On the properties of acyclic sign-skew-symmetric cluster algebras

Cluster algebras, introduced by Fomin and Zelevinsky in [FZ02] for providing a combinatorial framework to study total positivity and canonical bases associated by Lusztig to algebraic groups, are certain commutative algebras generated by certain combinatorially defined generators (called cluster variables) which are grouped into overlapping clusters. Since cluster algebras were invented, many links between cluster algebras and other subjects have been discovered. For example, cluster algebras have close relations with Riemann surfaces and triangulations, Zamolodchikov’s periodicity properties concerning $Y$-systems, representation theory, Donaldson-Thomas invariant theory and scattering diagrams.

The initial data to construct a cluster algebra is a totally sign-skew-symmetric matrix in the first paper in the series of foundational papers of cluster theory. Since then, most authors prefer to study skew-symmetrizable cluster algebras and many important properties and conjectures were proved in the skew-symmetrizable case. Even though there are still many unkonwns for totally sign-skew-symmetric cluster algebras, we believe that their properties behave like the skew-symmetrizable case.

The most important property of cluster algebras is the Laurent phenomenon. Fomin and Zelevinsky conjectured the cluster variables written as Laurent polynomials of a cluster always have the positive coefficients, which is called the positivity conjecture. This conjecture was proved by Lee and Schiffler for the symmetric case [LS15], by Gross, Hacking, Keel and Kontsevich for the skew-symmetrizable case [GHKK18], Huang and Li for the acyclic sign-skew-symmetric cluster algebras. Recently, Li and Pan proved it for all totally sign-skew-symmetric case [LP].

In the fourth paper of the foundational papers [FZ07], Fomin and Zelevinsky introduced a lot of combinatorial objects including $c$-vectors and $g$-vectors (also $C$-matrices and $G$-matrices) to study cluster algebras. The well-known sign coherence conjecture was also proposed in [FZ07]. The dualities of $G$-matrix and $C$-matrix were systematically studied by Nakanishi and Zelevinsky in [NZ12]. In their paper, they shall assume the sign coherence conjecture holds for $c$-vectors for all skew-symmetrizable cluster algebras. This was confirmed by Derksen, Weyman, and Zelevinsky in the skew-symmetric case [DWZ10] and by Gross, Hacking, Keel, and Kontsevich in the skew-symmetrizable case [GHKK18]. Thus the dualities obtained by Nakanishi and Zelevinsky in [NZ12] are true for skew-symmetrizable cluster algebras. Let $\mathcal{A}$ be a skew-symmetrizable cluster algebra of rank $n$. Roughly speaking, they proved the following results for skew-symmetrizable cluster algebras:

These claims were proved by Nakanishi and Zelevinsky [NZ12] (see also [Nak]) by assuming $c$-vectors are sign coherent. Even though the sign coherence conjecture of $c$-vectors was proved for the acyclic case by Huang and Li [HL18], and in full generality by Li and Pan [LP], the proofs in [NZ12] largely depends on the skew-symmetrizable property.

Reading studied the property of $G$-fan $\mathcal{F}_{G}$ consisting of all $G$-cones (non-negative spans of the column vectors of $G$-matrices) and their faces in [Rea14]. For the skew-symmetrizable cluster algebras, he also defined the mutation fan, and he proved the following result.

The bijection of $g$-vectors and cluster variables is a corollary of the positivity conjecture and the above claims as proved in [Nak].

Fomin and Zelevinsky also made a series of conjectures on the properties of exchange graphs of $\mathcal{A}$ as follows.

These conjectures were proved for skew-symmetrizable cluster algebras by Gekhtman, Shapiro, and Vainshtein [GSV08], Cao and Li [CL18], and Cao, Huang, and Li [CHL18].

In this paper, we show that under an condition (the Assumption in Section 4.2), the claims (1.1—1.9) hold for totally sign-skew-symmetric cluster algebras. In particular, they hold for acyclic sign-skew-symmetric cluster algebras.

In the next section, we recall basic theory of cluster algebras. In Section 3, we describe behaviors about cluster algebras under folding theory. In Section 4, we prove the properties of $G$-matrices for totally sign-skew-symmetric cluster algebras that satisfy a certain condition (which we conjecture it is always true). In the final section, we study the properties of exchange graphs.

Let $I$ be a countable set. We say an integer matrix $B\in Mat_{I\times I}(\mathbb{Z})$ is locally finite if for each $i\in I$, there are only finitely many nonzero entries $b_{ij}$ and $b_{ji}$, and say $B$ is sign-skew-symmetric if $b_{ij}b_{ji}\leq 0$ and $b_{ij}b_{ji}=0\Leftrightarrow b_{ij}=b_{ji}=0$ for all $i,j\in I$. A skew-symmetrizable matrix $B$ admits a diagonal matrix $D=\text{diag}(d_{i},i\in I)$ with positive integer diagonal entries such that $d_{i}b_{ij}=-d_{j}b_{ji}$, which is clearly sign-skew-symmetric. A quiver is a directed graph without loops or $2$-cycles, and since a locally finite skew-symmetric matrix naturally corresponds to a locally finite quiver, we do not distinguish them in the following sections.

A semifield $\mathbb{P}$ is a set equipped with a structure of abelian multiplicative group and a structure of abelian additive semigroup (the addition is usually denoted by $\oplus$) such that the addition $\oplus$ is distributive with respect to the multiplication in $\mathbb{P}$. The tropical semifield $\mathbb{P}=\mathrm{{trop}}(z_{j},j\in J)$ with an index set $J$ is a multiplicative group generated freely by the elements $z_{j},j\in J$, whose addition $\oplus$ is given by

Let $\mathbb{QP}$ be the quotient field of the integral domain $\mathbb{ZP}$.

For $k\in I$, define another triplet $(\mathbf{x^{\prime}},\mathbf{y^{\prime}},B^{\prime})=\mu_{k}(\mathbf{x},\mathbf{y},B)$ which is called the mutation of $(\mathbf{x},\mathbf{y},B)$ at $k$ and obtained by the following rules:

1. -

   $\mathbf{x^{\prime}}=(x^{\prime}_{i},i\in I)$ is given by

   $$x^{\prime}_{i}=\begin{cases}x_{i},&\text{$i\neq k$},\\ \frac{y_{k}\prod x_{i}^{[b_{ik}]_{+}}+\prod x_{i}^{[-b_{ik}]_{+}}}{(y_{k}\oplus 1)x_{k}},&\text{$i=k$.}\end{cases}$$

2. -

   $\mathbf{y^{\prime}}=(y^{\prime}_{i},i\in I)$ is given by

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

3. -

   $B^{\prime}=(b^{\prime}_{ij})$ is given by

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

where $[a]_{+}=\mathrm{max}\{a,0\}$.

Say two labeled seeds $\Sigma$ and $\Sigma^{\prime}$ define the same unlabeled seed if there is a bijection $\sigma$ of $I$ such that $x^{\prime}_{i}=x_{\sigma(i)}$, $y^{\prime}_{i}=y_{\sigma(i)}$, $b^{\prime}_{ij}=b_{\sigma(i)\sigma(j)}$ for all $i,j\in I$.

If every matrix obtained from a locally finite sign-skew-symmetric matrix by an arbitrary finite sequence of mutations is also sign-skew-symmetric, we call it a totally sign-skew-symmetric matrix. Clearly, skew-symmetrizable matrices are totally mutable. For a sign-skew-symmetric matrix $B=(b_{ij})_{i,j\in I}$, we may define a directed graph whose vertices are indexed by $I$, and there is an arrow from $i$ to $j$ if $b_{ij}>0$. We say $B$ is acyclic if the corresponding directed graph has no oriented cycles. Huang and Li also proved acyclic sign-skew-symmetric matrices are totally mutable [HL18].

Let $\mathbb{T}$ be an $|I|$-regular tree and its valencies emitting from each common vertex are labeled by $I$. A cluster pattern is an $|I|$-regular tree $\mathbb{T}$ such that for each vertex $t\in\mathbb{T}$, there is a seed $\Sigma_{t}=(\mathbf{x}_{t},\mathbf{y}_{t},B_{t})$ and for each edge labeled by $k\in I$, two seeds in the endpoints are obtained from each other by seed mutation at $k$. Note that a cluster pattern is determined by a totally sign-skew-symmetric matrix. The corresponding exchange graph is the graph whose vertices are unlabeled seeds and two unlabeled seeds are connected by an edge if they are related by a single mutation. The cluster algebra $\mathcal{A}=\mathcal{A}(\mathbf{x},\mathbf{y},B)$ associated to a totally sign-skew-symmetric matrix $B$ is the $\mathbb{ZP}$-subalgebra of $\mathcal{F}$ generated by all cluster variables on $\mathbb{T}$ corresponding to the matrix $B$. The cluster algebra $\mathcal{A}=\mathcal{A}(\mathbf{x},\mathbf{y},B)$ is said to have principal coefficients, if the semifield is the tropical semifield given by $\mathbb{P}=\mathrm{trop}(y_{i},i\in I)$.

Fomin and Zelevinsky proved the following Laurent property for cluster algebras, and they also conjectured cluster variables have non-negative Laurent expressions.

In the following discussion, we shall fix a seed, which we call the initial seed and is usually placed at $t_{0}\in\mathbb{T}$, to obtain other datum from it.

Let $\mathcal{A}$ be a cluster algebra with principal coefficients and initial seed $\Sigma=\Sigma_{t_{0}}$. Then by Theorem 2.2, we have that $x_{j,t}\in\mathbb{Z}[y_{i},x_{i}^{\pm 1},i\in I]$. The polynomial $F_{j,t}:=x_{j,t}|_{x_{i}=1,i\in I}$ is called the $F$-polynomial (with the initial seed at $t_{0}$). Since $y_{j,t}\in\text{trop}(y_{i},i\in I)$, we have that $y_{j,t}$ is of the form $y_{j,t}=\prod_{i\in I}y_{i}^{c_{ij,t}}$ for some integers $c_{ij,t},i\in I$. The column vector $c_{j,t}=(c_{ij,t},i\in I)^{T}$ is called the $c$-vectors and the matrix $C^{t_{0}}_{t}=(c_{j,t})_{j\in I}$ is called the $C$-matrix at $t$. The column vector $g_{j,t}$, the degree vector of $x_{j,t}$ with respect to the grading given in Theorem 2.2, is called the $g$-vector, and the matrix $G^{t_{0}}_{t}=(g_{j,t})_{j\in I}$ is called the $G$-matrix at $t$. Note that the definitions of $F$-polynomials, $C$-matrices and $G$-matrices depend on the choice of the initial seed. Also in [FZ07], Fomin and Zelevinsky obtained the mutation rules for $F$-polynomials, $C$-matrices and $G$-matrices.

For each edge $t\frac{k}{\quad\quad}t^{\prime}$ on $\mathbb{T}$, we have that

1. -

   $F_{i,t^{\prime}}=\begin{cases}F_{i,t},&\text{if $i\neq k$;}\\ (F_{k,t})^{-1}(\prod_{j\in I}y_{j}^{c_{jk,t}}F_{j,t}^{[b_{jk,t}]_{+}}+\prod_{j\in I}y_{j}^{-c_{jk,t}}F_{j,t}^{[-b_{jk,t}]_{+}}),&\text{otherwise.}\end{cases}$

2. -

   $c_{ij,t^{\prime}}=\begin{cases}-c_{ik,t},&\text{if $j=k$;}\\ c_{ij,t}+\mathrm{sign}(c_{ik,t})[c_{ik,t}b_{jk,t}]_{+},&\text{otherwise.}\end{cases}$

3. -

   $g_{ij,t^{\prime}}=\begin{cases}g_{ij,t},&\text{if $j\neq k$;}\\ -g_{ik,t}+\Sigma_{s\in I}g_{is,t}[-b_{sk,t}]_{+}-\Sigma_{s\in I}b_{is,t}[-c_{sk,t}]_{+},&\text{otherwise.}\end{cases}$

A nonzero vector is said to be sign-coherent if it has either all non-negative coordinates or all non-positive coordinates. Fomin and Zelevinsky conjectured every $c$-vectors is sign coherent and each row vector of every $G$-matrix is sign-coherent. The sign coherence conjecture were solved for skew-symmetrizable cluster algebras in [GHKK18] and for acyclic sign-skew-symmetric cluster algebras in [HL18, CHL22].

Recall that a quiver is a directed graph (may be infinite) which has no loops or $2$-cycles. It is well-known that a quiver is naturally associated to a skew-symmetric matrix.

The following Lemma follows easily from the definition.

Assume that $Q$ is $\Gamma$-admissible, denote $[i]$ the $\Gamma$-orbit for each $i\in Q_{0}$, $\bar{I}$ the set of $\Gamma$-orbits, and $\mu_{[i]}:=\prod_{j\in[i]}\mu_{j}$ the orbit mutation. The matrix $B^{\Gamma}=(b_{[i][j]})_{[i],[j]\in\bar{I}}$ is defined by the following rule:

The following Lemma follows easily from the definition.

Notice that for a locally finite quiver $Q$, the cluster algebra $\mathcal{A}(\mathbf{x},Q)$, $F$-polynomials, $c$-vectors, and $g$-vectors could be defined naturally. If $Q$ is globally foldable with respect to $\Gamma$ and for any mutable $i\in Q_{0}$, then define $\mu_{[i]}(x_{j})$ by the rules: $\mu_{[i]}(x_{j})=\mu_{j}(x_{j})$ if $j\in[i]$, and $\mu_{[i]}(x_{j})=x_{j}$ if $j\notin[i]$.

Let $Q$ be a locally finite quiver without frozen vertices. The corresponding framed quiver $\tilde{Q}$ is obtained from $Q$ by adding frozen vertices $Q^{\prime}_{0}:=\{i^{\prime},i\in Q_{0}\}$ and arrows $Q^{\prime}_{1}:=\{i^{\prime}\rightarrow i,i\in Q_{0}\}$. If $Q$ is globally foldable with respect to $\Gamma$, by Lemma 3.5, $\tilde{Q}$ is globally foldable with respect to $\Gamma$ with $\Gamma$ fixing frozen vertices.

Combining Theorem 3.9 and Theorem 3.10, we have the following results.

The tropical dualities for $G$-matrices and $C$-matrices were studied by Nakanishi and Zelevinky for the skew-symmetrizable cluster algebras [NZ12]. Notice that Nakanishi and Zelevinsky’s proof depends on the sign coherence of $c$-vectors. In this section, we consider the dual phenomenon for the totally sign-skew-symmetric case. In this section and the next section, we always assume that $|I|=n<\infty$.

As we mentioned in the Introduction, Fomin and Zelevinsky proposed a series of conjectures on $G$-matrices and $C$-matrices in their paper [FZ07]. Let us recall them.

In the skew-symmetrizable case, these conjectures are proved by a series of papers [DWZ10, GHKK18]. In the totally sign-skew-symmetric case, the Conjectures (i-iv) were proved by Huang and Li [HL18] and by Cao, Huang and Li [CHL22] for the acyclic sign-skew-symmetric case using the folding theory and categorification, and the Conjecture 4.1 (i) was proved by Li and Pan in full generality. In the following sections, we show that under an assumption Conjecture (ii-iv) are true. In particular, for the acyclic sign-skew-symmetric matrix, we obtain these results. Moreover, we show the duality between $G$-matrices and $C$-matrices, which has been only proved for the skew-symmetrizable case by Nakanishi and Zelevinsky [NZ12]. The essential idea of the proofs is developed by Nakanishi and Zelevinsky.