},\mathbf{y},\mathbf{z},B)$ of degree $R$ in the same way as other $x$- and $y$-variables as

[TABLE]

If $\sigma$ is strongly compatible with $R$, then $\sigma\mathbf{z}=\mathbf{z}$. Namely, the action is trivial.

We have the following supplementary result to Proposition 8.11.

Proposition 8.14**.**

In the same situation as Proposition 8.10, if $G_{t_{1}}=\sigma G_{t_{2}}$, then we have

[TABLE]

Proof.

Recall that $v$-variables and $z$-variables mutate by the same rule (7.6) and (7.15). Also, for each $t$, $\mathbf{v}_{t}$ is a tuple of independent formal variables. Therefore, by (8.47), we have $\mathbf{z}_{t_{1}}=\mathbf{z}_{t_{2}}$. Furthermore, since $\sigma$ is strongly compatible with $R$ by Proposition 8.13 (2), we have $\mathbf{z}_{t_{2}}=\sigma\mathbf{z}_{t_{2}}$. ∎

9. Synchronicity phenomenon in GCA

9.1. $xy/GC$-synchronicity

We are ready to give the $xy/GC$ synchronicity for GCA, which are analogous to the one in Section 5.2.

Theorem 9.1** ($xy/GC$ synchronicity).**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$, and let $\mathbf{\Gamma}(\mathbf{B}_{R},t_{0})=\{\Gamma_{t}=(\mathbf{F}_{t}(\mathbf{u},\mathbf{v}),G_{t},C_{t},\mathbf{v}_{t},\allowbreak B_{t})\mid t\in\mathbb{T}_{n}\}$ be the $FGC$-pattern of $\mathbf{B}_{R}$ with any initial point $t_{0}$. Then, for $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, the following three conditions are equivalent:

(a). $G_{t_{1}}=\sigma G_{t_{2}}$.

(b). $C_{t_{1}}=\sigma C_{t_{2}}$.

(c). $\mathbf{x}_{t_{1}}=\sigma\mathbf{x}_{t_{2}}$.

Moreover, one of Conditions (a)–(c) implies the following conditions:

(d). $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$.

(e). $\mathbf{z}_{t_{1}}=\mathbf{z}_{t_{2}}=\sigma\mathbf{z}_{t_{2}}$.

Proof.

((a) $\Longleftrightarrow$ (b)). This was already stated in (8.5).

((a) $\Longrightarrow$ (c), (d), (e)). This was proved in Propositions 8.11 and 8.14.

((c) $\Longrightarrow$ (b)). This is proved in the same manner as Theorem 5.2 by using a cluster pattern of degree $R$ without coefficients, where we replace Lemma 4.7 with Lemma 8.7. (This is only where we need Conjecture 8.6.)

∎

Let us introduce a condition for the implication (d) $\Longrightarrow$ (a), (b).

Definition 9.2**.**

Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$, and $\check{\mathbf{\Sigma}}_{R}[t^{\prime}_{0}]=\{\check{\Sigma}_{t}=(\check{\mathbf{x}}_{t},\check{\mathbf{y}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with $Y$-principal coefficients at $t^{\prime}_{0}$, such that they share a common $B$-pattern $\mathbf{B}_{R}$ of degree $R$. Let $\langle\mathbf{y}_{t^{\prime}_{0}},\mathbf{z}_{t^{\prime}_{0}}\rangle$ be the subsemifield in $\mathbb{P}$ generated by $\mathbf{y}_{t^{\prime}_{0}}$ and $\mathbf{z}_{t^{\prime}_{0}}$. Then, we say that $\mathbf{\Sigma}_{R}$* covers $\check{\mathbf{\Sigma}}_{R}[t^{\prime}_{0}]$* if there is a semifield homomorphism such that

[TABLE]

Now we present the second half of the statement of the $xy/GC$ synchronicity, which does not rely on Conjecture 8.6.

Theorem 9.3** ($xy/GC$ synchronicity).**

Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},\allowbreak B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$ which covers a cluster pattern of degree $R$ with $Y$-principal coefficients at some $t^{\prime}_{0}$. Then, Condition (d) implies Conditions (a)–(c) in Theorem 9.1.

Proof.

The proof is the same as Theorem 5.5. ∎

Corollary 9.4**.**

Let $D$ be a common skew-symmetrizer of $\mathbf{B}$.

(1). Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$. If $\mathbf{x}_{t_{1}}=\sigma\mathbf{x}_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$ and strongly compatible with $R$.

(2). Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},\allowbreak B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of $R$ with coefficients in any semifield $\mathbb{P}$ which covers a cluster pattern of $R$ with $Y$-principal coefficients at some $t^{\prime}_{0}$. If $\mathbf{y}_{t_{1}}=\sigma\mathbf{y}_{t_{2}}$ occurs for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is compatible with $D$ and strongly compatible with $R$.

Proof.

This follows from Propositions 8.4, 8.13 and Theorems 9.1, 9.3. ∎

Remark 9.5**.**

In parallel to Definition 9.2, for ${{\mathbf{\Sigma}}}_{R}$ therein and a cluster pattern $\tilde{\mathbf{\Sigma}}_{R}[t^{\prime}_{0}]=\{\tilde{\Sigma}_{t}=(\tilde{\mathbf{x}}_{t},\tilde{\mathbf{y}}_{t},\tilde{\mathbf{z}}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ of degree $R$ with principal coefficients at $t^{\prime}_{0}$, we say that $\mathbf{\Sigma}_{R}$* covers $\tilde{\mathbf{\Sigma}}_{R}[t^{\prime}_{0}]$* if there is a semifield homomorphism such that

[TABLE]

In this case, one can prove Theorem 9.1 without assuming Conjecture 8.6 by using a cluster pattern of degree $R$ with principal coefficients and the reduction (7.29).

9.2. More synchronicity results

For a seed of $\Sigma=(\mathbf{x},\mathbf{y},\mathbf{z},B)$ of degree $R$ and a permutation $\sigma\in S_{n}$, we define the action of $\sigma$ on $\Sigma$ by

[TABLE]

Let $\mathbf{\Sigma}_{R}=\{\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$. Suppose that $t_{1},t_{2}\in\mathbb{T}_{n}$ are connected in $\mathbb{T}_{n}$ as

[TABLE]

so that

[TABLE]

After observing Theorems 9.1 and 9.3, the following definition is natural.

Definition 9.6** ($\sigma$-periodicity).**

We call a sequence of mutations (9.11) of seeds of degree $R$ a $\sigma$-period if

[TABLE]

Then, all results in Section 5.3 are naturally generalized as we present below. The proofs are the same as before, so that we omit them. Note that the assumption of Conjecture 8.6 is necessary only for the statements involving $x$-variables. See also Remark 9.5 for the condition when this assumption can be avoided.

Theorem 9.7**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$. Then, the periodicity of $x$-variables $\mathbf{x}_{t}$ depends only on the $B$-pattern $\mathbf{B}_{R}$ of degree $R$ therein.

Theorem 9.8**.**

Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$ such that $\mathbf{\Sigma}_{R}$ covers a cluster pattern of degree $R$ with $Y$-principal coefficients at some $t^{\prime}_{0}$. Then, the periodicity of $y$-variables $\mathbf{y}_{t}$ depends only on the $B$-pattern $\mathbf{B}_{R}$ of degree $R$ therein.

Theorem 9.9**.**

Assume that Conjecture 8.6 is true. The $x$-variables in Theorem 9.7 and the $y$-variables in Theorem 9.8 with a common $B$-pattern $\mathbf{B}_{R}$ of degree $R$ share the same periodicity.

Theorem 9.10**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$. Then, the $x$-variables $\mathbf{x}_{t}$ and the seeds $\Sigma_{t}$ share the same periodicity. Furthermore, it depends only on the $B$-pattern $\mathbf{B}_{R}$ of degree $R$ therein.

Theorem 9.11**.**

Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$ such that $\mathbf{\Sigma}_{R}$ covers a cluster pattern of degree $R$ with $Y$-principal coefficients at some $t^{\prime}_{0}$. Then, the $y$-variables $\mathbf{y}_{t}$ and the $Y$-seeds $(\mathbf{y}_{t},B_{t})$ share the same periodicity. Furthermore, it depends only on the $B$-pattern $\mathbf{B}_{R}$ of degree $R$ therein.

Theorem 9.12**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$ such that $\mathbf{\Sigma}$ covers a cluster pattern of degree $R$ with $Y$-principal coefficients at some $t^{\prime}_{0}$. Then, the seeds $\Sigma_{t}$ and the $Y$-seeds $(\mathbf{y}_{t},B_{t})$ share the same periodicity. Furthermore, it depends only on the $B$-pattern $\mathbf{B}_{R}$ of degree $R$ therein.

9.3. Synchronicity with companion patterns

Another main result in Part II is the following one. See also Theorem 5.25.

Theorem 9.13**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ be a cluster pattern of degree $R$ with coefficients in any semifield $\mathbb{P}$. Let ${}^{L}{{\mathbf{\Sigma}}}=\{{}^{L}{\Sigma}_{t}=({}^{L}{\mathbf{x}}_{t},{}^{L}{\mathbf{y}}_{t},RB_{t})\mid t\in\mathbb{T}_{n}\}$ and ${}^{R}{{\mathbf{\Sigma}}}=\{{}^{R}{\Sigma}_{t}=({}^{R}{\mathbf{x}}_{t},{}^{R}{\mathbf{y}}_{t},B_{t}R)\mid t\in\mathbb{T}_{n}\}$ be cluster patterns with coefficients in any semifields ${}^{L}\mathbb{P}$ and ${}^{R}\mathbb{P}$, respectively. Then, the periodicity of seeds ${\Sigma}_{t}$ coincides with the common periodicity of seeds ${}^{L}{\Sigma}_{t}$ and ${}^{R}{\Sigma}_{t}$.

Proof.

This is a corollary of Theorem 7.10 together with Theorems 5.2, 5.11, 9.1, and 9.10. ∎

In other words, the seeds of a GCA and the seeds of its companion CA’s share the same exchange graph.

As a corollary, we have a strong version of Theorem 5.25 (1) when $B$ is an integer matrix. (We do not now if the statement still holds for a non-integer matrix $B$.)

Corollary 9.14**.**

Let $R\mathbf{B}$ and $\mathbf{B}R$ be a conjugate pair such that $B$ is an integer (skew-symmetrizable) matrix. Let ${{\mathbf{\Sigma}}}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},RB_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},B_{t}R)\mid t\in\mathbb{T}_{n}\}$ be cluster patterns with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. If $\Sigma_{t_{1}}=\sigma\Sigma_{t_{2}}$ or $\Sigma^{\prime}_{t_{1}}=\sigma\Sigma^{\prime}_{t_{2}}$ for some $t_{1},t_{2}\in\mathbb{T}_{n}$ and a permutation $\sigma\in S_{n}$, then $\sigma$ is strongly compatible with $R$.

9.4. Synchronicity under dualities

For completeness, we also give analogous results to the ones in Section 5.5.

Theorem 9.15**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}_{R}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},B^{T}_{t})\mid t\in\mathbb{T}_{n}\}$ be cluster patterns of degree $R$ with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. Then, the seeds in ${\mathbf{\Sigma}}_{R}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}_{R}$ share the same periodicity.

Proof.

$RB_{t}$ and $B^{T}_{t}R$ are related by the transposition duality in Section 5.5. Then, the claim follows from Theorems 9.13 and 5.20. ∎

Theorem 9.16**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}_{R}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},-B_{t})\mid t\in\mathbb{T}_{n}\}$ be cluster patterns of degree $R$ with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. Then, the seeds in ${\mathbf{\Sigma}}_{R}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}_{R}$ share the same periodicity.

Proof.

$RB_{t}$ and $-RB$ are related by the chiral duality in Section 5.5. Then, the claim follows from Theorems 9.13 and 5.21. ∎

Theorem 9.17**.**

Assume that Conjecture 8.6 is true. Let ${{\mathbf{\Sigma}}}_{R}=\{{\Sigma}_{t}=({\mathbf{x}}_{t},{\mathbf{y}}_{t},\mathbf{z}_{t},B_{t})\mid t\in\mathbb{T}_{n}\}$ and ${{\mathbf{\Sigma}}}^{\prime}_{R}=\{{\Sigma}^{\prime}_{t}=({\mathbf{x}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},{\mathbf{y}}^{\prime}_{t},-B^{T}_{t})\mid t\in\mathbb{T}_{n}\}$ be cluster patterns of degree $R$ with coefficients in any semifields $\mathbb{P}$ and $\mathbb{P}^{\prime}$, respectively. Then, the seeds in ${\mathbf{\Sigma}}_{R}$ and the seeds in ${\mathbf{\Sigma}}^{\prime}_{R}$ share the same periodicity.

Proof.

$RB_{t}$ and $-B^{T}_{t}R$ are related by the Langlands duality in Section 5.5. Then, the claim follows from Theorems 9.13 and 5.22. ∎