Uniform column sign-coherence and the existence of maximal green sequences

Abstract

In this paper, we prove that each matrix in $M_{m\times n}({\mathbb {Z}}_{\ge 0})$ is uniformly column sign-coherent (Definition 2.2 (ii)) with respect to any $n\times n$ skew-symmetrizable integer matrix (Corollary 3.3 (ii)). Using such matrices, we introduce the definition of irreducible skew-symmetrizable matrix (Definition 4.1). Based on this, the existence of maximal green sequences for skew-symmetrizable matrices is reduced to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.

1 Introduction

C-matrices (respectively, G-matrices) [7] are important research objects in the theory of cluster algebras. It is known that C-matrices (respectively, G-matrices) are column (respectively, row) sign-coherent (see Definition 2.2 (i)). In this paper, we consider the matrices which have the similar property with C-matrices. This property is called uniform column sign-coherence (see Definition 2.2 (ii)). By the definition of uniform column sign-coherence and a result in [9] (see Theorem 2.4 below), we know that $I_n$ is uniformly column sign-coherent using the terminology in this paper.

The motivation to consider the uniform column sign-coherence comes from Proposition 3.7. This proposition indicates if some submatrix of a skew-symmetrizable matrix B is uniformly column sign-coherent, then there is another non-trivial submatrix of B which is invariant under any particular sequence of mutations (see Proposition 3.7 for details).

It is natural to ask when a matrix is uniformly column sign-coherent. This is actually a hard question. However, we can turn our mind to the other side to think about how to produce new uniformly column sign-coherent matrices from a given one. Theorem 3.2 in this paper is an answer to this. As a corollary, matrices in $M_{m\times n}({\mathbb {Z}}_{\ge 0})$ are proved to be uniformly column sign-coherent (Corollary 3.3).

Maximal green sequences are particular sequences of mutations of skew-symmetrizable matrices introduced by Keller [10]. Such particular sequences have numerous applications, including the computations of spectrums of BPS states, Donaldson–Thomas invariants, tilting of hearts in derived categories, and quantum dilogarithm identities.

A very important problem in cluster algebra theory is the existence of maximal green sequences for a given skew-symmetrizable matrix B. Most of the results on this topic have been given on a case-by-case basis, for example, quivers of finite type or acyclic quivers by Brüstle et al. [2], quivers from specific triangulations of various marked surfaces [1, 4, 8, 11, 13]. Generally, the existence of maximal green sequences is not mutation invariant (see [12]). In some special cases, many other authors proved that the existence of maximal green sequences is mutation invariant, for example, for finite-type quiver by Brüstle et al. [2], and for any quiver Q of finite mutation type by Mills [13]. The authors in [3] proved Rotation Lemma which says that if B admits a maximal green sequence, so does any skew-symmetrizable matrices along this sequence.

Now we give the question that we focus on in this paper. In [12, Theorem 9], Muller proved that if B has a maximal green sequence, so does any principal submatrix of B. Conversely, can we get the information about the existence of maximal green sequences for B from its proper submatrices? Our answer to this question is given in Theorem 4.5, whose proof depends on the uniform column sign-coherence.

Thanks to Theorem 4.5 in this paper and [12, Theorem 9], we reduce the existence of maximal green sequences for skew-symmetrizable matrices to the existence of maximal green sequences for irreducible skew-symmetrizable matrices (Definition 4.1). We also give a characterization for irreducible skew-symmetrizable matrices (Proposition 4.2).

Note that a very special case of Theorem 4.5 has been given in [8, Theorem 3.12]. In detail, the authors proved that if both quivers $Q_1$ and $Q_2$ have a maximal green sequences, then so does the quiver Q which is a “t-colored” direct sum of quivers $Q_1$ and $Q_2$. They believe that this result also holds for any direct sum of $Q_1$ and $Q_2$ ( [8, Remark 3.13]) but they did not have a proof. Theorem  4.5 in this paper actually gives an affirmative answer to this.

This paper is organized as follows: In Sect. 2 some basic definitions are given. In Sect. 3 we give a method to produce uniformly column sign-coherent matrices from a given one (Theorem 3.2). Thus, we prove that each matrix in $M_{m\times n}({\mathbb {Z}}_{\ge 0})$ is uniform column sign-coherent (Corollary 3.3). In Sect. 4 we give the definition of irreducible skew-symmetrizable matrices and their characterization. Then we reduce the existence of maximal green sequences for skew-symmetrizable matrices to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.

2 Preliminaries

Recall that an integer matrix $B_{n\times n}=(b_{ij})$ is called skew-symmetrizable if there is a positive integer diagonal matrix S such that SB is skew-symmetric, where S is said to be a skew-symmetrizer of B. In this case, we say that B is S-skew-symmetrizable. For an $(m+n)\times n$ integer matrix ${\tilde{B}}=(b_{ij})$, the square submatrix $B=(b_{ij})_{1\le i,j\le n}$ is called the principal part of ${\tilde{B}}$. Abusing terminology, we say that ${\tilde{B}}$ itself is skew-symmetrizable or skew-symmetric if its principal part B is so.

Definition 2.1

Let ${\tilde{B}}_{(m+n)\times n}=(b_{ij})$ be S-skew-symmetrizable, the mutation of ${\tilde{B}}$ in the direction $k\in \{1,2,\ldots ,n\}$ is the $(m+n)\times n$ matrix $\mu _k({\tilde{B}})=(b_{ij}^{\prime })$, where

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

(1)

It is easy to see that $\mu _k({\tilde{B}})$ is still S-skew-symmetrizable, and $\mu _k(\mu _k({\tilde{B}}))={\tilde{B}}$.

Definition 2.2

• (i) For $m,n>0$, an $m\times n$ integer matrix A is called column sign-coherent (respectively, row sign-coherent) if any two nonzero entries of A in the same column (respectively, row) have the same sign.

• (ii) Let $B_{1}$ be an $n\times n$ skew-symmetrizable matrix, and $B_{2}\in M_{m\times n}({\mathbb {Z}})$ be a column sign-coherent matrix. $B_{2}$ is called uniformly column sign-coherent with respect to$B_{1}$ if for any sequence of mutations $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}$, the lower $m\times n$ submatrix of $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\begin{pmatrix}B_{1}\\ B_{2}\end{pmatrix}$ is column sign-coherent.

Remark 2.3

• (i) Note that the uniform column sign-coherence of $B_{2}$ is invariant up to permutation of its row vectors, by the equality (1).

• (ii) Roughly, the uniform column sign-coherence means that the column sign-coherence is invariant after a sequence of mutations.

Given an S-skew-symmetrizable matrix ${\tilde{B}}=\begin{pmatrix} B\\ I_n\end{pmatrix}\in M_{2n\times n}({\mathbb {Z}})$, let ${\tilde{B}}_\sigma =\begin{pmatrix} B_\sigma \\ C_\sigma \end{pmatrix}$ be the matrix obtained from ${\tilde{B}}$ by a sequence of mutations $\sigma :=\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}$. Recall that the lower part $C_\sigma$ of ${\tilde{B}}_\sigma$ is called a C-matrix of B, see [7]. Note that the matrix ${\tilde{B}}=\begin{pmatrix} B\\ I_n\end{pmatrix}$ is used to define cluster algebra with principal coefficients in [7], but we do not talk much about cluster algebra here.

Theorem 2.4

([9]) Using the above notations, each C-matrix of a skew-symmetrizable matrix B is column sign-coherent.

Remark 2.5

By Definition 2.2, this theorem means that $I_n$ is uniformly column sign-coherent with respect to the skew-symmetrizable matrix B.

Thanks to Theorem 2.4, one can define the sign functions on the column vectors of a C-matrix of a skew-symmetrizable matrix B. For a sequence of mutations $\sigma :=\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}$, denote by $\begin{pmatrix}B_{\sigma }\\ C_{\sigma }\end{pmatrix}:=\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\begin{pmatrix}B\\ I_n\end{pmatrix}$. If the entries of j-th column of $C_\sigma$ are all nonnegative (respectively, nonpositive), the sign of the j-th column of $C_\sigma$ is defined as $\varepsilon _\sigma (j)=1$ (respectively, $\varepsilon _\sigma (j)=-1$).

Definition 2.6

Let $C_\sigma$ be the C-matrix of B given by a sequence of mutations $\sigma$, a column index $j\in \{1,\ldots ,n\}$ of $C_\sigma$ is called green (respectively, red) if $\varepsilon _\sigma (j)=1$ (respectively, $\varepsilon _\sigma (j)=-1$).

Note that, by Theorem 2.4, the column index of a C-matrix$C_\sigma$ is either green or red.

Definition 2.7

Let B be a skew-symmetrizable matrix, and $\mathbf{k}=(k_1,\ldots ,k_s)$ be a sequence of column indices of B. Denote by $C_{\sigma _j}$ the C-matrix of B given by $\sigma _j:=\mu _{k_j}\ldots \mu _{k_2}\mu _{k_1}$.

• (i) $\mathbf{k}=(k_1,\ldots ,k_s)$ is called a green-to-red sequence of B if each column index of the C-matrix $C_{\sigma _s}$ is red, i.e., $C_{\sigma _s}\in M_{n\times n}({\mathbb {Z}}_{\le 0})$.

• (ii) $\mathbf{k}=(k_1,\ldots ,k_s)$ is called a green sequence of B if $k_i$ is green in the C-matrix $C_{\sigma _{i-1}}$ for $i=2,3,\ldots ,s$.

• (iii) $\mathbf{k}=(k_1,\ldots ,k_s)$ is called maximal green sequence of B if it is both a green sequence and a green-to-red sequence of B.

Example 2.8

Let $B=\begin{pmatrix}0&{}1&{}-1\\ -1&{}0&{}1\\ 1&{}-1&{}0 \end{pmatrix}$, and $\mathbf{k}=(2,3,1,2)$.

Hence, $\mathbf{k}=(2,3,1,2)$ is a maximal green sequence of B.

3 Uniform column sign-coherence of $B_2$

In this section, we give a method to produce uniformly column sign-coherent matrices from a known one (Theorem 3.2). Then it is shown that all nonnegative matrices and rank $\le 1$ column sign-coherent matrices are uniformly column sign-coherent (Corollary 3.3 and Corollary 3.4).

Lemma 3.1

Let $P=(p_{ij})\in M_{p\times m}({\mathbb {Z}}_{\ge 0}),p,m>0$, and $B_1$ be an $n\times n$ skew-symmetrizable matrix. If $B_2\in M_{m\times n}({\mathbb {Z}})$ is column sign-coherent, then for $1\le k\le n$,

$$\begin{aligned} \mu _k\left( \begin{pmatrix}I_n&{} \quad 0\\ 0&{} \quad P\end{pmatrix}\begin{pmatrix}B_1\\ B_2\end{pmatrix}\right) =\mu _k\begin{pmatrix}B_1\\ PB_2\end{pmatrix}=\begin{pmatrix}I_n&{} \quad 0\\ 0&{} \quad P\end{pmatrix}\mu _k\begin{pmatrix}B_1\\ B_2 \end{pmatrix}. \end{aligned}$$

Proof

Denote by $\begin{pmatrix}B_1\\ B_2\end{pmatrix}=(b_{ij})$, $\mu _k\begin{pmatrix}B_1\\ B_2\end{pmatrix}=(b_{ij}^{\prime })$, $\begin{pmatrix}B_1\\ PB_2\end{pmatrix}=(a_{ij})$, $\mu _k\begin{pmatrix}B_1\\ PB_2\end{pmatrix}=(a_{ij}^{\prime })$. Clearly, the principal parts of $\mu _k\begin{pmatrix}B_1\\ PB_2\end{pmatrix}$ and $\begin{pmatrix}I_n&{}0\\ 0&{}P\end{pmatrix}\mu _k\begin{pmatrix}B_1\\ B_2 \end{pmatrix}$ are equal. It suffices to show the lower parts of $\mu _k\begin{pmatrix}B_1\\ PB_2\end{pmatrix}$ and $\begin{pmatrix}I_n&{}0\\ 0&{}P\end{pmatrix}\mu _k\begin{pmatrix}B_1\\ B_2 \end{pmatrix}$ are equal. We know that for $i>n, a_{ij}=\sum \nolimits _{l=1}^mp_{il}b_{n+l,j}$. By Eq. (1), for $i>n$,

$$\begin{aligned} a_{ij}^{\prime }= & {} a_{ij}+{{\,\mathrm{sgn}\,}}(a_{ik})\max (a_{ik}b_{kj},0) =\sum \limits _{l=1}^mp_{il}b_{n+l,j}\\&+{{\,\mathrm{sgn}\,}}\left( \sum \limits _{l=1}^mp_{il}b_{n+l,k}\right) \max \left( \sum \limits _{l=1}^mp_{il}b_{n+l,k}b_{kj},0\right) . \end{aligned}$$

Because $B_2$ is column sign-coherent and $P\in M_{p\times m}({\mathbb {Z}}_{\ge 0})$, we know that $(p_{il_1}b_{n+l_1,k})(p_{il_2}b_{n+l_2,k})\ge 0, 1\le l_1,l_2\le m$. Thus, if $p_{il_1}b_{n+l_1,k}\ne 0$, then ${{\,\mathrm{sgn}\,}}(p_{il_1}b_{n+l_1,k})={{\,\mathrm{sgn}\,}}(\sum \nolimits _{l=1}^mp_{il}b_{n+l,k})$. So

$$\begin{aligned} a_{ij}^{\prime }= & {} \sum \limits _{l=1}^mp_{il}b_{n+l,j}+{{\,\mathrm{sgn}\,}}\left( \sum \limits _{l=1}^mp_{il}b_{n+l,k}\right) \max \left( \sum \limits _{l=1}^mp_{il}b_{n+l,k}b_{kj},0\right) \\= & {} \sum \limits _{l=1}^mp_{il}b_{n+l,j}+\sum \limits _{l=1}^m {{\,\mathrm{sgn}\,}}(p_{il}b_{n+l,k})\max (p_{il}b_{n+l,k}b_{kj},0)\\= & {} \sum \limits _{l=1}^mp_{il}(b_{n+l,j}+{{\,\mathrm{sgn}\,}}(b_{n+l,k})\max (b_{n+l,k}b_{kj},0)),\\= & {} \sum \limits _{l=1}^mp_{il}b_{n+l,j}^{\prime }. \end{aligned}$$

Then the result follows. $\square$

Theorem 3.2

Let $P\in M_{p\times m}({\mathbb {Z}}_{\ge 0})$ for $p,m>0$, and $B_1$ be an $n\times n$ skew-symmetrizable matrix. If $B_2\in M_{m\times n}({\mathbb {Z}})$ is uniformly column sign-coherent with respect to $B_1$, then so is $PB_2$.

Proof

For any sequence of mutations $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}$, the lower part of $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\begin{pmatrix}B_1\\ B_2 \end{pmatrix}$ is column sign-coherent, by the uniform column sign-coherence of $B_2$ with respect to $B_1$. Clearly, the lower part of $\begin{pmatrix}I_n&{} \quad 0\\ 0&{} \quad P\end{pmatrix}\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\begin{pmatrix}B_1\\ B_2 \end{pmatrix}$ is also column sign-coherent. By Lemma 3.1, we have

$$\begin{aligned} \mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\left( \begin{pmatrix}I_n&{} \quad 0\\ 0&{} \quad P\end{pmatrix}\begin{pmatrix}B_1\\ B_2\end{pmatrix}\right) =\begin{pmatrix}I_n&{}0\\ 0&{}P\end{pmatrix}\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\begin{pmatrix}B_1\\ B_2 \end{pmatrix}. \end{aligned}$$

So the lower part of $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}\left( \begin{pmatrix}I_n&{}0\\ 0&{}P\end{pmatrix}\begin{pmatrix}B_1\\ B_2\end{pmatrix}\right)$ is also column sign-coherent. Thus, $PB_2$ is uniformly column sign-coherent with respect to $B_1$. $\square$

Corollary 3.3

Let $B_1$ be an $n\times n$ skew-symmetrizable matrix. Then any matrix $P\in M_{m\times n}({\mathbb {Z}}_{\ge 0})$ is uniformly column sign-coherent with respect to $B_1$.

Proof

By Remark 2.5, $I_n$ is uniformly column sign-coherent with respect to $B_1$. Then the result follows from Theorem 3.2 since $P=PI_n$. $\square$

Corollary 3.4

Let $B_1$ be an $n\times n$ skew-symmetrizable matrix, and $B_2$ be an $m\times n$ column sign-coherent integer matrix. If $\mathrm{rank}(B_2)\le 1$, then $B_2$ is uniformly column sign-coherent with respect to $B_1$.

Proof

Because $\mathrm{rank}(B_2)\le 1$, $B_2$ has the form of

$$\begin{aligned} B_2=\begin{pmatrix}c_1\\ \vdots \\ c_n\end{pmatrix}\alpha , \end{aligned}$$

where $\alpha$ is a row vector and $c_1, c_2,\ldots ,c_m\in {\mathbb {Q}}$. Because $B_2$ is column sign-coherent, we can assume that $c_1,c_2,\ldots ,c_m\ge 0$. Clearly, $\alpha$ is uniformly column sign-coherent with respect to $B_1$. Then by Theorem 3.2, $B_2$ is uniformly column sign-coherent with respect to $B_1$. $\square$

By the two corollaries, we can construct many matrices which are uniformly column sign-coherent with respect to a given skew-symmetrizable matrix. Now we give an example showing that there does exist a matrix which is column sign-coherent but not uniformly column sign-coherent.

Example 3.5

It can be seen that $\begin{pmatrix}2&{}-1\\ 1&{}-2\end{pmatrix}$ is column sign-coherent but it is not uniformly column sign-coherent with respect to $\begin{pmatrix}0&{}1\\ -1&{}0\end{pmatrix}$.

It is natural to consider the following problem.

Problem 3.6

For a given skew-symmetrizable matrix $B_1$, which matrices are uniformly column sign-coherent with respect to $B_1$?

In the following proposition, we give a characterization for those matrices which are uniformly column sign-coherent with respect to $B_1$.

Proposition 3.7

Let $B=\begin{pmatrix}{B_1}&{}{B_{3}}\\ {B_2}&{}{B_{4}} \end{pmatrix}$ be a skew-symmetrizable matrix with $B_1\in M_n({\mathbb {Z}})$ and $B_4\in M_m({\mathbb {Z}})$, $m>0$. Then $B_2$ is uniformly column sign-coherent with respect to $B_1$ if and only if $B_{4}$ is invariant under any sequence of mutations $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}$ with $1\le k_i\le n, i=1,2,\ldots ,s$.

Proof

Let $B=(b_{ij})$, and $\mu _k(B)=(b_{ij}^{\prime }), 1\le k\le n$. We know for any i, j,

$$\begin{aligned} b_{ij}^{\prime }=b_{ij}+{{\,\mathrm{sgn}\,}}(b_{ik})\max (b_{ik}b_{kj},0). \end{aligned}$$

Then $b_{ij}^{\prime }=b_{ij}$ if and only if $b_{ik}b_{kj}\le 0$, and then if and only if $b_{ik}b_{jk}\ge 0$ because either $b_{kj}b_{jk}<0$ or $b_{kj}=b_{jk}=0$ holds.

So $B_{4}$ is invariant under the mutation $\mu _k(B)=(b_{ij}^{\prime }), 1\le k\le n$ if and only if $b_{ij}^{\prime }=b_{ij}$ for $n+1\le i,j\le n+m$, and then if and only if $b_{ik}b_{jk}\ge 0$ for $n+1\le i,j\le n+m, 1\le k\le n$, which means that $B_2$ is column sign-coherent. The result follows. $\square$

4 The existence of maximal green sequences

Based on the discussion about uniform column sign-coherence, in this section, we reduce the existence of maximal green sequences for skew-symmetrizable matrices to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.

4.1 Irreducible skew-symmetrizable matrices

In this subsection, we give the definition of irreducible skew-symmetrizable matrices and their characterization.

Let $B=(b_{ij})_{n\times n}$ be a matrix, and $n_1, n_2$ be two positive integers. For $1\le i_1<\cdots <i_{n_2}\le n$ and $1\le j_1<\cdots <j_{n_1}\le n$, denote by $B_{j_1,\ldots ,j_{n_1}}^{i_1,\ldots ,i_{n_2}}$ the submatrix of B with entries $b_{ij}$, where $i=i_1,\ldots ,i_{n_2}$ and $j=j_1,\ldots ,j_{n_1}$. If $n_2<n$ or $n_1<n$, the corresponding submatrix $B_{j_1,\ldots ,j_{n_1}}^{i_1,\ldots ,i_{n_2}}$ is a proper submatrix of B. If $n_2=n_1$ and $\{i_1,\ldots ,i_{n_2}\}=\{j_1,\ldots ,j_{n_1}\}$, the corresponding submatrix is a principal submatrix of B. Clearly, any principal submatrix of a skew-symmetrizable matrix is still skew-symmetrizable.

Definition 4.1

A skew-symmetrizable matrix $B=(b_{ij})_{n\times n}$ is called reducible, if B has a proper submatrix $B_{j_1,\ldots ,j_{n_1}}^{i_1,\ldots ,i_{n_2}}$ satisfying

• (i) $B_{j_1,\ldots ,j_{n_1}}^{i_1,\ldots ,i_{n_2}}$ is a nonnegative matrix, i.e., $B_{j_1,\ldots ,j_{n_1}}^{i_1,\ldots ,i_{n_2}}\in M_{n_2\times n_1}({\mathbb {Z}}_{\ge 0})$.

• (ii) $\{i_1,\ldots ,i_{n_2}\}\cup \{j_1,\ldots ,j_{n_1}\}=\{1,2,\ldots ,n\}$ and $\{i_1,\ldots ,i_{n_2}\}\cap \{j_1,\ldots ,j_{n_1}\}=\phi$.

Otherwise, B is said to be irreducible if such proper submatrix does not exist.

Clearly, B is reducible if and only if up to renumbering the row–column indices of B, B can be written as a block matrix as follows

$$\begin{aligned} B=\begin{pmatrix}{B_1}&{}{B_{3}}\\ {B_2}&{}{B_{4}} \end{pmatrix} \end{aligned}$$

with $B_1\in M_{n_1}({\mathbb {Z}})$ and $B_4\in M_{n_2}({\mathbb {Z}})$ such that the proper submatrix $B_2$ of B is a nonnegative matrix, i.e., $B_2\in M_{n_2\times n_1}({\mathbb {Z}}_{\ge 0})$.

In the skew-symmetric case the definition of irreducibility for quiver version has been given in [8].

For a skew-symmetrizable matrix B, we can encode the sign pattern of entries of B by the quiver $\Gamma (B)$ with the vertices $1,2,\ldots ,n$ and the arrows $i\rightarrow j$ for $b_{ij}>0$. We call $\Gamma (B)$ the underlying quiver of B . If $\Gamma (B)$ is an acyclic quiver, then B is said to be acyclic. If $\Gamma (B)$ is a connected quiver, then B is said to be connected. Clearly, if B is an irreducible skew-symmetrizable matrix, then it must be connected.

For a quiver Q, if there exists a path from a vertex a to a vertex b, then a is said to be a predecessor of b, and b is said to be a successor of a. For a vertex a in Q, denote by M(a), N(a) the set of predecessors of a and the set of successors of a, respectively. By viewing a vertex a as a trivial path from a to a, we know that $a\in M(a)\cap N(a)$.

Proposition 4.2

Let $B=(b_{ij})_{n\times n}$ be a connected skew-symmetrizable matrix. Then B is irreducible if and only if each arrow of the quiver $\Gamma (B)$ is in some oriented cycle.

Proof

Suppose that B is reducible, then B can be written as a block matrix

$$\begin{aligned} B=\begin{pmatrix}{B_1}&{}\quad {B_{3}}\\ {B_2}&{}\quad {B_{4}} \end{pmatrix} \end{aligned}$$

with $B_1\in M_{n_1}({\mathbb {Z}})$ and $B_4\in M_{n_2}({\mathbb {Z}})$ such that the proper submatrix $B_2\in M_{n_2\times n_1}({\mathbb {Z}}_{\ge 0})$, up to renumbering the row–column indices of B. Since B is connected, $B_2$ cannot be a zero matrix. So there exist $i>n_1, j\le n_1$ such that $b_{ij}\ne 0$. In fact $b_{ij}>0$, since $B_2\in M_{n_2\times n_1}({\mathbb {Z}}_{\ge 0})$. We know that the arrow $i\rightarrow j$ is not in any oriented cycle of $\Gamma (B)$, because $B_2\in M_{n_2\times n_1}({\mathbb {Z}}_{\ge 0})$.

Suppose that there exists an arrow $i\rightarrow j$ is not in any oriented cycle of $\Gamma (B)$. We know that i cannot be a successor of j, i.e., $i\notin N(j)$. Let $n_1$ be the number of elements of N(j). Clearly, $1\le n_1\le n-1$. We can renumber the row–column indices of B such that the elements of N(j) are indexed by $1,2,\ldots ,n_1$. B can be written as a block matrix

$$\begin{aligned} B=\begin{pmatrix}{B_1}&{}\quad {B_{3}}\\ {B_2}&{}\quad {B_{4}} \end{pmatrix}. \end{aligned}$$

We claim that $B_2\in M_{(n-n_1)\times n_1}({\mathbb {Z}}_{\ge 0})$. Otherwise, there exists $k_1>n_1$ and $k_2\le n_1$, i.e., $k_1\notin N(j), k_2\in N(j)$ such that $b_{k_1k_2}<0$. Thus, $k_1$ is a successor of $k_2$, so is a successor of j, by $k_2\in N(j)$. This contradicts $k_1\notin N(j)$. So $B_2\in M_{(n-n_1)\times n_1}({\mathbb {Z}}_{\ge 0})$ and B is reducible. The proof is finished. $\square$

Example 4.3

Let $B=\begin{pmatrix}0&{}1&{}-1\\ -2&{}0&{}2\\ 2&{}-2&{}0 \end{pmatrix}$. It is a skew-symmetrizable matrix with skew-symmetrizer $S=\mathrm{diag}\{2,1,1\}$. The underlying quiver $\Gamma (B)$ is as follows.

Since any arrow of $\Gamma (B)$ is in an oriented cycle, B is irreducible.

4.2 Reduction of the existence of maximal green sequences

In this subsection, we reduce the existence of maximal green sequences for skew-symmetrizable matrices to the existence of maximal green sequences for irreducible skew-symmetrizable matrices.

Lemma 4.4

Let B be a skew-symmetrizable matrix and $\sigma _{s+1}:=(k_1,\ldots ,k_{s+1})$ be a sequence of column indices of B. Denote by ${\tilde{B}}_{\sigma _{i}}=\begin{pmatrix}B_{\sigma _{i}}\\ C_{\sigma _{i}}\end{pmatrix}:=\mu _{k_{i}}\ldots \mu _{k_2}\mu _{k_1}\begin{pmatrix}B\\ I_n\end{pmatrix}$, $i=1,\ldots ,s+1$. If $k_{s+1}$ is a green column index of $C_{\sigma _s}$, then any green column index j of $C_{\sigma _s}$, with $j\ne k_{s+1}$, must be green in $C_{\sigma _{s+1}}$.

Proof

The proof is the same as that of Lemma 2.16 of [2]. For the convenience of readers, we give the proof here.

Because j and $k_{s+1}$ are green column indices of $C_{\sigma _s}$, we know that $(C_{\sigma _s})_{ij}\ge 0$ and $(C_{\sigma _s})_{ik_{s+1}}\ge 0$. By the definition of mutation, we have

$$\begin{aligned} (C_{\sigma _{s+1}})_{ij}= & {} (C_{\sigma _s})_{ij}+ {{\,\mathrm{sgn}\,}}((C_{\sigma _s})_{ik_{s+1}})\max ((C_{\sigma _s})_{ik_{s+1}}(C_{\sigma _s})_{k_{s+1}j},0)\\\ge & {} (C_{\sigma _s})_{ij}\ge 0. \end{aligned}$$

So, j is green in $C_{\sigma _{s+1}}$. $\square$

Theorem 4.5

(Direct sum formula) Let $B=\begin{pmatrix}{B_{1}}&{}\quad {B_{3}}\\ {B_{2}}&{}\quad {B_{4}} \end{pmatrix}=(b_{ij})$ be a skew-symmetrizable matrix with $B_1\in M_n({\mathbb {Z}})$ and $B_4\in M_m({\mathbb {Z}})$, and $\tilde{\mathbf{k}}$ be a sequence $\tilde{\mathbf{k}}=(k_1,\ldots ,k_s,k_{s+1},\ldots ,k_{s+p})$, with $1\le k_i\le n$, and $n+1\le k_j\le m+n$ for $i=1,\ldots ,s$, and $j=s+1,\ldots ,s+p$. If $B_{2}$ is a matrix in $M_{m\times n}({\mathbb {Z}}_{\ge 0})$, then $\tilde{\mathbf{k}}$ is a maximal green sequence of B if and only if $\mathbf{k}:=(k_1,\ldots ,k_s)$ (respectively, $\mathbf{j}:=(k_{s+1},\ldots ,k_{s+p})$) is a maximal green sequence of $B_{1}$ (respectively, $B_{4}$).

Proof

Let ${\tilde{B}}=\begin{pmatrix} {B_1}&{}{B_{3}}\\ {B_2}&{}{B_{4}}\\ I_n&{}0\\ 0&{}I_m\end{pmatrix}$ and $B_{\sigma _i}=\mu _{k_i}\ldots \mu _{k_2}\mu _{k_1}({\tilde{B}}), i=1,\ldots ,s, s+1,\ldots , s+p$. By $B_2\in M_{m\times n}({\mathbb {Z}}_{\ge 0})$ and Corollary 3.3, we know that $\begin{pmatrix}B_2\\ I_n\\ 0\end{pmatrix}$ is uniformly column sign-coherent with respect to $B_1$. By the same argument in Proposition 3.7, we know that the submatrix $\begin{pmatrix} {B_{4}}\\ 0\\ I_m\end{pmatrix}$ of ${\tilde{B}}$ is invariant under the sequence of mutations $\mu _{k_s}\ldots \mu _{k_2}\mu _{k_1}, 1\le k_i\le n$ for $i=1,2,\ldots ,s$. So for $i\le s$ the matrix $B_{\sigma _i}$ has the form of

$$\begin{aligned} B_{\sigma _i}=\begin{pmatrix} {B_{1;\sigma _i}}&{}\quad {B_{3;\sigma _i}}\\ {B_{2;\sigma _i}}&{}\quad {B_{4}}\\ C_{1;\sigma _i}&{}\quad 0\\ 0&{}\quad I_m\end{pmatrix}. \end{aligned}$$

(2)

$``\Longleftarrow ''$: Because $\mathbf{k}=(k_1,k_2,\ldots ,k_s)$ is a maximal green sequence of $B_1$, we know that $C_{1;\sigma _s}\in M_{n\times n}({\mathbb {Z}}_{\le 0})$. Thus, by the uniform column sign-coherence of $\begin{pmatrix}B_2\\ I_n\\ 0\end{pmatrix}$ with respect to $B_1$, we know that $\begin{pmatrix} {B_{2;\sigma _s}}\\ C_{1;\sigma _s}\\ 0\end{pmatrix}\in M_{(2m+n)\times n}({\mathbb {Z}}_{\le 0})$. By $B_{2;\sigma _s}\in M_{m\times n}({\mathbb {Z}}_{\le 0})$ and that the principal part of $B_{\sigma _s}$ is skew-symmetrizable, we can know $B_{3;\sigma _s}\in M_{n\times m}({\mathbb {Z}}_{\ge 0})$. Then by Corollary 3.3, we know that $\begin{pmatrix} {B_{3;\sigma _s}}\\ 0\\ I_m\end{pmatrix}\in M_{(2n+m)\times m}({\mathbb {Z}}_{\ge 0})$ is uniformly column sign-coherent with respect to $B_{4}$. By the same argument in Proposition 3.7 again, we know that the submatrix $\begin{pmatrix} {B_{1;\sigma _s}}\\ C_{1;\sigma _s}\\ 0\end{pmatrix}$ of $B_{\sigma _s}$ is invariant under the sequences of mutations $\mu _{k_{s+p}}\ldots \mu _{k_{s+2}}\mu _{k_{s+1}}(B_{\sigma _s}), n+1\le k_i\le n+m$ for $i=s+1,\ldots ,s+p$. So for $i\ge s+1$, the matrix $B_{\sigma _i}$ has the form of

$$\begin{aligned} B_{\sigma _i}=\begin{pmatrix} {B_{1;\sigma _s}}&{}\quad {B_{3;\sigma _i}}\\ {B_{2;\sigma _i}}&{}\quad {B_{4;\sigma _i}}\\ C_{1;\sigma _s}&{}\quad 0\\ 0&{}\quad C_{4;\sigma _i}\end{pmatrix}. \end{aligned}$$

Because $\mathbf{j}=(j_1,j_2,\ldots ,j_p)$ is a maximal green sequence of $B_{4}$, we know that $C_{4;\sigma _{s+p}}\in M_{m\times m}({\mathbb {Z}}_{\le 0})$. Thus, the lower part of $B_{\sigma _{s+p}}$ is $\begin{pmatrix} C_{1;\sigma _s}&{}0\\ 0&{}C_{4;\sigma _{s+p}}\end{pmatrix}\in M_{(m+n)\times (m+n)}({\mathbb {Z}}_{\le 0})$. It can be seen that $\tilde{\mathbf{k}}=(\mathbf{k},\mathbf{j})$ is a green sequence of B, so it is maximal.

$``\Longrightarrow ''$ By (2), $B_{\sigma _s}=\begin{pmatrix} {B_{1;\sigma _s}}&{}{B_{3;\sigma _s}}\\ {B_{2;\sigma _s}}&{}{B_{4}}\\ C_{1;\sigma _s}&{}0\\ 0&{}I_m\end{pmatrix}.$ Clearly, $\mathbf{k}=(k_1,\ldots ,k_s)$ is a green sequence of $B_1$ and $\mathbf{j}=(k_{s+1},\ldots ,k_{s+p})$ is a maximal green sequence of $B_{4}$.

We claim that each $l\in \{1,2,\ldots ,n\}$ is red in $C_{1;\sigma _s}$, i.e., $C_{1;\sigma _s}\in M_{n\times n}({\mathbb {Z}}_{\le 0})$, and thus, $\mathbf{k}=(k_1,\ldots ,k_s)$ is a maximal green sequence of $B_1$. Otherwise, there will exist a $l_0\in \{1,2,\ldots ,n\}$ which is green in $C_{1;\sigma _s}$. Thus, $l_0$ is green in $\begin{pmatrix} C_{1;\sigma _s}&{}0\\ 0&{}I_m\end{pmatrix}$ the lower part of $B_{\sigma _s}$. By Lemma 4.4 and $l_0\le n<k_{s+i}, i=1,2,\ldots ,p$, we know that $l_0$ will remain green in $\begin{pmatrix} C_{1;\sigma _{s+p}}&{}C_{3;\sigma _{s+p}}\\ C_{2;\sigma _{s+p}}&{}C_{4;\sigma _{s+p}}\end{pmatrix}$ the lower part of $B_{\sigma _{s+p}}$. It is impossible since $(k_1,\ldots ,k_{s},k_{s+1},\ldots ,k_{s+p})$ is a maximal green sequence of B. $\square$

When B is skew-symmetric and $B_2$ is a matrix over $\{0,1\}$, the above theorem has been actually given in [8, Theorem 3.12]. The authors of [8] believe that the result also holds for $B_2\in M_{m\times n}({\mathbb {Z}}_{\ge 0})$, but they did not have a proof. In fact, we have given the proof for this in the skew-symmetrizable case.

Remark 4.6

Note that the $``\Longleftarrow ''$ part of the proof of the above theorem also holds if we replace maximal green sequences with green-to-red sequences, and the proof is identical. We are thankful to Fan Qin for pointing out this.

Example 4.7

Let $B=\begin{pmatrix}0&{}-2\\ 3&{}0\end{pmatrix}$. Here $B_1=0=B_{4}$, $B_2=3\ge 0$. The column index set of $B_1$ is $\{1\}$, and the column index set of $B_{4}$ is $\{2\}$. It is known that (1) is a maximal green sequence of $B_1$ and (2) is a maximal green sequence of $B_{4}$. Then by Theorem 4.5, (1, 2) is a maximal green sequence of B. Indeed,

Example 4.8

Let     where $B_1$ is of order $3\times 3$ and $B_4$ is of order $2\times 2$. Clearly, B is skew-symmetrizable with skew-symmetrizer $S=\mathrm{diag}\{1,1,1,1,2\}$ and $B_2\in M_{2\times 3}({\mathbb {Z}}_{\ge 0})$. The column index set of $B_1$ is $\{1,2,3\}$, and the column index set of $B_{4}$ is $\{4,5\}$. By Example 2.8 (respectively, Example 4.7), (2, 3, 1, 2) (respectively, (4, 5)) is a maximal green sequence of $B_1$ (respectively, $B_{4}$). Then by Theorem 4.5, (2, 3, 1, 2, 4, 5) is a maximal green sequence of B. Indeed,

Denote by ${\tilde{B}}^{\prime }=\mu _2\mu _1\mu _3\mu _2({\tilde{B}})$. It is can be seen that the submatrix ${\tilde{B}}_{4,5}^{4,5,6,7,8,9,10}$ of ${\tilde{B}}$ is invariant along the mutation sequence (2, 3, 1, 2) and the submatrix $\tilde{B^{\prime }}_{1,2,3}^{1,2,3,6,7,8,9,10}$ of ${\tilde{B}}^{\prime }$ is invariant along the mutation sequence (4, 5).

The following lemma is the skew-symmetrizable version of [12, Theorem 9, Theorem 17] about induced subquivers. Although corresponding result in [12] was verified for the situation of quivers, or say, in skew-symmetric case, the method of its proof can be naturally extended to the skew-symmetrizable case.

Lemma 4.9

Let B be a skew-symmetrizable matrix. If B admits a maximal green sequence (respectively, green-to-red sequence), then any principal submatrix of B also has a maximal green sequence (respectively, green-to-red sequence).

Theorem 4.10

Let B be a skew-symmetrizable matrix. Then B has a maximal green sequence (respectively, green-to-red sequence) if and only if any irreducible principal submatrix of B has a maximal green sequence (respectively, green- to-red sequence).

Proof

It follows from Lemma 4.9, Theorem 4.5 and Remark 4.6. $\square$

Remark 4.11

By the above theorem, we can give our explanation of the existence of maximal green sequences for acyclic skew-symmetrizable matrices. Because any irreducible principal submatrix of an acyclic skew-symmetrizable matrix B is only a $1\times 1$ zero matrix, and it always has a maximal green sequence, we then know that by Theorem 4.10 any acyclic skew-symmetrizable matrix admits a maximal green sequence.

By Theorem 4.10, we reduce the existence of maximal green sequences (respectively, green-to-red sequences) for skew-symmetrizable matrices to the existence of maximal green sequences (respectively, green-to-red sequences) for irreducible skew-symmetrizable matrices B, i.e., those B whose all arrows of $\Gamma (B)$ are in oriented cycles, by Proposition 4.2. So it is natural to ask that

Problem 4.12

Which irreducible skew-symmetrizable matrices admit maximal green sequences (respectively, green-to-red sequences)?

Note that the existence of green-to-red sequences is mutation invariant [12] and acyclic skew-symmetrizable matrices always have a green-to-red sequences (Remark 4.11). So the irreducible skew-symmetrizable matrices which are mutation equivalent to acyclic matrices always admit a green-to-red sequences.

It is known that the existence of maximal green sequences for quivers of finite type, or quivers of finite mutation type is mutation invariant (see [2, 8, 13]). So the existence of maximal green sequences for irreducible subquivers of a quiver of finite type, or quivers of finite mutation type has a clear answer from these references.

In [2, 12], the authors have shown that the Markov quiver (Fig. 1) has no maximal green sequence. This is an example of irreducible quiver with no maximal green sequence. More generally, the authors in [2, Proposition 8.1] proved that if a quiver Q has a non-degenerate potential such that the corresponding quiver is Jacobi-infinite, then Q has no maximal green sequences.

Fig. 1

Markov quiver

4.3 An application

There are two ways to understand Theorem 4.5, i.e., the direct sum formula. On the one hand, by direct sum formula, we can reduce the existence of maximal green sequences for skew-symmetrizable matrices to the existence of maximal green sequences for irreducible cases (Theorem 4.10). On the other hand, by direct sum formula, we can use the known irreducible matrices which have maximal green sequences to construct more matrices which have maximal green sequences. The existence of maximal green sequence for many of these matrices is not known previously. This is the value of our direct sum formulas. In this subsection, we will talk about the second understanding in detail.

Let $\{B_\lambda :{\lambda \in \Lambda }\}$ be a set of irreducible skew-symmetrizable matrices, denote by $\langle B_\lambda :{\lambda \in \Lambda } \rangle$ the set of skew-symmetrizable matrices whose irreducible principal submatrices are all contained in the set $\{B_\lambda :{\lambda \in \Lambda }\}$. The following corollary follows directly from Theorem 4.10.

Corollary 4.13

If each $B_\lambda$ in $\{B_\lambda :{\lambda \in \Lambda }\}$ has a maximal green sequence, so does any skew-symmetrizable matrix in $\langle B_\lambda :{\lambda \in \Lambda } \rangle$.

The existence of maximal green sequences for quivers (or say skew-symmetric matrices) of finite type or from surfaces has a clear answer (see [2, 13]). Now we use Corollary 4.13 to give many skew-symmetric matrices which are not of finite type or from surfaces but each of them admits a maximal green sequence.

Let $B=\begin{pmatrix}0&{}1&{}-1\\ -1&{}0&{}1\\ 1&{}-1&{}0 \end{pmatrix}$, which is irreducible. By Example 2.8, $\mathbf{k}=(2,3,1,2)$ is a maximal green sequence of B. It is known that any skew-symmetric matrix from a surface has entries $\pm 2,\pm 1,0$ (see [5]), and any skew-symmetric matrix of finite type has entries $\pm 1,0$ (see [6]). Thus, $\langle B \rangle$ contains many matrices which are not of finite type or from surface. For example, $\begin{pmatrix}B&{}mI_3\\ -mI_3&{}B \end{pmatrix}\in \langle B \rangle$ is not of finite type or from surface for $m\ge 3$. By Corollary 4.13, we can get each of such matrices has a maximal green sequence. The existence of maximal green sequences for such matrices is not clear previously.