Cluster algebras and symmetrizable matrices

TL;DR

This paper explores the combinatorial properties of quasi-Cartan companions in acyclic skew-symmetrizable cluster algebras, revealing that their associated diagrams always contain an admissible cut of edges.

Contribution

It demonstrates that diagrams of skew-symmetrizable matrices in acyclic cluster algebras have an admissible cut, advancing understanding of their combinatorial structure.

Findings

Diagrams have an admissible cut of edges.

Properties of quasi-Cartan companions are characterized.

Results apply to all acyclic skew-symmetrizable cluster algebras.

Abstract

In this paper, we study combinatorial properties of quasi-Cartan companions defined by the c-vectors of acyclic skew-symmetrizable cluster algebras. In particular, we show that the diagram of any skew-symmetrizable matrix associated with an acyclic cluster algebra has an admissible cut of edges.

Equations

$$B_{ij}^{\prime }=\{\begin{array}{ll}-B_{ij}& \text{if i=k or j=k;}\\ B_{ij}+[B_{ik}{]}_{+}[B_{kj}{]}_{+}-[-B_{ik}{]}_{+}[-B_{kj}{]}_{+}& \text{otherwise.}\end{array}$$

$${\mathbf{c}}_i^{\prime }=\{\begin{array}{ll}-{\mathbf{c}}_i& \text{if i=k};\\ {\mathbf{c}}_i+[sgn({\mathbf{c}}_k)B_{k,i}{]}_{+}{\mathbf{c}}_k& \text{if i=k}.\end{array}$$

Full text

The author’s research was supported in part by the Turkish Research Council (TUBITAK)

1. Introduction

In the structure theory of cluster algebras, principle coefficients are parametrized by a family of integer vectors, called $\mathbf{c}$-vectors. Each $\mathbf{c}$-vector with respect to an acyclic initial seed is a real root of the corresponding root system; furthermore, the $\mathbf{c}$-vectors associated with any seed defines a symmetrizable quasi-cartan companion for the corresponding exchange matrix [8, Corollary 3.29]. In this paper, we study basic combinatorial properties of these companions. In particular, we show that $\mathbf{c}$-vectors define an admissible cut of edges in the associated diagrams.

To state our results, we need some terminology. Let us recall that an $n\times n$ integer matrix $B$ is skew-symmetrizable if there is a diagonal matrix $D$ with positive diagonal entries such that $DB$ is skew-symmetric. We denote by $\mathbb{T}_{n}$ an $n$-regular tree whose edges are labeled by the numbers $1,\dots,n$ such that the $n$ edges incident to each vertex have different labels. The notation t\!\begin{array}[]{c}\scriptstyle{k}\\[-7.22743pt] -\!\!\!-\!\!\!-\\[-7.22743pt] {}\scriptstyle\hfil\end{array}\!t^{\prime} indicates that vertices $t,t^{\prime}\in\mathbb{T}_{n}$ are connected by an edge labeled by $k$. We fix a vertex $t_{0}$ in $\mathbb{T}_{n}$ and assign the pair $(\mathbf{c}_{0},B_{0})$, where $\mathbf{c}_{0}$ is the tuple of standard basis and $B_{0}$ is a skew-symmetrizable matrix. Then, to every vertex $t\in\mathbb{T}_{n}$ we assign a pair, called a $Y$-seed, $(\mathbf{c}_{t},B_{t})$, where $\mathbf{c}_{t}=(\mathbf{c}_{1},...,\mathbf{c}_{n})$ with each $\mathbf{c}_{i}=\mathbf{c}_{i;t}=(c_{1},...,c_{n})\in\mathbb{Z}^{n}$ being non-zero and having either all entries nonnegative or all entries nonpositive; we write $sgn(\mathbf{c}_{i})=+1$ or $sgn(\mathbf{c}_{i})=-1$ respectively and call it a $\mathbf{c}$-vector. Furthermore, for any edge t\!\begin{array}[]{c}\scriptstyle{k}\\[-7.22743pt] -\!\!\!-\!\!\!-\\[-7.22743pt] {}\scriptstyle\hfil\end{array}\!t^{\prime}, the $Y$-seed $(\mathbf{c}^{\prime},B^{\prime})=(\mathbf{c}_{t^{\prime}},B_{t^{\prime}})$ is obtained from $(\mathbf{c},B)=(\mathbf{c}_{t},B_{t})$ by the $Y$-seed mutation $\mu_{k}$ defined as follows, where we denote $[b]_{+}=\max(b,0)$:

• •

The entries of the matrix $B^{\prime}=(B^{\prime}_{ij})$ are given by

[TABLE]

• •

The tuple $\mathbf{c}^{\prime}=(\mathbf{c}_{1}^{\prime},\dots,\mathbf{c}_{n}^{\prime})$ is given by

[TABLE]

By [4, Corollary 5.5], each $\mathbf{c}^{\prime}_{i}=(c^{\prime}_{1},...,c^{\prime}_{n})$ also has either all entries nonnegative or all entries nonpositive. The matrix $B^{\prime}$ is skew-symmetrizable with the same choice of $D$; we write $B^{\prime}=\mu_{k}(B)$ and call the transformation $B\mapsto B^{\prime}$ the matrix mutation. For the $Y$-seeds, we denote $\mu_{k}(\mathbf{c},B)=(\mathbf{c}^{\prime},B^{\prime})$; we call $(\mathbf{c}_{0},B_{0})$ the initial $Y$-seed. It is well known that mutation is an involutive operation.

Let us also recall that the diagram of a skew-symmetrizable $n\times n$ matrix ${B}$ is the directed graph $\Gamma({B})$ whose vertices are the indices $1,2,...,n$ such that there is a directed edge from $i$ to $j$ if and only if ${B}_{j,i}>0$, and this edge is assigned the weight $|B_{ij}B_{ji}|\,$. The diagram $\Gamma(B)$ is called acyclic if it has no oriented cycles. Then there is a corresponding generalized Cartan matrix $A$ such that $A_{i,i}=2$ and $A_{i,j}=-|B_{i,j}|$ for $i\neq j$. There is also the associated root system in the root lattice spanned by the simple roots $\alpha_{i}$ [6]. For each simple root $\alpha_{i}$, the corresponding reflection $s_{\alpha_{i}}=s_{i}$ is the linear isomorphism defined on the basis of simple roots as $s_{i}(\alpha_{j})=\alpha_{j}-A_{i,j}\alpha_{i}$. Then the real roots are defined as the vectors obtained from the simple roots by a sequence of reflections. It is well known that the coordinates of a real root with respect to the basis of simple roots are either all nonnegative or all nonpositive, see [6] for details.

On the other hand, an $n\times n$ matrix $A$ is called symmetrizable if there exists a symmetrizing diagonal matrix $D$ with positive diagonal entries such that $DA$ is symmetric. A quasi-Cartan companion (or ”companion” for short) of a skew-symmetrizable matrix $B$ is a symmetrizable matrix $A$ such that $A_{i,i}=2$, $|A_{i,j}|=|B_{i,j}|$ for all $i\neq j$.

A fundamental relation between $Y$-seeds and symmetrizable matrices has been given in [8, Corollary 3.29] as follows:

Theorem 1.1**.**

[8, Corollary 3.29]** Suppose that the initial seed $(\mathbf{c}_{0},B_{0})$ is acyclic. Then, for any $Y$-seed $(\mathbf{c}_{t},B_{t})$, $t\in\mathbb{T}_{n}$, each $\mathbf{c}$-vector $\mathbf{c}_{i}=\mathbf{c}_{i;t}$ is the coordinate vector of a real root with respect to the basis of simple roots in the corresponding root system. Furthermore, $A=A_{t}=(\langle\mathbf{c}_{j},{\mathbf{c}}^{\vee}_{i}\rangle)$, the matrix of the pairings between the roots and the coroots, is a quasi-Cartan companion of the skew-symmetrizable matrix $B=B_{t}$.

(The matrices $A_{t}$ are symmetrizable with the same choice of a symmetrizing matrix $D$, which is also skew-symmetrizing for all $B_{t}$.)

An important combinatorial property related to quasi-Cartan companions is admissibility [9, 10], which is a generalization of the notion of a generalized Cartan matrix. More precisely, a quasi-Cartan companion $A$ of a skew-symmetrizable matrix $B$ admissible if, for any oriented (resp. non-oriented) cycle $Z$ in $\Gamma(B)$, there is exactly an odd (resp. even) number of edges $\{i,j\}$ such that $A_{i,j}>0$. If $\Gamma(B)$ is acyclic, then the associated generalized Cartan matrix is admissible. Our first result generalizes this property by showing that the quasi-Cartan companions defined by $\mathbf{c}$-vectors are also admissible:

Theorem 1.2**.**

In the set-up of Theorem 1.1, the quasi-Cartan companion $A$ has the following properties:

• •

Every directed path of the diagram $\Gamma(B)$ has at most one edge $\{i,j\}$ such that $A_{i,j}>0$.

• •

Every oriented cycle of the diagram $\Gamma(B)$ has exactly one edge $\{i,j\}$ such that $A_{i,j}>0$.

• •

Every non-oriented cycle of the diagram $\Gamma(B)$ has an even number of edges $\{i,j\}$ such that $A_{i,j}>0$.

In particular, the quasi-Cartan companion $A$ is admissible. Furthermore, any admissible quasi-Cartan companion of $B$ can be obtained from $A$ by a sequence of simultaneous sign changes in rows and columns.

The special case of this theorem when $B$ is skew-symmetric was obtained in [10, Theorem 1.4] by the author. Let us also recall from [10] that a set $C$ of edges in $\Gamma(B)$ is called an ”admissible cut” if every oriented cycle contains exactly one edge that belongs to $C$ and every non-oriented cycle contains exactly an even number of edges in $C$. Thus, in the setup of the theorem, the $\mathbf{c}$-vectors define an admissible cut of edges: the set of edges $\{i,j\}$ in $\Gamma(B)$ such that $A_{i,j}>0$ is an admissible cut. For skew-symmetric matrices, this notion has been applied to the representation theory of algebras in [5, BRS].

Our next result is the following explicit description of the quasi-Cartan companions defined by the $\mathbf{c}$-vectors:

Theorem 1.3**.**

In the set-up of Theorem 1.1, the quasi-Cartan companion $A$ has the following properties:

• •

If $sgn(B_{j,i})=sgn(\mathbf{c}_{j})$, then $A_{j,i}=-sgn(\mathbf{c}_{j})B_{j,i}=-|B_{j,i}|$.

• •

If $sgn(B_{j,i})=-sgn(\mathbf{c}_{j})$, then $A_{j,i}=sgn(\mathbf{c}_{i})B_{j,i}=-sgn(\mathbf{c}_{i})sgn(\mathbf{c}_{j})|B_{j,i}|$.

In particular; if $sgn(\mathbf{c}_{j})=-sgn(\mathbf{c}_{i})$, then $B_{j,i}=sgn(\mathbf{c}_{i})A_{j,i}$.

Let us note that the special case of this theorem when $B$ is skew-symmetric was obtained in [10, Theorem 1.3] by the author. We will prove this more general theorem using [8, Corollary 3.29], which has been given above as Theorem 1.1. (Note that the statement [8, Corollary 3.29] was not present in the earlier versions of [8]).

Corollary 1.4**.**

In the setup of Theorem 1.3, suppose that t\!\begin{array}[]{c}\scriptstyle{k}\\[-7.22743pt] -\!\!\!-\!\!\!-\\[-7.22743pt] {}\scriptstyle\hfil\end{array}\!t^{\prime} in $\mathbb{T}_{n}$ Then, for $\mu_{k}(\mathbf{c},B)=(\mathbf{c}^{\prime},B^{\prime})$, we have the following: if $\mathbf{c}^{\prime}_{i}\neq\mathbf{c}_{i}$, then $\mathbf{c}^{\prime}_{i}=s_{\mathbf{c}_{k}}(\mathbf{c}_{i})$, where $s_{\mathbf{c}_{k}}$ is the reflection with respect to the real root $\mathbf{c}_{k}$ and $\mathbb{Z}^{n}$ is identified with the root lattice.

Let us also note that Theorem 1.3 could be useful for recognizing mutation classes of acyclic diagrams: a diagram that does not have an admissible quasi-Cartan companion can not be obtained from any acyclic diagram by a sequence of mutations. An example of such a diagram is given in Figure 1. (We refer to [9, Section 2] for properties of diagrams of skew-symmetrizable matrices). Another application of the admissibility property to the corresponding Weyl groups can be found in [11], where a fundamental class of relations have been shown to be satisfied by the reflections of the $\mathbf{c}$-vectors.

2. Proofs of main results

Let us first recall the following well-known property of root systems: For a generalized Cartan matrix $A$ with symmetrizing matrix $D={diag}(d_{1},...,d_{n})$, there is an invariant symmetric bilinear form $(,)$ defined on the simple roots as $(\alpha_{i},\alpha_{j})=d_{i}A_{i,j}=d_{j}A_{j,i}=(\alpha_{j},\alpha_{i})$. Let us note that, for any real root $\alpha$, the corresponding reflection $s_{\alpha}$ is defined on the real roots as $s_{\alpha}(\beta)=\beta-\langle\beta,{\alpha}^{\vee}\rangle\alpha$, with $\langle\beta,{\alpha}^{\vee}\rangle=\dfrac{2(\alpha,\beta)}{(\alpha,\alpha)}$. In particular, $s_{\alpha_{i}}(\alpha_{j})=\alpha_{j}-\langle\alpha_{j},{\alpha_{i}}^{\vee}\rangle\alpha_{i}=\alpha_{j}-A_{i,j}\alpha_{i}$.

Let us also recall the mutation of quasi Cartan companions [10, Definition 1.6]. Suppose that $B$ is a skew-symmetrizable matrix and let $A$ be a quasi-Cartan companion of $B$. Let $k$ be an index. For each sign $\epsilon=\pm 1$, ”the $\epsilon$-mutation of $A$ at $k$” is the quasi-Cartan matrix $\mu_{k}^{\epsilon}(A)=A^{\prime}$ such that for any $i,j\neq k$: $A^{\prime}_{i,k}=\epsilon sgn(B_{k,i})A_{i,k}$, $A^{\prime}_{k,j}=\epsilon sgn(B_{k,j})A_{k,j}$, $A^{\prime}_{i,j}=A_{i,j}-sgn(A_{i,k}A_{k,j})[B_{i,k}B_{k,j}]_{+}$. In the setup of Theorem 1.1, suppose that t\!\begin{array}[]{c}\scriptstyle{k}\\[-7.22743pt] -\!\!\!-\!\!\!-\\[-7.22743pt] {}\scriptstyle\hfil\end{array}\!t^{\prime} in $\mathbb{T}_{n}$ and let $A$ and $A^{\prime}$ be the associated quasi-Cartan companions. Then $A^{\prime}=\mu^{\epsilon}_{k}(A)$ for $\epsilon=sgn(\mathbf{c}_{k})$.

We first prove Theorem 1.3 for convenience:

Proof of Theorem 1.3. To prove the first part, let us suppose that $sgn(B_{j,i})=sgn(\mathbf{c}_{j})$. Let $\mu_{j}(\mathbf{c},B)=(\mathbf{c}^{\prime},B^{\prime})$ with $B^{\prime}=\mu_{j}(B)$. Then $\mathbf{c}_{i}^{\prime}=\mathbf{c}_{i}+[sgn(\mathbf{c}_{j})B_{j,i}]_{+}\mathbf{c}_{j}=\mathbf{c}_{i}+sgn(B_{j,i})B_{j,i}\mathbf{c}_{j}=\mathbf{c}_{i}+|B_{j,i}|\mathbf{c}_{j}$. We denote by $(,)$ the invariant symmetric bilinear form defined by $A_{0}$ on the root lattice and let $D=diag(d_{1},...,d_{n})$ be the symmetrizing matrix for $A_{0}$. Note that, by Theorem 1.1, we have the following: $2d_{i}=({\mathbf{c}^{\prime}_{i}},\mathbf{c}^{\prime}_{i})=(\mathbf{c}_{i},\mathbf{c}_{i})$, $2d_{j}=(\mathbf{c}_{j},\mathbf{c}_{j})$, $(\mathbf{c}_{j},\mathbf{c}_{i})=(\mathbf{c}_{i},\mathbf{c}_{j})=d_{i}A_{{i,j}}=d_{j}A_{j,i}$. Then $2d_{i}=({\mathbf{c}^{\prime}_{i}},\mathbf{c}^{\prime}_{i})=(\mathbf{c}_{i}+|B_{j,i}|\mathbf{c}_{j},\mathbf{c}_{i}+|B_{j,i}|\mathbf{c}_{j})=(\mathbf{c}_{i},\mathbf{c}_{i})+(\mathbf{c}_{i},|B_{j,i}|\mathbf{c}_{j})+(|B_{j,i}|\mathbf{c}_{j},\mathbf{c}_{i})+|B_{j,i}|(\mathbf{c}_{j},|B_{j,i}|\mathbf{c}_{j})=(\mathbf{c}_{i},\mathbf{c}_{i})+2|B_{j,i}|(\mathbf{c}_{j},\mathbf{c}_{i})+|{B}_{j,i}|^{2}(\mathbf{c}_{j},\mathbf{c}_{j})=2d_{i}+2|B_{j,i}|(\mathbf{c}_{j},\mathbf{c}_{i})+|B_{j,i}|^{2}2d_{j}=2d_{i}+2|B_{j,i}|d_{j}A_{j,i}+|B_{j,i}|^{2}2d_{j}=2d_{i}+2|B_{j,i}|d_{j}(A_{j,i}+|B_{j,i}|)$, implying that $A_{j,i}+|B_{j,i}|=0$, thus $A_{j,i}=-|B_{j,i}|=-sgn(B_{j,i})B_{j,i}=-sgn(\mathbf{c}_{j})B_{j,i}$.

To prove the second part of the theorem, let us suppose that $sgn(B_{j,i})=-sgn(\mathbf{c}_{j})$. Let $\mu_{i}(\mathbf{c},B)=(\mathbf{c}^{\prime},B^{\prime})$ with $B^{\prime}=\mu_{i}(B)$. Note that $sgn(B^{\prime}_{j,i})=-sgn(B_{j,i})$ and $|B^{\prime}_{j,i}|=|B_{j,i}|$ (by the definition of mutation). Let $A^{\prime}$ be the quasi-Cartan companion associated to the $Y$-seed $(\mathbf{c}^{\prime},B^{\prime})$ (Theorem 1.1 ), (Note then that $A^{\prime}=\mu^{\epsilon}_{i}(A)$ where $\epsilon=sgn(\mathbf{c}_{i})$).

For the proof, we first assume that $sgn(\mathbf{c}_{j})=-sgn(\mathbf{c}_{i})$. Then we have $sgn({\mathbf{c}}_{i})=sgn(B_{j,i})$, so ${\mathbf{c}^{\prime}}_{j}=\mathbf{c}_{j}$ and ${\mathbf{c}^{\prime}}_{i}=-\mathbf{c}_{i}$, implying $sgn({\mathbf{c}^{\prime}}_{j})=sgn({\mathbf{c}}_{j})=-sgn(B_{j,i})=sgn(B^{\prime}_{j,i})$, i.e. for the $Y$-seed $(\mathbf{c}^{\prime},B^{\prime})$, we have $sgn(B^{\prime}_{j,i})=sgn({\mathbf{c}^{\prime}}_{j})$. Thus, by the first part of the theorem, we have $-|B^{\prime}_{j,i}|=A^{\prime}_{j,i}=-A_{j,i}$. Thus $A_{j,i}=|B^{\prime}_{j,i}|=|B_{j,i}|=-sgn(\mathbf{c}_{i})sgn(\mathbf{c}_{j})|B_{j,i}|$.

Let us now assume that $sgn(\mathbf{c}_{j})=sgn(\mathbf{c}_{i})$. Then, since we have assumed $sgn(B_{j,i})=-sgn(\mathbf{c}_{j})$, we have $sgn({\mathbf{c}}_{i})=-sgn(B_{j,i})=sgn(B_{i,j})$. Then, by the first part of the theorem, we have $A_{i,j}=-|B_{i,j}|$. Thus, since $A$ is symmetrizable and a quasi-Cartan companion, we also have $A_{j,i}=-|B_{j,i}|$, which is equal to $-sgn(\mathbf{c}_{i})sgn(\mathbf{c}_{j})|B_{j,i}|$.

On the other hand, our assumption $sgn(B_{j,i})=-sgn(\mathbf{c}_{j})$ implies the following: $-sgn(\mathbf{c}_{i})sgn(\mathbf{c}_{j})|B_{j,i}|=-sgn(\mathbf{c}_{i})sgn(\mathbf{c}_{j})sgn(B_{j,i})B_{j,i}=\\ -sgn(\mathbf{c}_{i})sgn(\mathbf{c}_{j})(-sgn(\mathbf{c}_{j}))B_{j,i}=sgn(\mathbf{c}_{i})B_{j,i}$. This completes the proof.

Proof of Corollary 1.4. Let us note that for $\mu_{k}(\mathbf{c},B)=(\mathbf{c}^{\prime},B^{\prime})$ we have the following: $\mathbf{c}^{\prime}_{k}=-\mathbf{c}_{k}$; $\mathbf{c}^{\prime}_{i}=\mathbf{c}_{i}+[sgn(\mathbf{c}_{k})B_{k,i}]_{+}\mathbf{c}_{k}$ if $i\neq k$ by (1.2). On the other hand, $[sgn(\mathbf{c}_{k})B_{k,i}]_{+}\neq 0$ if and only if $sgn(\mathbf{c}_{k})B_{k,i}>0$ if and only if $sgn(\mathbf{c}_{k})=sgn(B_{k,i})$. Then, by Theorem 1.3, we have $[sgn(\mathbf{c}_{k})B_{k,i}]_{+}=-A_{k,i}$. Thus $\mathbf{c}^{\prime}_{i}=\mathbf{c}_{i}-A_{k,i}\mathbf{c}_{k}=s_{\mathbf{c}_{k}}(\mathbf{c}_{i})$ by the definition of a reflection. Also $\mathbf{c}^{\prime}_{k}=-\mathbf{c}_{k}=s_{\mathbf{c}_{k}}(\mathbf{c}_{k})$ This completes the proof of the statement.

Proof of Theorem 1.2. As we discussed in Section 1, the special case of this theorem when $B$ is skew-symmetric was obtained in [10, Theorem 1.4] by the author. The proof in [10] uses only the general properties of the mutations of skew-symmetrizable matrices with quasi-Cartan companions and the properties given in Theorem 1.3 (which was obtained for skew-symmetric matrices in [10, Theorem 1.3]; note that in this case the companion $A$ is symmetric and $A_{i,j}={\mathbf{c}_{i}}^{T}A_{0}\mathbf{c}_{j}$). Since we have proved Theorem 1.3 above for skew-symmetrizable matrices, the proof of [10, Theorem 1.4] also holds for the skew-symmetrizable matrices. Thus, for the proof of Theorem 1.2, we refer the reader to the proof of [10, Theorem 1.4].