sign-coherence of c-vectors and maximal green sequences for acyclic sign-skew-symmetric matrices

The problem posed by A. Berenstein, S. Fomin and A. Zelevinsky in [2] on whether any acyclic sign-skew-symmetric integer matrix is totally sign-skew-symmetric or not, was a great motivation for many mathematicians to study such matrices. M. Huang and F. Li gave an affirmative answer to this problem and proved in [10] that acyclic sign-skew-symmetric matrices are totally mutable. The authors in [10] also proved that every acyclic sign-skew-symmetric matrix can be covered by an (infinite) skew-symmetric matrix which is represented by an (infinite) cluster quiver and this covering can perform arbitrary steps of orbit-mutations. This (infinite) quiver is called an unfolding of this acyclic sign-skew-symmetric matrix. The existence of such an unfolding quiver for every acyclic sign-skew-symmetric matrix allows us to tackle problems related to an acyclic sign-skew-symmetric cluster algebra by promoting these problems to an (infinite) skew-symmetric cluster algebra. In this note we try to find an unfolding for the $c-$vectores of an acyclic sign-skew-symmetric matrix and prove that it always exists (see paragraph 3 of Remark 2.7). In other words, we prove that every extended acyclic sign-skew-symmetric matrix $\widetilde{B}=\begin{pmatrix}B\\ I_{n}\end{pmatrix}$ can be covered by an (infinite) extended skew-symmetric matrix $\widetilde{B^{\S}}=\begin{pmatrix}B^{\S}\\ I_{\infty}\end{pmatrix}$. The construction of this covering keeps the principal part as it was constructed in [10] which makes the ability of performing arbitrary steps of orbit-mutations remain valid. Using the unfolding method, we prove that the sign-coherence property holds for acyclic sign-skew-symmetric matrices, which together with the fact that these matrices are totally mutable, means that maximal green sequences are well-defined for acyclic sign-skew-symmetric matrices. Finally we prove that every acyclic sign-skew-symmetric matrix admits a maximal green sequence (see Theorem 3.10).

A skew-symmetric matrix is an integer matrix $B=(b_{ij})$ of the size $n\times n$, such that $b_{ij}=-b_{ji}$ for all $1\leq i,j\leq n$. A skew-symmetrizable matrix is an integer matrix $B=(b_{ij})$ of the size $n\times n$, such that $B=-(BD)^{T}$ for $D$ is a diagonal matrix with positive integers. $D$ is called the symmetrizing matrix. A sign-skew-symmetric matrix is an integer matrix $B=(b_{ij})$ of the size $n\times n$, such that either $b_{ij},b_{ji}=0$ or $b_{ij}.b_{ji}<0$ for any $1\leq i,j\leq n$.
The mutation of a matrix ${B}$ in direction $k$ where $1\leq k\leq n$ is the matrix $\mu_{k}(B)=B^{\prime}=(b_{ij}^{\prime})$ where:

Equation (1.1) is called the matrix mutation formula. The mutation is an involution i.e $\mu_{k}\mu_{k}(B)=B$. A skew-symmetric matrix $B=(b_{ij})\in Mat_{n\times n}(\mathbb{Z})$ can be represented by a directed diagram called a quiver with $n$ vertices such that there are $\mid b_{ij}\mid$ many arrows from $j$ to $i$ if $b_{ij}\geq 0$. $Q_{0}$ is the set of vertices in $Q$ and $Q_{1}$ is the set of arrows in $Q$. The mutation formula can be translated to the language of quivers such that for every $k\in Q_{0}$, the quiver mutation in direction $k$ is obtained by the following steps

1. (1)

   for each subquiver $i\rightarrow k\rightarrow j$ add a new arrow $i\rightarrow j$.

2. (2)

   reverse all arrows with source or target $k$.

3. (3)

   remove the arrows in a maximal set of pairwise disjoint 2-cycles.

$Q$ is finite if $Q_{0}$ and $Q_{1}$ are both finite. A vertex $i$ falls in the neighbourhood of a vertex $j$ if there is an arrow connecting $i$ and $j$.

We can easily check that the skew-symmetricity and the skew-symmetrizablity are invariant under mutation, whereas the sign-skew-symmetricity is not necessarily invariant under mutation. A sign-skew-symmetric matrix which remains sign-skew-symmetric under any arbitrary finite sequence of mutation is called totally sign-skew-symmetric matrix.
An $n\times n$ sign-skew-symmetric matrix $B$ can be associated with a (simple) quiver $\Delta(B)$ with vertices $1,\cdots,n$ such that for each pair $(i,j)$ with $b_{ij}<0$, there is exactly one arrow from vertex $i$ to vertex $j$. Trivially, $\Delta(B)$ has no loops and no 2-cycles. Recall that the sign-skew-symmetric matrix $B$ is called acyclic if $\Delta(B)$ is acyclic i.e, $\Delta(B)$ does not admit any directed cycles [10].

The mutation of a matrix $\widetilde{B}$ in direction $k$ where $1\leq k\leq n$ is the matrix $\mu_{k}(\widetilde{B})=\widetilde{B^{\prime}}=\begin{pmatrix}B^{\prime}\\ C^{\prime}\end{pmatrix}$ where $B^{\prime}$ is given as in Equation (1.1) and $C^{\prime}=(c_{ij}^{\prime})$ such that:

By convention $\mu_{k_{0}}(\widetilde{B})=\widetilde{B}$.($\mu_{k_{0}}$ means no mutation has been applied yet and any $c-$vector in $\widetilde{B}$ has its entries either all non-positive or all non-negative.)
If the entries of any $c-$vector in the matrix $\widetilde{B}^{\sigma_{j}}$ such that $0\leq j<\infty$ are either all non-positive or all non-negative, then we say that the sign-coherence property for $C-$matrix holds for the matrix $\widetilde{B}$.
The idea of the unfolding method of an acyclic sign-skew-symmetric matrix $B$ is to create an (infinite) quiver $Q$ which covers $B$ and can do orbit-mutations. We recall the way to create such quiver as it was mentioned in [10].
A locally-finite quiver is an infinite quiver which has finitely many arrows incident to each of its vertices. A locally-finite quiver $Q$ can be represented by an infinite and well-defined matrix $B^{\S}=(b^{\S}_{ij})$ called the adjacency matrix of $Q$ such that $b^{\S}_{ij}\geq 0$ if there are $\mid b^{\S}_{ij}\mid$ many arrows from $j$ to $i$ in $Q$.

Let $Q$ be a locally-finite quiver and $\Gamma$ be a subgroup of the symmetric group $S_{Q_{0}}$. If all the elements of $\Gamma$ are automorphisms of $Q$, then $\Gamma$ is said to be a group of automorphisms of this quiver. Let $Q$ be a locally-finite quiver equipped with a group of automorphisms $\Gamma$. We denote the orbits created under the action of $\Gamma$ by $\bar{i}$ such that $i\in Q_{0}$. A $\Gamma$-loop at $\bar{a}$ is an arrow $a\rightarrow h.a$ and a $\Gamma$-2 cycle at $\bar{a}$ is a pair of arrows $a\rightarrow j\rightarrow h.a$ such that $a,j\in Q_{0}$, $j\notin\bar{a}$ and $h\in\Gamma$.
Let $Q$ be a locally-finite quiver with $B^{\S}$ as its adjacency matrix and a group of automorphisms $\Gamma$ acting on it such that $Q$ does not admit a $\Gamma$-loop or a $\Gamma$-2 cycle at any of its orbits, the orbit-mutation in direction $\bar{k}$ is defined as follows

Since $Q$ is locally-acyclic, the summation in Equation (1.3) is well-defined and mutations in directions which belong to the same orbit commute since the quiver does not admit a $\Gamma-$loop, hence we get the fact

where ${\bar{k}}\big{\rvert}_{\{i,j\}}$ denotes the indices of $\bar{k}$ which are incident to $i$ or $j$ and $\prod$ denotes the the composition of mutations in directions $t\in{\bar{k}}\big{\rvert}_{\{i,j\}}$.

In [10] the authors proved the following Lemma.

In what follows, we recall the construction that M. Huang and F. Li set up in [10] to find a covering for acyclic sign-skew-symmetric matrices which can take arbitrary steps of orbit-mutation.

In this section, we modify Construction 1.7 to find an unfolding of the $c-$vectores of an extended sign-skew-symmetric matrix $\widetilde{B}=\begin{pmatrix}B\\ I_{n}\end{pmatrix}\in Mat_{2n\times n}(\mathbb{Z})$.
Let $Q$ be a locally-finite quiver, the locally-finite framed quiver $\widetilde{Q}$ is the quiver obtained from $Q$ by adding new vertices in a way that each vertex $a\in Q_{0}$ is connected to a new vertex $a^{\prime}$ by a single arrow $a\rightarrow a^{\prime}$ while $Q$ remains the same. The elements of the set $Q_{0}^{\prime}=\{a^{\prime}\mid a\in Q_{0}\}$ are called the frozen vertices. This quiver is represented by the extended infinite skew-symmetric matrix $\widetilde{B^{\S}}=\begin{pmatrix}B^{\S}\\ I_{\infty}\end{pmatrix}$. The bottom part of the matrix $\widetilde{B^{\S}}$ is called the $C$-matrix and $B^{\S}$ is called the principal part.
We extend the action of the group $\Gamma$ to the frozen vertices in the quiver $\widetilde{Q}$ such that for every $g\in\Gamma$, $g(a^{\prime})=g(b^{\prime})$ if and only if $g(a)=g(b)$, that is two frozen vertices lie in the same $\Gamma$-orbit if their mutable copies lie in the same orbit.
Let $\Gamma$ be a group of automorphisms acting on $B^{\S}$ such that $Q$ does not admit any $\Gamma-$loops or $\Gamma-$2 cycles, clearly $\Gamma$ is also a group of automorphisms of $I_{\infty}$. Hence $\Gamma$ is said to be a group of automorphisms of an extended matrix $\widetilde{B^{\S}}$ if it is a group of automorphisms of its principal part $B^{\S}$.
We define the orbit-mutation on the $C$-matrix in direction $\bar{k}$ where $k\in Q_{0}$ as follows

Again since $Q$ does not admit a $\Gamma-$loop, the orbit-mutation of the $C-$matrix can be defined as

Where $t\in{\bar{k}}\big{\rvert}_{\{i,j\}}$ denotes the indices of $\bar{k}$ which are incident to $i$ or $j$ and $\prod$ denotes the the composition of mutations in directions $t\in{\bar{k}}\big{\rvert}_{\{i,j\}}$ .
By the definition of orbit-mutation for an extended infinite skew-symmetric matrix $\widetilde{B^{\S}}=\begin{pmatrix}B^{\S}\\ I_{\infty}\end{pmatrix}$ given in (1.3) and (2.1), it is easy to check that if $\Gamma$ is a group of automorphisms of $\widetilde{B^{\S}}$, it will be a group of automorphisms of any extended infinite skew-symmetric matrix obtained from $\widetilde{B^{\S}}$ by any finite sequence of orbit-mutations.
Since the extended adjacency matrix $\widetilde{B^{\S}}=\begin{pmatrix}B^{\S}\\ I_{\infty}\end{pmatrix}$ of a locally-finite framed quiver $\widetilde{Q}$ is well-defined, we say that the sign-coherence property holds for a locally-finite framed quiver $\widetilde{Q}$ ($\widetilde{Q}$ is sign-coherent) if after performing any finite sequence of ordinary mutations $\mu_{k_{s}}\mu_{k_{s-1}}...\mu_{k_{1}}$ on $\widetilde{B^{\S}}$ where $k_{j}\in Q_{0}$ for every $1\leq j\leq s$, the entries of any $c-$vector in the matrix $\mu_{k_{s}}\mu_{k_{s-1}}...\mu_{k_{1}}(\widetilde{B^{\S}})$ are either all non-negative or all non-positive. In other words a locally-finite framed quiver $\widetilde{Q}$ is sign-coherent if the arrows connecting any mutable vertex with the frozen vertices in the quiver $\mu_{k_{s}}\mu_{k_{s-1}}...\mu_{k_{1}}(\widetilde{Q})$ are either all emerging from this mutable vertex or all reaching at this mutable vertex, where $k_{j}\in Q_{0}$ for every $1\leq j\leq s$ and $s<\infty$.

The sign-coherence property was proved for finite skew-symmetric matrices (finite quivers) in [5] and for finite skew-symmetrizable matrices in [9]. Here we prove that the sign-coherence property holds for locally-finite framed quivers.

Let $\widetilde{A}=\begin{pmatrix}A\\ C\end{pmatrix}$ be a matrix with $C$ as the $C-$matrix and its columns are the $c-$vectors. Suppose that this matrix is equipped with a group of automorphisms $\Gamma$ acting on its principal part $A$ such that the ordinary mutation and orbit-mutation are well-defined and $\widetilde{A}$ can do arbitrary steps of ordinary mutation and orbit-mutation. If all the entries of any $c-$vector are either all non-positive or all non-negative after performing any finite sequence of orbit-mutation i.e, any $c-$vector in the matrix $\mu_{\bar{i}_{m}}....\mu_{\bar{i}_{0}}(\widetilde{A})$ has entries which are all non-negative or all non-positive where $i_{j}$ is used to index the matrix $A$ for $0\leq j\leq m$ and $0\leq m<\infty$, then we say that the orbit-sign coherence property holds for this matrix. By convention $\mu_{\bar{i}_{0}}(\widetilde{A})=\widetilde{A}$.($\mu_{\bar{i}_{0}}$ means no orbit-mutation has been applied yet and any $c-$vector in $\widetilde{A}$ has its entries either all non-positive or all non-negative.)

We denote by $sgn(i)$ the sign of the column indexed by $i$ in the $C-$matrix and $sgn(i)=+$ when the entries of the column $i$ are non-negative while $sgn(i)=-$ when the entries of the column $i$ are non-positive.

When $sgn(s)=+$ for every $s\in\bar{i}$, then $sgn(\bar{i})=+$ and $\bar{i}$ is said to be a green orbit. Respectively, when $sgn(s)=-$ for every $s\in\bar{i}$, then $sgn(\bar{i})=-$ and $\bar{i}$ is said to be a red orbit.

Let ${\widetilde{B^{\S}}}=\begin{pmatrix}{B^{\S}}\\ {C^{\S}}\end{pmatrix}$ be the adjacency matrix of a locally-finite framed quiver $\widetilde{Q}$, in [10] the authors defined the folding matrix $B=(b_{\bar{i}\bar{j}})$ of a locally-finite quiver $Q$ endowed with a group of automorphisms $\Gamma$

$b_{\bar{i}\bar{j}}=\underset{k\in\bar{i}}{\sum}b^{\S}_{kj}$

Analogously we define the folding of the $C$-matrix

$c_{\bar{i}\bar{j}}=\underset{k\in\bar{i}}{\sum}c^{\S}_{kj}$

Clearly, when we have a finite number $n$ of orbits, the folding of the adjacency matrix $\widetilde{B^{\S}}$ of a locally-finite framed quiver $\widetilde{Q}$, is the extended matrix $\widetilde{B}=\begin{pmatrix}B\\ I_{n}\end{pmatrix}\in Mat_{2n\times n}(\mathbb{Z})$.

Now we construct an unfolding for a given extended sign-skew-symmetric matrix.

By convection $\mu^{\bar{\sigma}_{0}}(\widetilde{Q})=\widetilde{Q}$ and $\mu^{\sigma_{0}^{f}}(\widetilde{B})=\widetilde{B}$.