Mutation classes of skew-symmetric 3 × 3−matrices

Ibrahim Assem  Martin Blais  Thomas Brüstle

Audrey Samson^{∗}

Dedicated to the memory of A.V.Roiter

Abstract

In this paper, we establish a bijection between the set of mutation classes of mutation-cyclic skew-symmetric integral 3×3−matrices and the set of triples of integers (a, b, c) such that 2 ≤ a ≤ b ≤ c and ab ≥ c. We also give an algorithm allowing to verify whether a matrix is mutation-cyclic or not. We prove that given a, b, the two cases depend on whether c is large enough or not.

1 Introduction

Cluster algebras have been introduced and studied by Fomin and Zelevinsky in [4, 5, 2]. The class of acyclic cluster algebras has been defined in [2], and it was shown in [5] that every cluster algebra of finite type is acyclic and corresponds to a Dynkin quiver. Later, a fruitful connection between the study of acyclic cluster algebras and the representation theory of finite dimensional algebras has been established in [3]. This led to several important developments in representation theory, notably to the introduction of the class of cluster-tilted algebras which provides a generalization of classical tilting theory (see, for instance [3]). Since cluster-tilted algebras can be defined only for an acyclic cluster algebra, it is natural to ask for a criterion allowing to verify whether a cluster algebra is acyclic or not. The objective of the present note is to study the first non-trivial situation, that of the (coefficient-free) cluster algebras of rank three which are given by a square skew-symmetric matrix.

∗Ibrahim Assem is partially supported by NSERC of Canada and the Université de Sherbrooke. Martin Blais and Audrey Samson were working under a summer research fellowship of NSERC. Thomas Brüstle is partially supported by NSERC, by Bishop’s University and the Université de Sherbrooke.

In general, for any square skew-symmetric integral matrix, the associated coefficient-free cluster algebra depends not on the matrix itself, but rather on its mutation class within the set Skew_{n}(ℤ) of the skew-symmetric integral n × n−matrices (see [4]). Every matrix B = [b_{ij}] in Skew_{n}(ℤ) determines a quiver Q_{B} having {1, …, n} as set of points, and b_{ij} arrows from i to j whenever b_{ij} > 0. Thus, mutations on B can equivalently be expressed as mutations on the quiver Q_{B} (see (2.1)). The associated cluster algebra is called acyclic if there is a matrix in the mutation class of B whose quiver is acyclic, and otherwise it is called cyclic (see [5]). We say that a matrix B ∈ Skew_{n}(ℤ) is cyclic (or acyclic) if the quiver Q_{B} is so, and we say that B is mutation-cyclic (or mutation-acyclic) if the corresponding cluster algebra is cyclic (or acyclic, respectively).

In this paper, we consider the case where n = 3: this corresponds to quivers with three points. As we see in (2.2) below, the mutation class of a connected quiver with three points always contains a cyclic representative which is determined by three positive integral parameters a, b, c corresponding to the number of arrows on each side. Up to orientation, we may suppose that a ≤ b ≤ c. Accordingly, the cyclic matrices B ∈ Skew_{3}(ℤ) are determined, up to transposition and simultaneous permutation of rows and columns, by the three parameters a ≤ b ≤ c. Our first theorem characterizes mutationcyclic quivers (or, equivalently, matrices), in terms of these parameters. Note that we consider all matrices up to transposition and permutation.

Theorem 1.1 There exists a bijection between the set of mutation classes of mutation-cyclic matrices in Skew_{3}(ℤ) and the set of triples (a, b, c) of integers such that 2 ≤ a ≤ b ≤ c and ab ≥ 2c.

Our proof gives at the same time a handy algorithm allowing to verify whether a given matrix (or the corresponding quiver) is mutation-cyclic or not. In our second theorem, we show that for fixed a, b the two cases (mutation-cyclic and mutation-acyclic) depend on whether the third parameter c is large enough or not.

Theorem 1.2 Let 2 ≤ a ≤ b. Then there exists a unique integer c_{0} ∈ [ab − b, ab − 1] such that a cyclic matrix in Skew_{3}(ℤ) which is represented by the triple (a, b, c) with a ≤ b ≤ c is mutation-acyclic if and only if c > c_{0}.

The proofs of these theorems are purely combinatorial. The problem of characterizing the mutation-cyclic skew-symmetric 3×3−matrices is also considered in [1] from the geometrical point of view. In case the mutation class of a mutation-acyclic skew-symmetric 3 × 3−matrix is infinite, the corresponding cluster-tilted algebras are studied in [6] . The paper is organized as follows. In section 2, after a brief preliminary discussion, we prove Theorem 1.1 and state our algorithm. Section 3 contains the proof of Theorem 1.2 and ends with some examples.

2 Mutation classes of quivers with three points

2.1 Preliminaries

Since our intuition is graphical, we work with quivers rather than with matrices. For any quiver Q, we denote by |Q_{1}| the number of arrows of Q. The map B ↦ Q_{B} yields a bijection between Skew_{n}(ℤ) and the set Q_{n} of quivers with n points having neither loops nor cycles of length two. We begin by discussing the effect of certain matrix operations on the corresponding quivers. When constructing the associated cluster algebra, the matrix B is given only up to simultaneous row and column permutations. This corresponds to considering the quivers Q_{B} up to isomorphism. The transposition B ↦ B^{T} of the matrix B corresponds to forming the opposite quiver. Therefore, we can consider our quivers up to isomorphism and changes of orientation. We shall always do so, in order to formulate our results in a more concise way. The mutation at k of a matrix B as defined in [4] yields a mutation of a quiver Q ∈ Q_{n} at the point k as follows:

(1) All arrows passing through k are reversed.

(2) If Q has r_{ij} paths of length two from i to j passing through k, then we add r_{ij} arrows from i to j.

(3) If, in the resulting quiver, there are s_{ij} arrows from i to j, and t_{ij} arrows from j to i, with s_{ij} ≥ t_{ij}, we replace all these arrows by s_{ij} −t_{ij} arrows from i to j.

Every mutation is an involution on Q_{n}. Taking the reflexive and transitive closure yields an equivalence relation denoted by ∼. If Q′ is obtained from Q by a mutation at the point k, we also write Q ∼^{k} Q′.

2.2 Mutation classes in Q_{3}

When we deal with a cyclic quiver Q ∈ Q_{3}, we always represent it as

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

where a, b, c (called the parameters of Q) represent the number of arrows in the shown direction. Up to isomorphism of quivers and replacing the cyclic quiver Q by its opposite quiver, we can suppose that 0 < a ≤ b ≤ c.

Lemma 2.1 Let Q be a connected quiver in Q_{3}, then

(1) Q is mutation-equivalent to a cyclic quiver.

(2) Q is mutation-acyclic if and only if Q is mutation-equivalent to the

quiver Q′

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

where 0 < a ≤ b ≤ c and ab ≤ c.

Proof. (1) If Q is acyclic, it is of the form

\xymatrix{ & 1 \ar[dr]^{s} \\ 2 \ar[rr]^{t} \ar[ur]^{r} &  & 3}

where r, s, t represent the numbers of arrows in the shown direction. Since Q is connected, we may assume that at most one of r, s, t is zero. If r and s are non-zero, then mutation at 1 yields the quiver Q′′

\xymatrix{ & 1 \ar[dl]_{r} \\ 2 \ar[rr]^{rs+t} &  & 3 \ar[ul]_{s}}

which is cyclic because rs + t ≥ rs > 0. If r and t are non-zero, then mutation at 3 yields the quiver Q′′′

\xymatrix{ & 1 \\ 2 \ar[ur]^{r} &  & 3 \ar[ul]_{s} \ar[ll]^{t}}

and thus, as before, mutation at 2 yields a cyclic quiver. The case where s and t are non-zero is dual.

(2) The quiver Q′ from (2) is clearly mutation-acyclic, since mutating at 1 yields an acyclic quiver when ab ≤ c. Conversely, starting with an acyclic quiver Q, we obtain in the proof of (1) the quivers Q′′ and Q′′′ which satisfy the required condition. □

2.3 Mutation-finite quivers

A quiver in Q_{n} is called mutation-finite if its mutation class is a finite set. The results in this subsection can also be derived from [7] and [8]. We recall from [8] that a sign skew-symmetric matrix B (in the sense of [5]) is 4−bounded if every matrix B′ = (b′_{ij}) which is mutation-equivalent to B is sign skew-symmetric and |b′_{ij}b′_{ji}| ≤ 4 for all i and j.

Lemma 2.2 Let Q be a connected quiver in Q_{3}, then Q is mutation-finite if and only if any cyclic quiver Q′ in its mutation class has at most two parallel arrows.

Proof. Sufficiency. There are only four connected cyclic quivers in Q_{3} with at most two parallel arrows:

[figure: 1; 1; 2; 1; 3; 1; 1; 1; 2; 1; 3; 2; 1; 2; 2; 2; 3; 2; 1; 2; 2; 2; 3; 1]

By [8], the first three quivers are 4-bounded, hence mutation-finite. The last one does not satisfy the hypothesis, because mutation at 1 yields a cyclic quiver with 3 arrows from 3 to 2. Necessity. Assume Q′ ∼ Q with Q′ of the form

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

where 0 < a ≤ b ≤ c and c ≥ 3. Mutating at 3 yields a quiver Q′′ having bc − a arrows from 1 to 2. Since c ≥ 3 and b ≥ a, then bc − a ≥ 2a > a. Thus |Q′′_{1}| > |Q′_{1}|. Induction shows that we may repeat this procedure arbitrarily many times leading to quivers with more and more arrows. Therefore Q is not mutation-finite. □

Corollary 2.3 There are exactly three mutation classes of mutation-finite connected quivers in Q_{3}, given by the following representatives:

[figure: 1; 1; 2; 1; 3; 1; 1; 2; 1; 1; 3; 1; 2; 2; 2; 3; 2]

Of these, only the last quiver is mutation-cyclic. Moreover, it is the only representative in its mutation class.

2.4 Increasing the number of arrows

Consider a cyclic quiver given by parameters 0 < a ≤ b ≤ c. Then Lemma 2.1, (2) implies that the quiver is mutation-acyclic if a = 1. Thus, since we are interested in mutation-cyclic quivers, we may assume that (a, b, c) ∈ ℤ^{3} satisfy 2 ≤ a ≤ b ≤ c. The following lemma shows when such a quiver stays cyclic under one mutation. It also describes how the parameters change.

Lemma 2.4 Let Q be the cyclic quiver

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

with 2 ≤ a ≤ b ≤ c, and Q′ be obtained from Q by a mutation at a point k ∈ {1, 2, 3}.

(1) If k = 1, then Q′ is cyclic with |Q′_{1}| ≥ |Q_{1}| if and only if ab ≥ 2c.

(2) If k = 2, then Q′ is cyclic with the parameters satisfying 2 ≤ a ≤ c ≤ ac − b.

(3) If k = 3, then Q′ is cyclic with the parameters satisfying 2 ≤ b ≤ c ≤ bc − a.

Proof. Assume first k = 1.

i) If c ≥ ab, then |Q′_{1}| = a + b + (c − ab) = |Q_{1}| − ab < |Q_{1}|.

ii) If c < ab, then |Q′_{1}| = a + b + (ab − c) = |Q_{1}| + (ab − 2c).

Thus, Q′ is cyclic when ab > c, and |Q′_{1}| ≥ |Q_{1}| if and only if ab ≥ 2c.

Assume now k = 2. Then the numbers of arrows of Q′ are a, c and ac − b, respectively. Clearly 2 ≤ a ≤ c. Moreover, since a ≥ 2 and c ≥ b we have ac ≥ c + c ≥ b + c, thus ac − b ≥ c which is the inequality we wanted. The proof is similar for k = 3. □

Remark. Parts (2) and (3) of the preceding lemma show that mutation in a point opposite to one of the parameters a or b necessarily yields a new maximal parameter. This new parameter is strictly greater than the two other parameters unless a = 2 and b = c in (2) or a = b = c = 2 in case (3).

Proposition 2.5 Let Q be the cyclic quiver

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

with 2 ≤ a ≤ b ≤ c. If ab ≥ 2c, then Q is mutation-cyclic and, for any Q′ ∼ Q, we have |Q′_{1}| ≥ |Q_{1}|.

Proof. If Q′ ∼ Q, there exists a sequence of mutations

Q = Q^{(0)} ∼^{k_{1}} Q^{(1)} ∼^{k_{2}} ⋯ ∼^{k_{n}} Q^{(n)} = Q′

Without loss of generality, we may assume this sequence to be reduced, that is, two consecutive mutations in this sequence are not inverse to each other (thus, k_{i} ≠ k_{i+1} for all i ). We show by induction on n that the quiver Q^{(n)} is cyclic (and so are all quivers Q^{(i)} in the sequence) and that |Q_{1}^{(n)}| ≥ |Q_{1}^{(n−1)}|. Lemma 2.4 shows that Q^{(1)} is cyclic with |Q_{1}^{(1)}| ≥ |Q_{1}|, thus we consider the induction step. Suppose n ≥ 2, and let Q^{(n−1)} be the quiver

\xymatrix{ & 1′ \ar[dr]^{b′} \\ 2′ \ar[ur]^{a′} &  & 3′ \ar[ll]^{c′}}

which is cyclic by induction. Up to duality, we can suppose a′ ≤ b′ ≤ c′, and by induction, the numbers of arrows cannot decrease by mutations, thus 2 ≤ a′. The quiver Q^{(n)} is obtained from Q^{(n−1)} by mutation at k_{n}. If k_{n} ≠ 1′ then the statement follows by lemma 2.4. Assume now that k_{n} = 1′. Since the sequence above is supposed to be minimal, the mutation preceding k_{n} satisfies k_{n−1} ≠ 1′. This means that one of the values a′ or b′ has been changed when going from Q^{(n−2)} to Q^{(n−1)}. But we know that a′ ≤ b′ ≤ c′, therefore the remark following lemma 2.4 shows that we are in one of the cases a′ = 2 and b′ = c′ or a′ = b′ = c′ = 2. If a′ = b′ = c′ = 2, then all the quivers Q^{(i)} are isomorphic, and the statement holds. If a′ = 2 and b′ = c′, then mutation at 1′ transforms the number c′ in Q^{(n−1)} into a′b′ −c′ = 2b′ −b′ = b′ = c′, thus it stays the same, which implies that Q^{(n)} is cyclic and |Q′_{1}| ≥ |Q_{1}|, which is what we wanted to show. □

2.5 The main result

In the sequel, we call root (of its mutation class) a cyclic quiver Q ∈ Q_{3} whose parameters (a, b, c) satisfy 2 ≤ a ≤ b ≤ c and ab ≥ 2c.

The previous results yield the following algorithm which decides whether a given connected quiver Q ∈ Q_{3} is mutation-cyclic or not:

(1) If Q is acyclic, stop.

(2) Otherwise, Q is a cyclic quiver with ordered parameters 0 ≤ a ≤ b ≤ c. If ab ≤ c, then Q is mutation-acyclic by lemma 2.1, stop.

(3) Perform a mutation at the point opposite to c. If the number of arrows has decreased, go back to step (1). Otherwise, Q is a root by lemma 2.4, and thus Q is mutation-cyclic.

This procedure must clearly stop after finitely many steps, since we deal with strictly decreasing sequences of natural numbers.

Proof of theorem 1.1. We have shown in proposition 2.5 that every root is mutation-cyclic. Conversely, let Q be a mutation-cyclic quiver. Applying the algorithm above, we find a root in the mutation class of Q. Up to duality, this root is uniquely described by its parameters a ≤ b ≤ c. □

3 Separating the cyclic case from the acyclic

The proof of theorem 1.2 needs some preparation. As usual, we assume our quiver given by the 3 parameters (a, b, c) with a ≤ b ≤ c. We first consider in 3.1 the case where a = 2, then show in 3.2 that, for a and b given, the quiver is mutation-cyclic for c ≤ ab − b and mutation-acyclic for c ≥ ab − 1. The proof of the theorem depends on a particular difference equation which we solve in 3.3.

Lemma 3.1 The quiver Q

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{2} &  & 3 \ar[ll]^{c}}

with 2 ≤ b ≤ c is mutation-cyclic if and only if b = c.

Proof. Sufficiency follows from proposition 2.5, so we only show necessity. Asssume that Q is mutation-cyclic. By applying a sequence of mutations at the point opposed to the maximum, we reach a root Q′

\xymatrix{ & 1′ \ar[dr]^{b′} \\ 2′ \ar[ur]^{2} &  & 3′ \ar[ll]^{c′}}

with 2 ≤ b′ ≤ c′ and 2b′ ≥ 2c′ (because Q′ is a root). Thus b′ = c′. But mutations at 1′ or 2′ do not change the quiver Q′. This implies that Q′ = Q, and so b = b′ = c′ = c. □

Lemma 3.2 Let Q be the quiver

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

with 2 ≤ a ≤ b ≤ c.

(1) If c ≤ ab − b, then Q is mutation-cyclic.

(2) If c ≥ ab − 1, then Q is mutation-acyclic.

Proof. Mutating at 1 yields a quiver Q′ with ab − c arrows between 2 and 3.

(1) If ab − c ≥ b (or, equivalently, c ≤ ab − b), then the new maximal number is ab − c. Since the maximum did not change its position, it follows from lemma 2.4 that Q or Q′ is a root. In particular, Q is mutation-cyclic.

(2) If ab − c ≤ 1 (or, equivalently, c ≥ ab − 1), then Q is mutation-acyclic by lemma 2.1, (2).

□

Lemma 3.3 Consider the homogeneous difference equation

S_{n+2} = aS_{n+1} − S_{n}

(with a ≥ 2, S_{0} = 0 and S_{1} = 1), then

(1) S_{n} = \{ \begin{matrix}\frac{1}{2^{n}\sqrt{a^{2}−4}} [(a + \sqrt{a^{2} − 4})^{n} − (a − \sqrt{a^{2} − 4})^{n}] & if a ≥ 3 \\ n & if a = 2 \end{matrix}

(2) The sequence (S_{n})_{n≥0} is strictly increasing

(3) S_{n} ≥ a for any n ≥ 2.

Proof.

(1) This is straightforward and left to the reader.

(2) Use induction on n: clearly S_{0} < S_{1}. Assume S_{n} < S_{n+1}, then S_{n+2} = aS_{n+1} − S_{n} > aS_{n+1} − S_{n+1} ≥ S_{n+1} because a ≥ 2.

(3) Since S_{2} = a, this follows from (2). □

Proof of theorem 1.2. Let Q be the quiver

\xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

with 2 ≤ a ≤ b ≤ c. By lemma 3.1, the statement holds if a = 2 (with c_{0} = b). Assume a ≥ 3 and that such a c_{0} does not exist. Then there exist quivers Q, Q′

Q \xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} & & 3 \ar[ll]^{c}} Q′ \xymatrix{ & 1 \ar[dr]^{b} \\ 2 \ar[ur]^{a} & & 3 \ar[ll]^{c+1}}

with 3 ≤ a ≤ b ≤ c such that Q is mutation-acyclic and Q′ is mutationcyclic. We show by induction that the quiver Q never reaches an acyclic representative. This will yield a contradiction which implies our statement.

More precisely, we prove by induction on n ≥ 1 that, after n mutations at the point opposed to the maximum, we obtain respectively the quivers Q^{(n)} and Q^{′(n)}

Q^{(n)} \xymatrix{ & 1′ \ar[dr]^{b′} \\ 2′ \ar[ur]^{a} & & 3′ \ar[ll]^{c′}} Q^{′(n)} \xymatrix{ & 1′ \ar[dr]^{b′′} \\ 2′ \ar[ur]^{a} & & 3′ \ar[ll]^{c′′}}

with S_{n} = b′ − b′′, S_{n−1} = c′ − c′′, max(a, b′, c′) = c′ and max(a, b′′, c′′) = c′′ where S_{n} is as in lemma 3.3 and Q^{(n)} is still a cyclic quiver (note that the parameter a does not change). Assume first n = 1. After one mutation at the point opposed to the maximum, we obtain respectively the quivers

Q^{(1)} \xymatrix{ & 1 \ar[dl]_{a} \\ 2 \ar[rr]_{ab−c} & & 3 \ar[ul]_{b}} Q^{′(1)} \xymatrix{ & 1 \ar[dl]_{a} \\ 2 \ar[rr]_{ab−c−1} & & 3 \ar[ul]_{b}}

If Q^{(1)} is not cyclic, then ab − c ≤ 0, hence ab − c − 1 < 0 and so Q^{′(1)} is not cyclic either. This is a contradiction to the hypothesis that Q′ is mutationcyclic, hence Q^{(1)} is a cycle. In order to pursue the algorithm, we must have b > ab − c (otherwise, the next mutation yields Q^{(2)} = Q and so Q or Q^{(1)} is a root, contradicting the hypothesis that Q is mutation-acyclic). Therefore, b > ab − c − 1. Thus, in Q^{(1)} and Q^{′(1)}, the maxima correspond to the same sides. Moreover, S_{1} = (ab − c) − (ab − c − 1) = 1 and S_{0} = b − b = 0.

Assume the statement holds for n. Mutating at 1′ (which is opposed to the respective maxima c′ and c′′) yields respectively the quivers

Q^{(n+1)} \xymatrix{ & 1′ \ar[dl]_{a} \\ 2′ \ar[rr]_{ab′−c′} & & 3′ \ar[ul]_{b′}} Q^{′(n+1)} \xymatrix{ & 1′ \ar[dl]_{a} \\ 2′ \ar[rr]_{ab′′−c′′} & & 3′ \ar[ul]_{b′}}

We first note that (ab′ −c′)−(ab′′ −c′′) = a(b′ −b′′)−(c′ −c′′) = aS_{n} −S_{n−1} = S_{n+1}. Also, Q^{(n+1)} is a cycle. Indeed, if this is not the case, then ab′ − c′ ≤ 0. Since S_{n+1} ≥ 0 (by lemma 3.3), we have ab′′ − c′′ ≤ 0, contradicting the hypothesis that Q′ is mutation-cyclic.

We now determine the maximal parameter in Q^{(n+1)}.

1) Assume max(a, b′, ab′ − c′) = a. By lemma 3.3, we have

(ab′ − c′) − (ab′′ − c′′) = S_{n+1} ≥ a

hence ab′′ − c′′ ≤ (ab′ − c′) − a ≤ 0, contradicting the hypotesis that Q′ is mutation-cyclic.

2) Assume max(a, b′, ab′ − c′) = ab′ − c′. Then, mutating at the point opposed to the maximum yields Q^{(n+2)} = Q^{(n)}, thus Q is mutation-cyclic, a contradiction. Therefore max(a, b′, ab′−c′) = b′. We claim that also max(a, b′′, ab′′−c′′) = b′′. Assume first that a > b′′. Since ab′′ − c′′ ≥ 2, we have

a(ab′ − c′) = a[(ab′′ − c′′) + S_{n+1}] = a(ab′′ − c′′) + aS_{n+1}
≥ 2a + aS_{n+1}
> 2b′′ + aS_{n}
> 2b′′ + 2S_{n} = 2b′.

On the other hand, S_{n+1} = (ab′ − c′) − (ab′′ − c′′) ≥ 0 yields ab′ − c′ ≥ ab′′ − c′′ ≥ 2. These inequalities show, by Theorem 1.1, that Q^{(n+1)} is a root. Thus Q is mutation-cyclic, a contradiction. Therefore a ≤ b′′. Since b′ > ab′ − c′, we have b′′ = b′ − S_{n} > (ab′ − c′) − S_{n+1} = ab′′ − c′′. This shows that b′′ = max(a, b′′, ab′′−c′′), completes the proof of the induction statements and thus establishes the theorem. □

Example 1. Let Q be the quiver

\xymatrix{ & 1 \ar[dr]^{a} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

with 2 ≤ a ≤ c. Then we claim that c_{0} = a^{2} − 2. Indeed, applying a mutation at 1 yields the quiver Q′

\xymatrix{ & 1 \ar[dl]_{a} \\ 2 \ar[rr]_{a^{2}−c} &  & 3 \ar[ul]_{a}}

1) Suppose c ≤ a^{2} − 2. If a^{2} − c is the maximum, then the next mutation at the point opposed to the maximum yields Q so that Q or Q′ is a root and Q is mutation-cyclic. If not, then a is the maximum and moreover a(a^{2} − c) ≥ 2a so that Q′ is a root, whence Q is again mutation-cyclic.

2) Suppose c > a^{2} − 2. Then a^{2} − c ≤ 1 and so Q is mutation-acyclic by lemma 2.1, (2).

Example 2. Let Q be the quiver

\xymatrix{ & 1 \ar[dr]^{a+m} \\ 2 \ar[ur]^{a} &  & 3 \ar[ll]^{c}}

where 2 < a ≤ c and 1 ≤ m ≤ 4. Then we claim that c_{0} = a^{2} + am − 3. Indeed, applying a mutation at 1 yields the quiver Q′

\xymatrix{ & 1 \ar[dl]_{a} \\ 2 \ar[rr]_{a^{2}+m−c} &  & 3 \ar[ul]_{a+m}}

1) Assume c ≤ a^{2} + am − 3. If a^{2} + am − c is the maximum, then the next mutation at the point opposed to the maximum yields Q again, which is then mutation-cyclic. If not, then a + m is the maximum. If a ≥ 2m, then c ≤ a^{2} + am − 3 yields a(a^{2} + am − c) ≥ 3a = 2a + a ≥ 2(a + m). Thus, Q′ is a root and Q is mutation-cyclic. On the other hand, if a < 2m, then we have

3 ≤ a < 2m ≤ 8
a < a + m ≤ a + 4
a + m ≤ c ≤ a^{2} + am − c.

There are only finitely many quivers verifying these inequalities. A straightforward verification shows that in each case Q is mutation-cyclic, as desired.

2) Assume c > a^{2} + am − 3. If a^{2} + am − c ≤ 1, then by lemma 2.1, (2) Q is mutation-acyclic. Otherwise, a^{2} + am − c = 2 and then the same conclusion follows from (3.1).

References

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[3] A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572-612.

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[6] O. Kerner, Wild cluster tilted algebras of rank three, preprint, to appear in J. Pure Appl. Algebra.

[7] A. Seven, Recognizing Cluster Algebras of Finite type, arXiv:math.CO/0406545.

[8] M. St-Pierre, Every 4-bounded matrix is skew-symmetrizable, to appear in Ann. Sci. Math. du Québec.

Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec), J1K 2R1, Canada

E-mail address: ibrahim.assem@usherbrooke.ca

Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec), J1K 2R1, Canada

E-mail address: Martin.V.Blais@usherbrooke.ca

Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec), J1K 2R1, Canada and Department of Mathematics, Bishop’s University, 2600 College St., Sherbrooke, Quebec, Canada J1M 0C8

E-mail address: thomas.brustle@usherbrooke.ca and
tbruestl@ubishops.ca

Département de Mathématiques, Université de Sherbrooke, Sherbrooke (Québec), J1K 2R1, Canada

E-mail address: Audrey.Samson@USherbrooke.ca