Exchange Maps of Cluster Algebras

Every two seeds in a field of fractions $\mathcal{F}$ together with a symmetric group element gives rise to an automorphism of $\mathcal{F}$ called an exchange automorphism. For positive cluster algebras, we provide equivalent conditions for exchange automorphisms to be cluster isomorphisms of the corresponding cluster algebras. The equivalent conditions are given in terms of a symmetric group action on the set of seeds, which is also studied. Presentations of groups of cluster automorphisms of some seeds of ranks 1 and 2 are given.


Introduction
Cluster algebras are introduced by S. Fomin, and A. Zelevinsky in [7 ,8, 2, 9]. The original motivation for this theory is to create an algebraic framework to study total positivity and canonical basis in semisimple algebraic groups. It is inspired by the discovery of connection between total positivity and canonical basis, due to G Lusztig, [11] Great progress has been made in the theory, however we deviate from the original motivations.
A cluster algebra of rank n, A n (S), is a commutative subalgebra of an ambient field, generated by a distinguished set of generators called cluster variables. The cluster variables are grouped in overlapping sets (clusters) of transcendence basis of the ambient field. Any two clusters can be obtained from each other by applying some sequence of mutations. We call the pair (χ S , S C ), where χ S is the set of all cluster variables and S C denotes the class of all clusters, is called the cluster structure of A n (S).
In this work, we introduce the cluster automorphisms of A n (S), which are the field automorphisms which leave χ S invariant. It turned out to be, under certain conditions, leaving χ S invariant is equivalent to leaving the cluster structure invariant, (theorem 3.3). The group of all such automorphisms is called the cluster group of A n (S) and is denoted by C n (S).
Also, we study the action of the symmetric group on the set S, of all seeds of the field F = K(t 1 , t 2 , . . . , t n ) . We show that in the simply-laced cluster algebras, the orbits of such actions are subsets of the orbits of the mutations group action on S, theorem 2.4. However, the simply-laced hypothesis is necessary (example 2.5).
Every two seeds and a permutation group element define a field automorphism, we call such automorphism an exchange automorphism. The subgroup of Aut. K F , generated by the set of all exchange automorphisms, is called the exchange group of A n (S), and is denoted by m n (S).
The main result of this article is providing a description for the intersection of the cluster group C n (S) and the exchange group m n (S) for any coefficient free cluster 2 IBRAHIM SALEH algebra satisfying the Fomin-Zelevinsky positivity conjecture, in terms of the orbits of the symmetric group action on S and the cluster pattern data (theorem 3.3).
Through out the paper, K is a field and F = K(t 1 , t 2 . . . t n ) is the field of rational functions in n independent (commutative) variables over k. We always denote (b ij ) for the square matrix B, (c ij ) for C, etc., and [1, n] = {1, 2, . . . , n}.
(1)A pre-seed of rank n in F over K is a pair (X, B) where X = (x 1 , x 2 , . . . , x n ) ∈ F n , such that the {x 1 , x 2 , . . . , x n } is a transcendence basis of F over K and B = (b ij ) is an n×n, sign-skew-symmetric integral matrix. Sign-skew-symmetric means, b ij = b ji = 0 or b ij b ji < 0, ∀i, j ∈ [1, n].
(2)The diagram of a sign-skew-symmetric matrix B = (b ij ) is the weighted directed graph, Γ(B), with set of vertices [1, n], such that there is an edge from i to j if and only if b ij > 0, and this edge is assigned the weight |b ij b ji |.
(3)A pre-seed (X, B) is called connected if Γ(B) consists of exactly one connected component.

Definition 1.2 (Seed mutation).
For each fixed k ∈ {1, . . . , n}, and each given pre-seed (X, B) we define a new pair µ k (X, B) = (X * , B * ) by setting The operation µ k is called a mutation in k−direction.
Remark 1.3. If the pre-seed (X, B) is connected, then the mutation of (X, B) in any direction is still connected. Furthermore, one can see that µ 2 k = 1 for all k ∈ [1, n], and {x 1 , . . . , x i−1 , x * i , x i+1 , · · · , x n } is always a transcendence basis of F over K for all i ∈ [1, n]. However, B * need not be sign-skew-symmetric in general. So (µ k (X), µ k (B)) is not a pre-seed in general.
is called a seed if we always obtain a pre-seed after applying seed mutations to (X, B) in all possible directions and all possible sequences, i.e., the matrix µ i1 µ i2 . . . µ iq (B) is again a sign skew symmetric matrix for all possible choices of i 1 , i 2 . . . i q ∈ {1, 2, . . . n}. In this case µ i1 µ i2 . . . µ iq ((X, B)) is called mutation-equivalent to (X, B). It is clear that mutation-equivalence is an equivalence relation and pre-seed (X, B) is a seed if and only if any one (that all) pre-seed that is mutation-equivalent to (X, B) is a seed.
and the second condition is Cartan counterpart A(B) = (a ij ) of B, is of finite type as a Cartan matrix. The type of p is the same as the Cartan-Killing type of A(B). Where A(B) = (a ij ) is defined by Remark 1.6. A connected pre-seed (X, B), with B skew-symmetrizable matrix, is always a seed. skew-symmetrizable means there exist a diagonal positive integral matrix D = (d i ), i ∈ [1, n], such that DB is skew-symmetric.
Let S be the set of all seeds in F . Fix p = (X, B) ∈ S. Let S denote the mutation equivalence class of p, and S C = {Y ; (Y, A) ∈ S}. Elements of S C are called clusters and components of any cluster are called cluster variables. We call the pair, (X S , S C ) the cluster structure, where X S is the set of all cluster variables in S. Definition 1.7. (Cluster algebra) Let X S be the set of all cluster variables in S i.e. the union of all clusters in S C . The cluster algebra A n (S) of rank n, associated to the initial seed p = (X, B) (of rank n), is defined to be the Z−subalgebra of F generated by X S i.e.

(1.5)
A n (S) := Z[X S ] ⊂ F Definition 1.8. (Cluster pattern of A n (S) [8]). The cluster pattern T n (S) of the cluster algebra A n (S) is an regular n−ary tree whose edges are labeled by the numbers 1, 2, . . . , n such that the n edges emanating from each vertex receive different labels. The vertices are assigned to be the elements of S (the seeds) such that the endpoints of any edge are obtained from each other by seed mutation in the direction of the edge label.
One can see, the cluster pattern of A n (S) can be completely determined by any seed in S. Definition 1.9. A cluster algebra, A n (S), is called of finite type, if S is a finite set. Equivalently if X S is finite.
The details for the following two theorems are available in [8], [16] and [7]. Theorem 1.10. (Finite type classification) For a cluster algebra A n (S), the following are equivalent: • A n (S) is of finite type; • for every seed (X, B) in S, the entries of the matrix B = (b ij ) satisfy the inequalities |b ij b ji | ≤ 3, for all i, j ∈ [1, n]; • S contains a distinguished seed. Theorem 1.11. Every finite type Cartan matrix corresponds to one and only one, up to field automorphism, finite type cluster algebra. Furthermore, a cluster algebra A n (S) is of finite type if and only if S contains a distinguished seed and cluster type of A n (S) is the same as the Cartan-Killing type of the Cartan counter part the distinguished seed. Remark 1.12. If there is a seed (X, B) in S such that, there is a linear ordering of {1, 2, · · · n} , where b ij ≥ 0 for all i < j, then the seed (X, B) is called acyclic seed, and A n (S) is called acyclic cluster algebra, and in this case we have; , and A n (S) is finitely generated as an algebra.
. , x ± n ] for any cluster X = (x 1 , x 2 , . . . , x n ) ∈ S C . More precisely, every non zero element y ∈ A n (S), can be uniquely written as where (α 1 , α 2 , . . . , α n ) ∈ Z n , and P (x 1 , x 2 , . . . , x n ) is an element of the ring of polynomials Z[x 1 , x 2 , . . . , x n ], is not divisible by any cluster variable x i . It is further conjectured that in the case of y is a cluster variable, the polynomials P (x 1 , x 2 , . . . , x n ) have nonnegative integer coefficients(known as Fomin-Zelevinsky positivity conjecture). The conjecture has been proved in many cases including classical type cluster algebras [8], rank two affine cluster algebras as in [15], acyclic cluster algebra [3], cluster algebras arising from spaces [14], and more.

Cluster groups
Definition 2.1. Let Aut K (F ) denote the automorphism group of F over K. An automorphism φ ∈ Aut K (F ) is called a cluster isomorphism from a cluster algebra A n (S) onto a cluster algebra B n (S ′ ) over F , if φ sends every cluster variable in A n (S) onto a cluster variable in A n (S ′ ). In particular, φ is called cluster automorphism of A n (S), if it leaves X S invariant. (Where X S is the set of all cluster variables of A n (S)).
The subgroup of Aut K (F ) of all cluster automorphisms of a cluster algebra A n (S) is called the cluster group of A n (S) and is denoted by C n (S).

Remarks 2.2.
(1) One can see that any cluster automorphism of a cluster algebra A n (S) is an algebra automorphism of A n (S) over K, i.e. C n (S) is a subgroup of Aut K A n (S), where Aut K A n (S) denotes the automorphism group of the algebra A n (S) over K.
(2) If ψ : A n (S) → B n (S ′ ) is a cluster isomorphism, then ψ induces a group isomorphism between C n (S), and C n (S ′ ). To see this last fact, we define the group isomorphism by π : C n (S) → C n (S ′ ), given by φ → ϕ, where ϕ(y) = ψ(φ(ψ −1 (y))) for y a cluster variable in B n (S ′ ). A routine check shows that π is a group isomorphism.
Let Σ n be the symmetric group on n letters. One can see that Σ n ⊂ Aut K (F ) as follows: let T = (t 1 , t 2 , . . . , t n ), where {t 1 , t 2 , . . . , t n } is a transcendence basis of ON THE AUTOMORPHISMS OF CLUSTER ALGEBRAS 5 F over K, σ ∈ Σ n , and f = f (t 1 , t 2 , . . . t n ) ∈ F . We have; In the following we introduce an action of the symmetric group Σ n on S .
. . x n ) be a fixed cluster, and let σ ∈ Σ n . For any . We define σ X (p), as follows; Before stating the next theorem, we need to develop some notations. For a seed p = (Y, B), the neighborhood of a cluster variable y i is defined to be the subset of {y 1 , y 2 , · · · , y n } of all cluster variables y j , where b ij = 0. We denote this N p (y i ). For every connected integral skew-symmetric matrix, B = (b ij ), it is convenient to assign a quiver. We define Q B as Q B = (Q 1B , Q 2B ), where Q 1B denotes the vertices and Q 2B denotes the arrows. The vertices Q 1B = {1, . . . , n} and there is b ij arrows from i to j if and only if b ij > 0. The mutation operation of the matrix B can be translated to the associated quiver. Let µ k (Q B ) denote the mutation at k of Q B . First, all the arrows incident to k in Q B are reversed in µ k (Q B ). Second, for each pair of sets of arrows, one set is of the incoming arrows say from j to k, and the other is of the outgoing arrows from k say to i, we add number of arrows from j to i equals to the number of the product of the cardinal numbers of both sets. Last step, is canceling all two-cycles between j and i.
Then for any σ ∈ Σ n , σ X (p) is mutation-equivalent to p.
where σ ij stands for a transposition element of Σ n , that sends every k ∈ {1, 2, . . . , n} to itself except i and j. We have Now, since the symmetric group is generated by transpositions, and remarking that all seeds are connected, we are left to show the identity (2.3).
Sketch of proof of identities (2.3): Note that, each simply-laced sign skew symmetric matrix must be skew symmetric. So, if it is connected it associates to a (connected) quiver, which reduces to proving (2.3) on (X, Q B ) instead. In the following we provide a proof for the identity (2.3) in some cases as examples.
The proof of all other cases follow similarly: We provide a proof for A 3 -type, a general A n -type case is quite similar, (ii) For the exchange inside the n-cycles We prove it for A 3 −type, a general n-cycle is quite similar: (remark that; the number 2 written over the arrows from 3 to 2 and from 2 to 3 in third and fourth steps respectively, refers to double arrows).
(iii) Exchange of external vertex with adjacent one which is a vertex in an n-cycle We provide calculations for n = 4 case.
Connected cycles and different quivers shapes are similar.
For non simply-laced type seeds, the above result is not necessarily true, we provide the following counter example.
In the following, we show that there is no sequence of mutations If we could show that, there is no sequence of mutations that sends the entry b 23 to zero, we will be done. We do this by showing that every sequence of mutations sends b 23 to an odd number. First we show by induction on the length of the sequence of mutations that, any sequence of mutations sends b 13 and b 12 to even numbers. For single element sequences: one can see that, only µ 2 and µ 3 may change b 13 and b 12 respectively: that is µ 2 and µ 3 send b 13 and b 12 to 2 respectively. Now, assume that every sequence of mutations of length k sends b 13 and b 12 to an even number, and let µ i k+1 µ i k . . . µ i1 be a sequence of length k + 1. So if

IBRAHIM SALEH
Then b ′ 23 = 2d for some integer number d. Now we have Since b ′ 13 is an even number then µ i k+1 (b ′ 13 ) must be an even too. This shows that any sequence of mutations will send b 13 to an even number. In a similar way one can show that any sequence of mutation sends b 12 to an even number. Secondly, we show that every sequence of mutations sends |b 23 | to an odd number. We show this by induction on the number of occurrences of µ 1 in the sequence. Note that any sequence not containing µ 1 will not change |b 23 |. Sequences contains only one copy of µ 1 : Without loss of generality, let µ i1 µ i2 . . . µ i k be a sequence of mutations such that µ i k = µ 1 , and µ ij = µ 1 , ∀j ∈ [1, k]. This becomes clear by considering that the possible change in |b 23 | appears only after applying µ 1 , and there is no change in |b 23 | due to µ 2 or µ 3 . Then using same notation as in (2.4), we have However b ′ 21 and b ′ 13 are both even numbers, so b ′ 21 |b ′ 13 |+|b ′ 21 |b ′ 13 2 must be even, and b ′ 23 is an odd number. Sequences contains more than one copy of µ 1 : Assume that any sequence of mutations, with µ 1 repeated k− times sends b 23 to an odd number. Let µ it µ i2 . . . µ i1 be a sequence of mutations containing µ 1 , k + 1−times, then we can assume that µ it = µ 1 . Let , then one can see that b ′ 23 is an odd number and b ′ 12 and b ′ 13 are both even numbers. Then, a sum of an odd and even numbers, b ′′ 23 is an odd number. Definition (2.3) gives rise to an equivalence relation on S, as we will see in the following definition.  = (X, B) and p ′ = (Y, B ′ ) be any two seeds, and σ ∈ n . So, if T pp ′ ,σ ∈ Aut. K (F ), given by; x i → y σ(i) . Then, we have; In particular, p ∼ p ′ if and only if for some permutation σ, T pp ′ ,σ sends µ i (X) to µ σ(i) (Y ).
Which is equivalent to saying, if B and B ′ are σ-similar, then for any k ∈ [1, n], µ k (B), and µ σ(k) (B ′ ) are also σ-similar. An induction process generalizes this fact to an arbitrary sequence of mutations, Part (1) tells us that p i1i k−1 and p ′ σ(i1)σ(i k−1 ) are σ-similar. Then lemma 2.7 implies that ). So, it remains to show that To get to this, let q = (Z, D), and q ′ = (T, C) be any two σ-similar seeds, and let Where Z = (z 1 , z 2 , . . . , z n ), and T = (t 1 , t 2 , . . . , t n ). Next we show that (2.14) T q1q ′ 1 ,σ (µ k µ i (Z)) = T qq ′ ,σ (µ k µ i (Z)). Let z j be a cluster variable in Z, then for j = i, both of T q1q ′ 1 ,σ , and T qq ′ ,σ leave z j unchanged. Now, let j = i, and we have However, Hence, T p1p ′ 1 ,σ (z i ) = t σ(i) . This shows that T q1q ′ 1 ,σ , and T qq ′ ,σ have the same action on every cluster variable in Z, and since cluster variables from the µ k µ i (Z) are integral laurent polynomials of cluster variables from Z, this gives (2.14). For equation (2.13) we use induction on the length of the mutation sequence. Assume that equation (2.13) is true for any sequence of mutation of length less than or equal k − 1. Now we have; where the first equality is by identity (2.14), and the second, by the induction hypotheses.
Theorem 2.9. Let A n (S) be a cluster algebra, and (X, B) be a self σ-similar seed in S for some σ ∈ Σ n . Then, σ X is a cluster automorphism.
Example 2.10. Let A n (S) be a cluster algebra of A n −type. Then, C n (S) is not trivial.
To see that, fix an initial seed (X, B) such that B = (b ij ) where either b ij ≥ 0, ∀i > j or b ij ≤ 0, ∀i > j, such seed is always exist in any cluster algebra of A n − type . The following permutation is a cluster automorphism with symmetric group action defined, with respect to the initial cluster X; Now one can see that (X, B) is self σ-similar.

Exchange groups
Every path in the cluster pattern defines a field automorphism, which we codify in the following definition. In this section, we study the intersection of the group generated by all such automorphisms and the cluster group.
Definitions 3.1. Let p = (X, B), and p ′ = (Y, B ′ ) be any two vertices in the exchange graph g n (S) of the cluster algebra A n (S). For any σ ∈ n , The field automorphism T pp ′ ,σ : F → F induced by x i → y σ(i) is called an exchange automorphism.
The subgroup of Aut. K (F ) generated by set of all exchange automorphisms is called the exchange group of A n (S), and is denoted by m n (S).
Remark 3.2. let A n (S) be a simply laced cluster algebra, and fix an initial seed p = (X, B). Then, every symmetric group element can be seen as a field automorphism, (as in the paragraph proceeding definition (2.3)), taking T = X. From Theorem 2.4, every symmetric group element (in the above sense) corresponds to a path in the exchange graph of A n (S). So, the symmetric group elements can be seen as exchange automorphisms.
Before we state the main results we sharpen the notations of the neighbors and monomials of the cluster variables. Let x i0 be a cluster variable in p = (X, B), i.e.X = (x 1 , . . . , x i0−1 , x i0 , x i0+1 , . . . , x n ). The set of neighbors of x i0 at the seed p, denoted by N p (x i0 ) and defined as, . We denote f p,xi 0 = m p,+ (x i0 ) + m p,− (x i0 ), one can see that f p,xi 0 is not divisible by x i , ∀i ∈ [1, n].
The following theorem provides a description for C n (S), through m n (S) and the equivalent classes of ∼. In the proof of the theorem, we assume that the positivity conjecture is true, Theorem 1.13. However, a proof without the positivity conjecture can be written in rank two and finite type cluster algebras. Theorem 3.3. Let A n (S) be a cluster algebra satisfies the positivity conjecture, (theorem 1.13). Let p = (X, B) and p ′ = (Y, B ′ ) be any two vertices in the exchange graph of A n (S). Then, for the field automorphism T pp ′ ,σ (Definition 3.1.), the following are equivalent (1) T pp ′ ,σ is a cluster automorphism, (2) p and p ′ are σ-similar, (3) T pp ′ ,σ permutes the clusters. Furthermore for any seed q = (Z, D), we have; Proof.
Now we have that f p ′ ,y σ(i) is also a sum of two monomials in variables from the cluster Y , with positive exponents. However, P is a polynomial with positive integers coefficients, and not divisible by any cluster variable from Y ′ = µ σ(i) (Y ). Then it must be either a sum of at least two monomials in variables from Y ′ , or it is a positive integer. Equation (3.5) says that, the first option for P is impossible because T pp ′ ,σ (f p,x ) and f p ′ ,y σ(i) sums of exactly two monomials, so P must be an integer. Because the coefficients of T pp ′ ,σ (f p,x ) and f p ′ ,y σ(i) are all ones, which must be exactly 1. Hence, Therefore, Now, lemma 2.7 implies p and p ′ are σ-similar.
(3)⇒(1) Permuting the clusters implies leaving χ S invariant, because every cluster variable is contained in some cluster.
The following is a corollary of the proof of Theorem 3.3, and is actually the generalization of the statement of the same theorem, to the level of cluster isomorphism.
Corollary 3.4. Let A n (S), and A n (S ′ ) be any two cluster algebras over F . If p = (X, B) ∈ S, p ′ = (Y, B ′ ) ∈ S ′ , and σ ∈ n . Then the following are equivalent (1) the field automorphism φ pp ′ ,σ : F → F , given by x i → y σ(i) is a cluster isomorphism from A n (S) onto A n (S ′ ), (2) p and p ′ are σ-similar, (3)φ pp ′ ,σ sends every cluster in S C onto a cluster in S ′ C . In particular, two cluster algebras are cluster isomorphic if and only if they contain two σ-similar seeds for some permutation σ.
Proof. Follow the proof of Theorem 3.3, mutatis mutandis. Corollary 3.5. If A n (S) is a cluster algebra of simply-laced type then C n (S) = 1.
Proof. This follow from Theorem 2.4 and theorem 3.3. However, the converse is not necessarily true. Consider the the cluster algebra given in example 2.5, and the cluster automorphism is the transposition σ 23 . A routine check shows that, σ 23 corresponds to the sequence of mutations µ 2 µ 3 µ 2 µ 3 µ 2 . Then Theorem 2.4 implies the transposition σ 23 is a cluster automorphism, while the cluster algebra is not simply-laced.
Conjecture 3.6. The set of all cluster variables χ S can be complectly determined by [p] as follows; In the following we calculate the cluster and exchange groups for some cluster algebras of low ranks.
Example 3.7. Cluster and exchange groups of rank 1 In this case, F = K(t), and Aut K F is isomorphic to the projective linear group P GL 2 (K), and ]. So the cluster group and the exchange groups associated to A 1 (S) are the subgroup of Aut. K (F ) generated by the automorphism T 1 : K(x) → K(x) induced by x → 1 x , and then; x t−1 x t+1 = x m t + 1, if t is odd x n t + 1, otherwise. Thus, the cluster algebra A(S)(m.n) corresponding to θ is the subalgebra of F = K(x 1 , x 2 ) generated by {x t ; t ∈ Z} , however, since θ is acyclic seed then A(S)(m.n) = Z[x 0 , x 1 , x 2 , x 3 ] ⊂ K(x 1 , x 2 ). Theorem 3.9. [11] The sequence (3.9) of the cluster variables {x t } t∈Z in A(S)(m.n) is periodic if and only if mn ≤ 3, and is of period 5 (resp., 6,8) if mn = 1 (resp., 2, 3).
In the following, let C 2 (m, n) denote the cluster group associated to A(S)(m, n). Lemma 3.10. There is a cluster isomorphism between the cluster algebra A 2 (S)(m, n) and A 2 (S)(n, m).
Proof. Fix (x 1 , x 2 ) and (y 1 , y 2 ) as initial clusters for A 2 (S)(m, n) and A 2 (S)(n, m) respectively. Consider the following cluster isomorphism σ 12 : A 2 (S)(m, n) → A 2 (S)(n, m) given by: x 1 → y 2 and x 2 → y 1 , one can see that this automorphism induces a one to one correspondence between the sets of all clusters of A 2 (S)(m, n) and A 2 (S)(n, m). Corollary 3.11. C 2 (m, n) ∼ = C 2 (n, m).
Proof. From the previous lemma, and part (2) in remarks 3.2.

Acknowledgement
I would like to thank A. Zelevinsky for his valuable comments and remarks on this article during his visit to Kansas state university January 25-26, 2010. I would like to thank Z Lin, N. Rojkovskaia, and Z. Fan for their valuable discussion and remarks. I have presented this article in the "Graduate student conference in Algebra and Representation theory Kansas State university May 24-25,2009". Before posting this paper on arxiv, Ibrahim Assem, Ralf Schiffler and Vasilisa Shramchenko have posted their paper "Cluster Automorphisms" [1].