aa r X i v : . [ m a t h . R T ] A p r TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 00, Number 0, Pages 000–000S 0002-9947(XX)0000-0
EXCHANGE MAPS OF CLUSTER ALGEBRAS
IBRAHIM SALEH
Abstract.
Every two labeled seeds in a field of fractions F together with apermutation give rise to an automorphism of F called an exchange map. Weprovide equivalent conditions for exchange maps to be cluster isomorphismsof the corresponding cluster algebras. The conditions are given in terms of anaction of the quiver automorphisms on the set of seeds. Introduction
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in [5, 6, 3,7]. The original motivation was to find an algebraic combinatorial framework tostudy canonical basis and total positivity. One of the unique characterizations ofcluster algebras is the way their generators (the cluster variables) are related. Thecluster variables are grouped in overlapping sets called clusters and each clusterforms a (commutative) free generators set of an ambient field. Attached with eachcluster is a valued quiver and the cluster together with the valued quiver form apair called a seed. A new seed is produced from an existing one using an operationcalled mutation, which defines an equivalence relation on the set of all seeds. Theequivalence classes are called mutation classes and each mutation class characterizesa cluster algebra.In an effort to explore the automorphisms that preserve the mutation classesof coefficient-free skew-symmetric cluster algebras, the authors in [2] introducedand studied cluster automorphisms , which are Z -automorphisms of cluster algebrasthat send a cluster to another and commute with mutation. It was proved in [2,Corollary 2.7] that the cluster automorphisms are exactly the Z -algebra automor-phisms that map each cluster to a cluster. Inspired by this result and by the strongisomorphisms introduced in [6], we mean by a cluster isomorphism a Z -algebraisomorphism that induces a bijection between the two sets of clusters of the twocluster algebras.Any two seeds ( X, Q ) and (
Y, Q ′ ) of rank n in a field of fractions F and a per-mutation σ ∈ S n , define an automorphism over F , induced by sending the clustervariable x i in X to the cluster variable y σ ( i ) in Y for every i . Such filed automor-phism is called exchange map , (Definition 3.10). The aim of this work is to findequivalent conditions on exchange maps to be cluster isomorphisms.Quiver automorphisms act on the set of all seeds which gives rise to a relationbetween seeds called σ -similarity (Definition 3.5). Theorem 1.1.
Two cluster algebras of the same rank n are cluster isomorphic ifand only if there exists a permutation σ ∈ S n such that the cluster algebras containtwo σ -similar seeds. c (cid:13) XXXX American Mathematical Society
Positive cluster algebras are the cluster algebras that satisfy the positivity con-jecture [5]. The class of positive cluster algebras includes all skew-symmetric clusteralgebras [11], acyclic cluster algebras [10] and cluster algebras arising from trian-gulations of surfaces [12].
Theorem 1.2.
An exchange map from a positive cluster algebra A to a positivecluster algebra B is a cluster isomorphism if and only if it sends every clustervariable in A to a cluster variable in B . The article is organized as follows. Section 2 is devoted to cluster algebras asso-ciated with valued quivers. In the first three subsections of section 3, we introducean action of quiver automorphisms on the set of all seeds, define the σ -similarityrelation and provide some equivalent conditions for two seeds to be σ -similar. Insubsection 3.4 we prove the main results (Theorems 3.14-3.15). We finish section3, by providing presentations of groups of cluster automorphisms of some clusteralgebras of types B and G .Throughout the paper, K is a field with zero characteristic and F = K ( t , t . . . t n )is the field of rational functions in n independent (commutative) variables over K .Let Aut K ( F ) denote the automorphism group of F over K and Let S n be thesymmetric group on n letters. We always denote ( b ij ) for the square matrix B and[1 , n ] = { , , . . . , n } . Acknowledgements.
The author is very grateful to the referee for the commentsand suggestions which were very useful in finishing the paper in the final form.2.
Cluster algebras associated with valued quivers
For more details about the material of this section refer to [4, 9, 5, 6, 7].2.1.
Valued quivers. • A valued quiver of rank n is a quadruple Q = ( Q , Q , V, d ), where – Q is a set of n vertices labeled by [1 , n ]; – Q is called the set of arrows of Q and consists of ordered pairs ofvertices, that is Q ⊂ Q × Q ; – V = { ( v ij , v ji ) ∈ N × N | ( i, j ) ∈ Q } , V is called the valuation of Q ; – d = ( d , · · · , d n ), where d i is a positive integer for each i , such that d i v ij = v ji d j , for every i, j ∈ [1 , n ].In the case of ( i, j ) ∈ Q , then there is an arrow oriented from i to j andin notation we shall use the symbol · i ( v ij ,v ji ) / / · j . If v ij = v ji = 1 we simplywrite · i / / · j .In this paper, we moreover assume that ( i, i ) / ∈ Q for every i ∈ Q , and if( i, j ) ∈ Q then ( j, i ) / ∈ Q • If v ij = v ji for every ( v ij , v ji ) ∈ V then Q is called equally valued quiver . • A valued quiver morphism φ from Q = ( Q , Q , V, d ) to Q ′ = ( Q ′ , Q ′ , V ′ , d ′ )is a pair of maps ( σ φ , σ ) where σ φ : Q → Q ′ and σ : Q → Q ′ such that σ ( i, j ) = ( σ φ ( i ) , σ φ ( j )) and ( v ij , v ji ) = ( v ′ σ φ ( i ) σ φ ( j ) , v ′ σ φ ( j ) σ φ ( i ) ) for each( i, j ) ∈ Q . If φ is invertible then it is called a valued quiver isomorphism .In particular φ is called a quiver automorphism of Q if it is a valued quiverisomorphism from Q to itself. XCHANGE MAPS OF CLUSTER ALGEBRAS 3
Remarks . (1) Every (non valued) quiver Q without loops nor 2-cycles cor-responds to an equally valued quiver which has an arrow ( i, j ) if thereis at least one arrow directed from i to j in Q and with the valuation( v ij , v ji ) = ( m, m ), where m is the number of arrows from i to j .(2) Every valued quiver of rank n corresponds to a skew symmetrizable integermatrix B ( Q ) = ( b ij ) i,j ∈ [1 ,n ] given by(2.1) b ij = v ij , if ( i, j ) ∈ Q , , if neither ( i, j ) nor ( j, i ) is in Q , − v ij , if ( j, i ) ∈ Q . Conversely, given a skew symmetrizable n × n matrix B , a valued quiver Q B can be easily defined such that B ( Q B ) = B . This gives rise to a bijectionbetween the skew-symmetrizabke n × n integral matrices B and the valuedquivers with set of vertices [1 , n ], up to isomorphism fixing the vertices. • The mutation of valued quivers is defined through Fomin-Zelevinsky’s mu-tation of the associated skew-symmetrizable matrix. The mutation of askew symmetrizable matrix B = ( b ij ) on the direction k ∈ [1 , n ] is given by µ k ( B ) = ( b ′ ij ), where(2.2) b ′ ij = ( − b ij , if k ∈ { i, j } ,b ij + sign( b ik ) max(0 , b ik b kj ) , otherwise. • The mutation of a valued quiver Q can be described using the mutationof B ( Q ) as follows: Let µ k ( Q ) be the valued quiver obtained from Q byapplying mutation at the vertex k . We obtain Q ′ and V ′ , the set of thearrows and the valuation of µ k ( Q ) respectively, by altering Q and V of Q ,based on the following rules(1) replace the pairs ( i, k ) and ( k, j ) with ( k, i ) and ( j, k ) respectively andswitch the components of the ordered pairs of their valuations;(2) if ( i, k ) , ( k, j ) ∈ Q , but ( j, i ) / ∈ Q and ( i, j ) / ∈ Q (respect to ( i, j ) ∈ Q ) add the pair ( i, j ) to Q ′ , and give it the valuation ( v ik v kj , v ki v jk )(respect to change its valuation to ( v ij + v ik v kj , v ji + v ki v jk ));(3) if ( i, k ), ( k, j ) and ( j, i ) in Q , then we have three cases(a) if v ik v kj < v ij , then keep ( j, i ) and change its valuation to ( v ji − v jk v ki , − v ij + v ik v kj );(b) if v ik v kj > v ij , then replace ( j, i ) with ( i, j ) and change its val-uation to ( − v ij + v ik v kj , | v ji − v jk v ki | );(c) if v ik v kj = v ij , then remove ( j, i ) and its valuation. • One can see that; µ k ( Q ) = Q and µ k ( B ( Q )) = B ( µ k ( Q )) for each vertex k . Example . Let Q = · , / / · , ~ ~ ⑦⑦⑦⑦⑦⑦⑦⑦ · , O O . Applying mutation at the vertices 1 and 2produces the valued quivers µ ( Q ) = · , (cid:15) (cid:15) · , o o · , > > ⑦⑦⑦⑦⑦⑦⑦⑦ and µ ( Q ) = · · , o o · , o o . IBRAHIM SALEH
Cluster algebras. [7, Definition 2.3]
A labeled seed of rank n in F is a pair( X, Q ) where X = ( x , . . . , x n ) is an n -tuple elements of F forming a free generatingset and Q is a valued quiver of rank n . In this case, X is called a cluster .We will refer to labeled seeds simply as seeds, when there is no risk of confusion.The definition of clusters above is a bit different from the definition of clusters givenin [3], [5] and [6]. Definition 2.3 (Seed mutations) . Let p = ( X, Q ) be a seed in F , and k ∈ [1 , n ]. Anew seed µ k ( X, Q ) = ( µ k ( X ) , µ k ( Q )) is obtained from ( X, Q ) by setting µ k ( X ) =( x , . . . , x ′ k , . . . , x n ) where x ′ k is defined by the so-called exchange relation :(2.3) x ′ k x k = f p,x k , where f p,x k = Y ( i,k ) ∈ Q x v ik i + Y ( k,i ) ∈ Q x v ki i . And µ k ( Q ) is the mutation of Q at the vertex k . The elements of F obtained byapplying iterated mutations on elements of X are called cluster variables . Definitions 2.4 (Mutation class and cluster algebra) . • The equivalence classof a seed (
X, Q ) under mutation is called the mutation class of (
X, Q ) andit will be denoted by Mut(
X, Q ). • Let X be the union of all clusters in Mut( X, Q ). The (coefficient free) cluster algebra A = A ( X, Q ) is the Z -subalgebra of F generated by X . Theorem 2.5 (5, Theorem 3.1, Laurent Phenomenon) . The cluster algebra A ( X, Q ) is contained in the integral ring of Laurent polynomials Z [ x ± , . . . , x ± n ] . More pre-cisely, every non zero element y in A ( X, Q ) can be uniquely written as (2.4) y = P ( x , x , . . . , x n ) x α · · · x α n n , where α , . . . , α n are integers and P ( x , x , . . . , x n ) is a polynomial with integercoefficients which is not divisible by any of the cluster variables x , . . . , x n .Conjecture Positivity Conjecture ) . If y is a cluster variable, then the polyno-mial P ( x , x , . . . , x n ), in (2.4), has positive integer coefficients. Definition 2.7.
Positive cluster algebras are the cluster algebras that satisfy thepositivity conjecture.3. isomorphisms of cluster algebras
Quiver automorphisms action on seeds.
Let φ be a quiver automorphismof Q = ( Q , Q , V, d ). Then φ induces a permutation σ φ ∈ S n . We can obtain anew valued quiver φ ( Q ) = ( Q ′ , Q ′ , V ′ , d ′ ) from Q as follows • Q ′ is obtained by permuting the vertices of Q using σ φ ; • Q ′ = { ( σ φ ( i ) , σ φ ( j )) | ( i, j ) ∈ Q } ; • For every ( σ φ ( i ) , σ φ ( j )) ∈ Q ′ we give the valuation ( v σ φ ( i ) σ φ ( j ) , v σ φ ( j ) σ φ ( i ) ); • d ′ = ( d σ φ (1) , · · · , d σ φ ( n ) ). XCHANGE MAPS OF CLUSTER ALGEBRAS 5
Example . Consider the valued quiver Q = · , (cid:15) (cid:15) (2 , / / · · , > > ⑦⑦⑦⑦⑦⑦⑦⑦ with d = (1 , , φ with underlying permutation σ φ = (123). Then φ ( Q ) = · , (cid:15) (cid:15) (2 , / / · · , > > ⑦⑦⑦⑦⑦⑦⑦⑦ and φ ( d ) = (4 , , Question . For which valued quivers Q and a quiver automorphism φ are therea sequence of mutations µ such that φ ( Q ) = µ ( Q )?The following example shows that there are cases in which such a sequence ofmutations does not exist. Example . Consider the valued quiver Q = · , / / · / / · , and the quiverautomorphism with underlying permutation (12). In the following, we will showthat there is no sequence of mutations µ , such that µ ( Q ) = · , / / · / / · orequivalently there is no sequence of mutations µ such that µ ( B ( Q )) = − − .Here we have B ( Q ) = − − . The proof is written in terms of thematrix B ( Q ) = ( b ij ). If we could show that there is no sequence of mutationsthat sends the entry b to zero, we will be done. We do this by showing thatevery sequence of mutations sends b to an odd number. First we show, by induc-tion on the length of the sequence of mutations, that any sequence of mutationssends b and b to even numbers. For sequences containing only one mutation:one can see that, only µ and µ would change b and b respectively, that is µ ( b ) = µ ( b ) = 2.Now, assume that every sequence of mutations of length k sends b and b to aneven number, and let µ i k +1 µ i k . . . µ i be a sequence of length k + 1. So if(3.1) µ i k . . . µ i (( b ij )) = ( b ′ ij ) . Then b ′ = 2 m for some integer number m . We have µ i k +1 ( b ′ ) = b ′ + sign( b ′ ) max(0 , b ′ b ′ )= b ′ + sign( b ′ ) max(0 , mb ′ )which is a sum of two even numbers. This shows that any sequence of mutationswill send b to an even number. In a similar way one can show that any sequenceof mutation sends b to an even number.Secondly, we show that every sequence of mutations sends | b | to an odd number,noting that the possible change in | b | appears only after applying µ . We showthis by induction on the number of occurrences of µ in the sequence. IBRAHIM SALEH
Sequences containing only one copy of µ : Without loss of generality, let µ i µ i . . . µ i k be a sequence of mutations such that µ i k = µ , and µ i j = µ , ∀ j ∈ [1 , k − b ′ = ± b ′ ) max(0 , b ′ b ′ ) . However b ′ and b ′ are both even numbers, so sign( b ′ ) max(0 , b ′ b ′ ) must be aneven number, and then b ′ is an odd number. Sequences containing more than one copy of µ : Assume that any sequenceof mutations with µ repeated k times sends b to an odd number.Let µ i t µ i . . . µ i be a sequence of mutations containing k + 1-copies of µ . Thenwe can assume that µ i t = µ . Let(3.3) µ i t . . . µ i (( b ij )) = ( b ′′ ij ) and µ i t − . . . µ i (( b ij )) = ( b ′ ij ) , then one can see that b ′ is an odd number and b ′ and b ′ are both even numbers.Then,(3.4) b ′′ = b ′ + sign( b ′ ) max(0 , b ′ b ′ ) , a sum of an odd and an even number, hence b ′′ is an odd number. (cid:3) Let T = { t , t , . . . , t n } be a free generating set of F and σ ∈ S n . We define an F -automorphism σ T , by σ T ( r ( t , . . . t n )) = r ( t σ (1) , . . . t σ ( n ) ) for r ( t , . . . t n ) ∈ F . Definition 3.4.
Fix a cluster X in F . A quiver automorphism φ , with underlyingpermutation σ φ ∈ S n , acts on the set of all seeds of F with respect to X as follows:for any seed q = ( Y, Γ) define(3.5) φ X ( q ) := ( φ X ( Y ) , φ (Γ)) , where φ X ( Y ) = (( σ φ ) X ( y ) , . . . , ( σ φ ) X ( y n )). We write φ ( p ) and φ ( Y ) instead of φ X ( q ) and φ X ( Y ) respectively if there is no chance of confusion. Definitions 3.5. (1) Let σ ∈ S n . Two valued quivers Q and Q ′ are said tobe σ -similar if σ is the underlying permutation of a quiver isomorphismbetween Q and one of the valued quivers Q ′ or ( Q ′ ) op .(2) Two seeds ( X, Q ) and (
Y, Q ′ ) are said to be σ -similar if Q and Q ′ are σ -similar. Remark . (a) Q and Q ′ are σ -similar if and only if B ( Q ) = ǫσ ( B ( Q ′ )), for ǫ ∈ {− , +1 } .(b) The σ -similarity relation defines an equivalence relation on the set of allseeds of F . Lemma 3.7.
Two quivers Q and Q ′ are σ -similar if and only if (3.6) Y ( i,k ) ∈ Q t v ik i + Y ( k,i ) ∈ Q t v ki i = Y ( σ ( i ) ,σ ( k )) ∈ Q t v σ ( i ) σ ( k ) σ ( i ) + Y ( σ ( k ) ,σ ( i )) ∈ Q t v σ ( k ) σ ( i ) σ ( i ) , ∀ k ∈ [1 , n ] Proof.
One can see that Q and Q ′ are σ -similar if and only if ( v ij ) i,j ∈ [1 ,n ] =( v ′ σ ( i ) σ ( j ) ) i,j ∈ [1 ,n ] and one of the following two conditions is satisfied(3.7) ( i, j ) ∈ Q if and only if ( σ ( i ) , σ ( j )) ∈ Q ′ , for every i, j ∈ [1 , n ]; XCHANGE MAPS OF CLUSTER ALGEBRAS 7 or(3.8) ( i, j ) ∈ Q if and only if ( σ ( j ) , σ ( i )) ∈ Q ′ , for every i, j ∈ [1 , n ] . Since { t , . . . , t n } is a transcendance basis of F , then one of the conditions (3.7) or(3.8) is satisfied if and only if one of the following two conditions is satisfied(1) Y ( i,k ) ∈ Q t v ik i = Y ( σ ( i ) ,σ ( k )) ∈ Q t v σ ( i ) σ ( k ) σ ( i ) and Y ( k,i ) ∈ Q t v ki i = Y ( σ ( k ) ,σ ( i )) ∈ Q t v σ ( k ) σ ( i ) i , ∀ k ∈ [1 , n ];or(2) Y ( i,k ) ∈ Q t v ik i = Y ( σ ( k ) ,σ ( i )) ∈ Q t v σ ( k ) σ ( i ) i and Y ( k,i ) ∈ Q t v ki i = Y ( σ ( i ) ,σ ( k )) ∈ Q t v σ ( i ) σ ( k ) σ ( i ) , ∀ k ∈ [1 , n ] . Which is equivalent to (3.6). (cid:3)
Main definitions.
Let p = ( X, Q ) and p ′ = ( Y, Q ′ ) be two seeds of rank n . Definition 3.8.
Let f be an element of Aut K ( F ). • f is called a cluster variables preserver from A ( X, Q ) to B ( Y, Q ′ ), if it sendsevery cluster variable in A to a cluster variable in B . In particular, f isa cluster variables preserver of a cluster algebra A ( X, Q ) if it leaves X ,the set of all cluster variables of A , invariant. For simplicity we call suchautomorphisms the X - preservers . • [1 , , f is said to be a cluster isomorphism from A ( X, Q ) to B ( Y, Q ′ ) ifit induces a one to one correspondence from the set of all clusters of A tothe set of all clusters of B . In particular, f : A → A is called a clusterautomorphism of A if it permutes the clusters of A . Remark . An element f of Aut K ( F ) is a cluster isomorphism from A ( X, Q ) to B ( Y, Q ′ ) if and only if it induces a one to one correspondence between the mutationclasses Mut( X, Q ) and Mut(
Y, Q ′ ). This is due to the fact that for every seed( X, Q ) the quiver Q is uniquely defined by the cluster X which has been proved in[8, Theorem 3].3.3. Exchange maps.Definition 3.10.
Let σ ∈ S n and p = ( X, Q ) and p ′ = ( Y, Q ′ ) be two labeled seedsin F . Then, the map T pp ′ ,σ , induced by x i y σ ( i ) , is called an exchange map .One can see that exchange maps are elements of Aut K ( F ). Lemma 3.11.
Let σ ∈ S n . Then Q and Q ′ are σ − similar if and only if T pp ′ ,σ ( µ k ( x k )) = µ σ ( k ) ( y σ ( k ) ) , for all k ∈ [1 , n ] . IBRAHIM SALEH
Proof. ⇒ ) Assume that Q and Q ′ are σ - similar valued quivers. Then v ij = v ′ σ ( i ) σ ( j ) for every i, j ∈ [1 , n ] and one of the conditions (3.7) or (3.8) is satisfied. Hence T pp ′ ,σ ( µ k ( x k )) = T pp ′ ,σ x k Y ( i,k ) ∈ Q x v ik i + Y ( k,i ) ∈ Q x v ki i = 1 y σ ( k ) Y ( i,k ) ∈ Q y v ik σ ( i ) + Y ( k,i ) ∈ Q y v ki σ ( i ) = 1 y σ ( k ) Y ( σ ( i ) ,σ ( k )) ∈ Q ′ y v ′ σ ( i ) σ ( k ) σ ( i ) + Y ( σ ( k ) ,σ ( i )) ∈ Q ′ y v ′ σ ( k ) σ ( i ) σ ( i ) = µ σ ( k ) ( y σ ( k ) ) . ⇐ ) Suppose that Q and Q ′ are not σ -similar. Then (3.6) is not satisfied. Hence T pp ′ ,σ ( f p,x k ) = f p ′ ,y σ ( k ) , for some k ∈ [1 , n ]. Therefore T pp ′ ,σ ( µ i ( x k )) = µ σ ( k ) ( y σ ( k ) )for some k ∈ [1 , n ]. (cid:3) Lemma 3.12.
For every σ ∈ S n and any square matrix B , we have (3.9) σ ( µ k ( B )) = µ σ ( k ) ( σ ( B )) , for all k ∈ [1 , n ] . In particular, for every valued quiver Q (3.10) σ ( µ k ( Q )) = µ σ ( k ) ( σ ( Q )) , for all ∈ [1 , n ] . Proof.
Let σ ( B ) = ( b ∗ ij ), µ σ ( k ) ( σ ( B )) = ( b ij ), µ k ( B ) = ( b ′ ij ), and σ ( µ k ( B )) = ( b ⋆ij ).We obtain the matrix σ ( B ) from B , by relocating the entries of B using σ . Indeed,the entry b ∗ ij = b σ − ( i ) σ − ( j ) . b ij = ( − b ∗ ij , if σ ( k ) ∈ { i, j } ,b ∗ ij + sign( b ∗ ik )max(0 , b ∗ ik b ∗ kj ) , otherwise= ( − b σ − ( i ) σ − ( j ) , if k ∈ { σ − ( i ) , σ − ( j ) } b σ − ( i ) σ − ( j ) + sign( b σ − ( i ) σ − ( k ) )max(0 , b σ − ( i ) σ − ( k ) b σ − ( k ) σ − ( j ) ) , otherwise= b ′ σ − ( i ) σ − ( j ) = b ⋆ij . This proves (3.9). Identity (3.10) is immediate by using B = B ( Q ) in (3.9). (cid:3) Theorem 3.13.
Let ( X, Q ) and ( Y, Q ′ ) be two σ -similar seeds. Then for anysequence of mutations µ i k , µ i k − , . . . , µ i , the following are true (1) µ i k µ i k − . . . µ i ( X, Q ) and µ σ ( i k ) µ σ ( i k − ) . . . µ σ ( i ) ( Y, Q ′ ) are σ - similar, (2) T pp ′ ,σ ( µ i k µ i k − . . . µ i ( X )) = µ σ ( i k ) µ σ ( i k − ) . . . µ σ ( i ) ( Y ) .Proof. To prove part (1), assume that Q and Q ′ are two σ -similar valued quivers.Then B ( Q ) = ǫσ ( B ( Q ′ )) for ǫ ∈ {− , +1 } . Hence from (3.9), we have µ k ( B ( Q )) = µ k ( ǫσ ( B ( Q ′ ))) = ǫσ ( µ σ ( k ) ( B ( Q ′ ))) , for every k ∈ [1 , n ] . XCHANGE MAPS OF CLUSTER ALGEBRAS 9
Therefore, µ k ( B ( Q )) and µ σ ( k ) ( B ( Q ′ )) are σ -similar for every k ∈ [1 , n ]. An in-duction process generalizes this fact to any sequence of mutations µ i k , µ i k − , . . . , µ i .For part (2), let p i i k = µ i k µ i k − . . . µ i ( p ) and p ′ σ ( i ) σ ( i k ) = µ σ ( i k ) µ σ ( i k − ) . . . µ σ ( i ) ( p ′ ).Part (1) of this theorem tells us that p i i k − and p ′ σ ( i ) σ ( i k − ) are σ -similar. ThenLemma 3.11 implies that(3.11) T p i ik p ′ σ ( i σ ( ik ) ,σ ( µ i k ( µ i k − . . . µ i ( X ))) = µ i σ ( k ) ( µ i σ ( k − . . . µ σ ( i ) ( Y )) . So, it remains to show that(3.12) T p i ik p ′ σ ( i σ ( ik ) ,σ ( µ i k ( µ i k − . . . µ i ( X ))) = T pp ′ ,σ ( µ i k ( µ i k − . . . µ i ( X )) . This will be proved by induction on the length of the sequence of mutations. Firstwe will show (3.12) in the general setting for sequences of mutations of length 2.Let q = ( Z, D ) and q ′ = ( T, C ) be any two σ -similar seeds, and let q = µ i ( Z, D ) =( Z ′ , D ′ ) , q ′ = µ σ ( i ) ( T, C ) = ( T ′ , C ′ ). We show that(3.13) T q q ′ ,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 q q ′ ,σ , and T qq ′ ,σ willleave z j unchanged. Now, let j = i . Then, using the notation f p,x k introduced inEquation (2.3) above, we have T q q ′ ,σ ( µ i ( z i )) = T q q ′ ,σ (cid:18) f q,z i z i (cid:19) = f q ′ ,t σ ( i ) T p p ′ ,σ ( z i ) . However, T q q ′ ,σ ( µ i ( z i )) = µ σ ( i ) ( t σ ( i ) ) = f q ′ ,t σ ( i ) t σ ( i ) . Hence, T p p ′ ,σ ( z i ) = t σ ( i ) . This shows that T q q ′ ,σ , and T qq ′ ,σ have the same actionon every cluster variable in Z , and since cluster variables from the cluster µ k µ i ( Z )are integral Laurent polynomials of cluster variables from Z , this gives (3.13).For equation (3.12) we use induction on the length of the mutation sequence. As-sume that equation (3.12) is true for any sequence of mutations of length less thanor equal k −
1. Now we have; T p i ik p ′ i ik ,σ ( µ i k µ i k − . . . µ i ( X )) = T p i ik − p ′ i ik − ,σ ( µ i k µ i k − . . . µ i ( X ))= T pp ′ ,σ ( µ i k µ i k − . . . µ i ( X )) , where the first equality is by (3.13) and the second is by the induction hypotheses. (cid:3) Main theorem.
Equivalent conditions for exchange maps to be cluster auto-morphisms are provided in this subsection.
Theorem 3.14.
Let p = ( X, Q ) and p ′ = ( Y, Q ′ ) be two labeled seeds in F , and σ ∈ S n . Then p and p ′ are σ -similar if and only if T pp ′ ,σ is a cluster isomorphismfrom A ( X, Q ) to B ( Y, Q ′ ) . In particular, two cluster algebras of the same rank n are cluster isomorphic if and only if there exists a permutation σ ∈ S n such thatthe cluster algebras contain two σ -similar seeds.Proof. Assume that p and p ′ are σ -similar. Let ( Z, D ) ∈ Mut(
X, Q ). Then thereis a sequence of mutations µ i , µ i , . . . , µ i k such that Z = µ i µ i . . . µ i k ( X ). Thenpart (1) of Theorem 3.13 implies that µ i . . . µ i k ( X, B ) and µ σ ( i ) . . . µ σ ( i k ) ( Y, B ′ )are σ -similar too. But, from Theorem 3.13 part (2), we have T pp ′ ,σ ( µ i ( µ i . . . µ i k ( X ))) = µ σ ( i ) ( µ σ ( i ) . . . µ σ ( i k ) ( Y )) , and since the right hand side is a cluster, then T pp ′ ,σ sends Z to a cluster inMut( Y, Q ′ ). So, T pp ′ ,σ sends every cluster in Mut( X, Q ) to a cluster in Mut(
Y, Q ′ ).Since the map µ i . . . µ i t µ σ ( i ) . . . µ σ ( i t ) , with t a non-negative integer, is a oneto one correspondence on the set of all sequences of mutations. Thus T pp ′ ,σ definesa one to one correspondence from the set of all clusters of Mut( X, Q ) to the set ofall clusters of Mut(
Y, Q ′ ).Assume that T pp ′ ,σ is a cluster isomorphism. Then T pp ′ ,σ ( µ i ( X )) is a cluster inMut( Y, Q ′ ); which shares n − µ σ ( i ) ( Y ). Then from[8, Theorem 3], each one of the two clusters can be obtained from the other by apply-ing one mutation which must be µ σ ( i ) . But µ σ ( i ) ( y σ ( i ) ) = y σ ( i ) and T pp ′ ,σ ( µ i ( x i )) = T pp ′ ,σ ( f p,xi ) y σ ( i ) . Then the two clusters coincide, hence T pp ′ ,σ ( µ i ( x i )) = µ σ ( i ) ( y σ ( i ) ) forevery i ∈ [1 , n ]. Therefore, p and p ′ are σ -similar, thanks to Lemma 3.11. (cid:3) Theorem 3.15.
Let p = ( X, Q ) and p ′ = ( Y, Q ′ ) be two labeled seeds, of the samerank n , in F and σ ∈ S n . Then, if both of A ( X, Q ) and B ( Y, Q ′ ) are positivecluster algebras, then the following are equivalent: (1) T pp ′ ,σ is a cluster variables preserver from A ( X, Q ) to B ( Y, Q ′ ) ; (2) p and p ′ are σ -similar; (3) T pp ′ ,σ is a cluster isomorphism from A ( X, Q ) to B ( Y, Q ′ ) .Proof. (1) ⇒ (2). To show that p and p ′ are σ -similar, we only need to show that T pp ′ ,σ ( µ i ( x i )) = µ σ ( i ) ( y σ ( i ) ) , for all i ∈ [1 , n ], thanks to Lemma 3.11.Let z = T pp ′ ,σ ( µ i ( x i )) and ξ = µ σ ( i ) ( y σ ( i ) ). Then(3.14) z = T pp ′ ,σ ( f p,x ) y σ ( i ) , and ξ = f p ′ ,y σ ( i ) y σ ( i ) . Both of T pp ′ ,σ ( f p,x ) and f p ′ ,y σ ( i ) are elements in the ring of polynomials Z [ y σ (1) , · · · , y σ ( i − , y σ ( i +1) , . . . , y σ ( n ) ] such that neither of them is divisible by y σ ( j ) for any j ∈ [1 , n ]. From the assumption that T pp ′ ,σ is a cluster variables preserverfrom cluster algebra A ( X, Q ) to the cluster algebra B ( Y, Q ′ ), then z must be acluster variable in B ( Y, Q ′ ). Hence, by Laurent phenomenon, z is an element ofthe ring of Laurent polynomials in the variables of the cluster µ σ ( i ) ( Y ) with integercoefficients. More precisely, z can be written uniquely as(3.15) z = P ( y σ (1) , y σ (2) , . . . , y σ ( i − , ξ, y σ ( i +1) , . . . , y σ ( n ) ) y α σ (1) . . . y α i − σ ( i − ξ α i y α i +1 σ ( i +1) . . . y α n σ ( n ) , where α i ∈ Z , for all i ∈ [1 , n ] and P is not divisible by any of the cluster variables y σ (1) , y σ (2) , . . . y σ ( i − , ξ, y σ ( i +1) , . . . , y σ ( n ) . Comparing z from (3.14) and (3.15), weget XCHANGE MAPS OF CLUSTER ALGEBRAS 11 (3.16) T pp ′ ,σ ( f p,x ) · y α σ (1) . . . y α i − σ ( i − . . . ξ α i . . . y α i +1 σ ( i +1) . . . y α n σ ( n ) = P · y σ ( i ) . Since f p,x is not divisible by any cluster variable x i for any i ∈ [1 , n ] as well then T pp ′ ,σ ( f p,x ) is not divisible by y i for all i ∈ [1 , n ]. More precisely T pp ′ ,σ ( f p,x ) is asum of two monomials in cluster variables from Y \ { y σ ( i ) } , with positive exponents.Therefore, α j = 0 for all j ∈ [1 , n ] \ { i } . Then equation (3.16) is reduced to T pp ′ ,σ ( f p,x ) (cid:18) f p ′ ,y σ ( i ) y σ ( i ) (cid:19) α i = P · y σ ( i ) . For α i , we break it down into three cases; (1) if α i ≥
0, then y α i +1 σ ( i ) divides either T pp ′ ,σ ( f p,x ) or f p ′ ,y σ ( i ) which is a contradiction, (2) if α i < −
1, then y − α i − σ ( i ) divideseither P or f p ′ ,y σ ( i ) which again is a contradiction, (3) assume α i = −
1. Hence(3.16) ends up to(3.17) T pp ′ ,σ ( f p,x ) = P · f p ′ ,y σ ( i ) , where f p ′ ,y σ ( i ) is a sum of two monomials in cluster variables from Y \ { y σ ( i ) } , withpositive exponents. However, P is a polynomial with positive integers coefficients,and not divisible by any cluster variable from Y ′ = µ σ ( i ) ( Y ). From equation (3.16)and since T pp ′ ,σ ( f p,x ) is a sum of two monomials, then P must be an integer. Finally,since the coefficients of T pp ′ ,σ ( f p,x ) and f p ′ ,y σ ( i ) are all ones, then P = 1.Hence(3.18) T pp ′ ,σ ( f p,x ) = f p ′ ,y σ ( i ) . Therefore(3.19) T pp ′ ,σ ( µ i ( x i )) = µ σ ( i ) ( y σ ( i ) ) , for all i ∈ [1 , n ] . (2) ⇒ (3) from Theorem 3.14 and (3) ⇒ (1) is immediate. (cid:3) Corollary 3.16. If p and p ′ are two labeled seeds in A ( X, Q ) and σ ∈ S n .Then (1) p and p ′ are σ -similar if and only if T pp ′ ,σ is a cluster automorphism of A . (2) If A is positive cluster algebra. Then, the following are equivalent (a) T pp ′ ,σ is a X -preserver; (b) p and p ′ are σ -similar; (c) T pp ′ ,σ is a cluster automorphism.Proof. Special cases of Theorems 3.14 and 3.15 by taking B ( Y, Q ′ ) = A ( X, Q ) (cid:3) The exchange group and the group of cluster automorphisms.
Defi-nition 3.8 and 3.10 give rise to the following two subgroups of
Aut K ( F ). Definitions 3.17. (a) The exchange group, denoted by EAut A ( X, Q ), is thesubgroup of
Aut K ( F ) generated by { T pp ′ ,σ | p, p ′ ∈ Mut(
X, Q ) , σ ∈ S n } .(b) [2] The cluster automorphisms group Aut A ( X, Q ) is the subgroup of
Aut K ( F )that consists of all cluster automorphisms of A ( X, Q ). Corollary 3.18. (1) Aut A ( X, Q ) = { T pp ′ ,σ | p, p ′ are σ − similar in A ( X, Q ) , σ ∈ S n } . (2) Aut A ( X, Q ) = EAut A ( X, Q ) if and only if the σ -similarity relation on Mut(
X, Q ) has only one equivalence class.Examples . (1) If ( X, Q ) is a seed of rank 2 then Aut A ( X, Q )=EAut A ( X, Q ).(2) Let Q = · , (cid:15) (cid:15) (cid:15) (cid:15) (2 , / / · · , > > ⑦⑦⑦⑦⑦⑦⑦⑦ . Then Aut A ( X, Q )=EAut A ( X, Q ).In [2], the authors computed the cluster automorphism groups for cluster algebrasof Dynkin and Euclidean types. Using Corollary 3.18 and part (1) of Example 3.19,we provide presentations for exchange groups and the cluster automorphisms groupsof cluster algebras of types B and G . Example . (1) The group of cluster automorphisms
Aut( X, B ).Aut A ( X, B ) = { T , T | T = T = 1 , ( T T ) = 1 } . (2) The group of cluster automorphisms
Aut ( X, G ).Aut A ( X, G ) = { T , T | T = T = 1 , ( T T ) = 1 } . References [1] Ibrahim Assem, Gregoire Dupont and Ralf Schiffler ”On a category of cluster algebras”, J.Pure and Applied Algebra (3) 553-582 (2013).[2] Ibrahim Assem, Ralf Schiffler and Vasilisa Shramchenko ”Cluster Automorphisms”, Proc.London Math. Soc. (3) (2012) 1271-1302.[3] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and doubleBruhat cells, Duke Math. J. (2005), no. 1, 152.[4] V. Dlab, C. M. Ringel, Indecomposable Representations of Graphs and Algebras, Mem. Amer.Math. Soc.,
Am. Math. Soc, Providence (1976).[5] S. Fomin and A. Zelevinsky, Cluster algebras I. Foundations, J. Amer. Math. Soc., :497-529(electronic), (2002).[6] S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. (2003), 63-121.[7] S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. (2007),no. 1, 112-164, DOI 10.1112/S0010437X06002521. MR2295199 (2008d:16049).[8] M. Gekhtman, M. Shapiro and A. Vainshtein, On the properties of the exchange graph of acluster algebra, Math. Res. Lett.15