Canonically Positive Basis of Cluster Algebras of Type A

CANONICALLY POSITIVE BASIS OF CLUSTER (1) ALGEBRAS OF TYPE A2

GIOVANNI CERULLI IRELLI

(1) Abstract. We study cluster algebras of type A2 with every choice of coeﬃcients. We ﬁnd linear basis of such algebras which are called canonically positive: positive linear combinations of the elements of such basis coincide with the cone of positive elements of the cluster algebra.

Contents 1. Introduction and main results 1 1.1. Main properties of the elements of B 4 (1) 2. Cluster algebras of type A2 8 2.1. Background on cluster algebras 8

2.2. Algebraic structure of AP 10 2.3. Explicit expression of the coeﬃcient tuples 13 3. Proof of theorem 1.2 15 4. Proof of theorem 1.3 18 4.1. Proof of proposition 1.1 18 4.2. Proof of proposition 1.3 22 5. Proof of theorem 1.1 22 5.1. Linear independence of B 23 5.2. Positivity of the elements of B 23

5.3. The set B spans AP over ZP 23 5.4. The elements of B are positive indecomposable 30 Acknowledgements 36 References 36

1. Introduction and main results Cluster algebras have been introduced and studied in a series of papers [5], [6], [1] and [7] by S. Fomin and A. Zelevinsky. In this paper we study a (1) particular class of such algebras called of type A2 and this allows us to be completely self–contained, even if our main reference remains [7]. Recall that a semifield P = (P, ·, ⊕) is an abelian multiplicative group (P, ·) endowed with an auxiliary addition ⊕ : P×P → P which is associative, commutative and a(b ⊕ c) = ab ⊕ ac for every a, b, c ∈ P. The main example of a semifield is a tropical semifield: by definition a tropical semifield T rop(yj : j ∈ J) is an abelian multiplicative group freely generated by the elements {yj : j ∈ J} (for some set of indices J) endowed 1 2 GIOVANNI CERULLI IRELLI with the auxiliary addition ⊕ given by:

Y aj Y bj Y min(aj ,bj ) yj ⊕ yj := yj . j j j It can be shown (see [5, Section 5]) that every semifield P is torsion-free as a multiplicative group and hence its group ring ZP is a domain. Given a semifield P, let QP be the field of fractions of ZP and FP = QP(x1, x2, x3) be the field of rational functions in three indipendent variables with coefficients in QP. A rank three cluster algebra with coefficients in P is a ZP–subalgebra of FP generated by some recursively generated rational functions called cluster variables. The recursions at every step are called exchange relations and are governed by skew–symmetrizable integer matrices which are called exchange matrices. Every such matrix has naturally associated a generalized Cartan matrix called its Cartan counterpart. In [6] it is shown that if an exchange matrix has Cartan counterpart of finite type, then the corresponding cluster algebra has finitely many cluster variables and it is hence called of finite type. If there is a Cartan counterpart of affine type then all the exchange matrices have Cartan counterpart either of the same type or of wild type. (1) Here we study rank three cluster algebras of type A2 with coefficients in every semifield P. More precisely (see section 2.1 for some background on cluster algebras) we consider a semifield P and the cluster algebra AP with initial seed   0 1 1 (1) Σ := {H =  −1 0 1  , x = {x1, x2, x3}, y = {y1, y2, y3}}. −1 −1 0

The algebra AP is hence a ZP–subalgebra of FP := QP(x1, x2, x3) and the elements y1, y2 and y3 are arbitrarily chosen elements of P. More explicitly the algebra AP might be presented by generators and relations in the following way (cf. lemma 2.1): we consider the family {y1;m : m ∈ Z} ⊂ P of coefficients defined by the initial conditions: y = 1 , y = y , y = y1y2 1;0 y3 1;1 1 1;2 y1⊕1 together with the recursive relations: y1;m+2y1;m+1 y1;my1;m+3 = (y1;m+1 ⊕ 1)(y1;m+2 ⊕ 1) for m ≥ 1 and m ≤ −4 (a solution of this recurrence is given in (43)). The cluster variables of AP are the elements w, z and xm (m ∈ Z) defined as follows: (2) w := y2x1+x3 , z := y1y3x1x2+y1+x2x3 (y2⊕1)x2 (y1y3⊕y1⊕1)x1x3 while xm is defined recursively by the exchange relation: xm+1xm+2 + y1;m (3) xmxm+3 = . y1;m ⊕ 1

By definition AP := ZP[w, z, xm : m ∈ Z] ⊂ FP. Every cluster variable s1 can be completed to a set C = {s1, s2, s3} of cluster variables which form a free generating set of the field FP, so that FP ' QP(s1, s2, s3). Such a set C is called a cluster of AP. The clusters of AP are (1) CLUSTER ALGEBRAS OF TYPE A2 3 the sets {xm, xm+1, xm+2}, {x2m+1, w, x2m+3} and the set {x2m, z, x2m+2} for every m ∈ Z. The following figure 1 shows (a piece of) the “exchange graph” of AP. By definition it has clusters as vertices and an edge between

• • • •w • • •

··· x−1 x1 x3 x5 ··· • • •• •• •• •• •• •

··· x0 x2 x4 ···

• •••••z

Figure 1. The exchange graph of AP two clusters if they share exactly two cluster variables. In this figure cluster variables are associated with regions: there are infinitely many bounded regions labeled by the xm’s, and there are two unbounded regions labeled respectively by w and z. Each cluster {s1, s2, s3} corresponds to the vertex of the three regions s1, s2 and s3. The exchange graph of AP appears also (1) in [5] as the exchange graph of a coefficient–free cluster algebra of type A2 (i.e. P = 1) and it is called the “brick wall”. We have already observed that FP ' QP(s1, s2, s3) for every cluster {s1, s2, s3} of AP and hence every element of AP can be expressed as a rational function in {s1, s2, s3}. By the Laurent phenomenon proved in [5], such a rational function is actually a Laurent polynomial. Following [11] we say that an element of AP is positive if its Laurent expansion in every cluster has coefficients in Z≥0P. Positive elements form a semiring, i.e. sums and products of positive elements are positive. We say that a ZP–basis B of AP is canonically positive if the semiring of positive elements coincides with the Z≥0P–linear combinations of elements of it. If a canonically positive basis exists, it is composed by positive indecomposable elements, i.e. positive elements that cannot be written as a sum of positive elements. Moreover such a basis is unique up to rescaling by elements of P. Such problem arises naturally in the general theory of cluster algebras and it is still open. We de- scribe briefly some results in this direction without any aim of completeness: recall that a monomial in cluster variables belonging to the same cluster is called a cluster monomial. For example the cluster monomials of AP are the a b c a b c a b c monomials xmxm+1xm+2, x2m+1w x2m+3 and x2mz x2m+2 for every non– negative integers a,b,c and for every m ∈ Z. In [2] P. Caldero and B.Keller prove that cluster monomials form a basis of coefficient–free cluster algebras of finite type (i.e. with a finite number of cluster variables). Moreover in [10] the authors prove that such cluster monomials are positive. It is not known if such cluster monomials are positive indecomposable. The problem has been solved in rank two cluster algebras of finite and affine type by P. Sherman and A. Zelevinsky in [11]. In this case they prove that in finite type cluster monomials have such property. They also notice that passing from finite type to infinite type, cluster monomials do not generate the cluster algebra anymore. They hence complete them to a basis by adding some other elements. 4 GIOVANNI CERULLI IRELLI

We are now going to ﬁnd such elements for the algebra AP.

Deﬁnition 1.1. Let W := (y2 ⊕ 1)w and Z := (y1y3 ⊕ y1 ⊕ 1)z. We deﬁne elements {un| n ≥ 0} of AP by the initial conditions: 2 (4) u0 = 1, u1 = ZW − y1y3 − y2, u2 = u1 − 2y1y2y3 together with the recurrence relation

(5) un+1 = u1un − y1y2y3un−1, n ≥ 2. The following is the main theorem of the paper. Theorem 1.1. The set k k (6) B := {cluster monomials} ∪ {unw , unz | n > 1, k > 0} is a canonically positive basis of AP. It is unique up to rescaling by elements of P. 1.1. Main properties of the elements of B. The elements of the canonically positive basis B have some distinguished properties. We start with (1) the connection with the root system of type A2 . By the already mentioned Laurent phenomenon [5, Theorem 3.1] every element b of AP is a Laurent polynomial in {x , x , x } of the form Nb(x1,x2,x3) for some primitive, i.e. not 1 2 3 d1 d2 d3) x1 x2 x3 divisible by any xi, polynomial Nb ∈ ZP[x1, x2, x3] in x1, x2 and x3, and some integers d1, d2, d3. We consider the root lattice Q generated by an (1) affine root system of type A2 . The choice of a simple system {α1, α2, α3}, 3 with coordinates {e1, e2, e3}, identifies Q to Z . We usually write elements of Q as column vectors with integer coefficients and we denote them by a bold t type letter. The map b 7→ d(b) = (d1, d2, d3) is hence a map between AP and Q; it is called the denominator vector map in the cluster {x1, x2, x3}. The following result provides a parameterization of B by Q. Recall that given t ◦ δ := (1, 1, 1) , the minimal positive imaginary root, and Π = {α1, α3}, a ◦ basis of simple roots for a root system ∆ of type A2, the positive real roots of Q are of the form α + nδ with n ≥ 0 if α is a positive root of ∆◦ and n ≥ 1 if α is a negative root of ∆◦ (see e.g. [9, Proposition 6.3]). Theorem 1.2. The denominator vector map d : A → Q : b 7→ d(b) in the cluster {x1, x2, x3} restricts to a bijection between B and Q. Under this (1) bijection positive real roots of the root system of type A2 correspond to the set of cluster variables together with the set {unw, unz| n ≥ 1}. Moreover for every cluster C = {c1, c2, c3}, the set {d(c1), d(c2), d(c3)} of corresponding denominator vectors is a Z–basis of Q. In lemma 2.2 denominator vectors of elements of B are given. Figure 2 shows the qualitative positions of denominator vectors of cluster variables (different from the initial ones) in Q: it shows the intersection between the plane P = {e1 + e2 + e3 = 1} and the positive octant Q+ of the real vector space QR generated by Q; a point labelled by d(s) denotes the intersection between P and the line generated by the denominator vector of the cluster variable s. A dotted line joins two cluster variables that belong to the same cluster. Such lines form mutually disjoint triangles. We notice that some of the points of P ∩ Q+ do not lie in one of these triangles. All (1) CLUSTER ALGEBRAS OF TYPE A2 5 such points belongs to the line between d(w) and d(z) that is denoted by a double line in the figure. This is the intersection between P and the “imaginary” cone generated by d(w) and d(z). The denominator vector k k of unw and unz , for all n, k ≥ 0, lie on the imaginary cone. We notice that such a figure appears also in [4] where the authors analyze canonical (1) decomposition of representations of the acyclic quiver of type A2 . This is because denominator vectors of cluster variables are all the positive real “Schur roots”, but we do not use this fact here.

(0, 0, 1)

d(x0) •8 ÖÖ 88 ÖÖ 88 ÖÖ 88 d(ÖÖx−2) • 88 ÖÖ 88 ÖÖ 88 Ö d(x−4) • 8 ÖÖ 88 ÖÖ 88 ÖÖ d(x−6) • 88 ÖÖ 88 d(z) ÖÖ • 88 •ÖRR • •8 Ö RRR • •ll d8(x−1) Ö RRR d(x−2m) • ll 8 Ö RRR • d(x−3) 8 Ö RRRR ••ll 8 ÖÖ RRRR ••• d(x−5) 88 Ö RRR lld(x ) 8 Ö d(x2m) RlRlRR −(2m+1) 8 ÖÖ ••lll RRRR 88 Ö d(x10) ••ll•l RRR 8 Ö lll • RRRR 8 Ö d(x8) l•l • RRR 88 ÖÖ lll • d(x2m+1) RRRR 8 Ö ll•l • RRRR 8 ÖÖ d(x6) lll RRRR 88 Ö lll d(x ) RRR 8 Ö l•ll • 9 RRRR 8 d(x ) Ö lll RRR 8 4 ÖÖ ll • d(x7) RRR 88d(w) •Ölll ••• RRR d(x5) (1, 0, 0) (0, 1, 0)

Figure 2. “Cluster triangulation” of the intersection between the positive octant Q+ and the plane P = {e1 + e2 + e3 = 1}. A dotted line joins two points corresponding to cluster variables belonging to the same cluster. The double line between w and z denotes the intersection with the “imaginary” cone CIm := Z≥0d(w) + Z≥0d(z) which contains the k k denominator vector of all the elements unw and unz .

Our next result provides explicit formulas for the elements of B in every cluster of AP. By the symmetries of the exchange relations it is suﬃcient to consider only the two clusters {x1, x2, x3} and {x1, w, x3} and only cluster variables xm with m ≥ 2 (see remark 2.1).

Theorem 1.3. Let P be any semiﬁeld. Then the following formulas hold. In the cluster {x1, x2, x3}: for every m ≥ 1

P e1−e3m−1−e3e1−1yexε(e) + xm−1x2m−2 e e2−e3 m−1−e1 e3 2 3 x2m+1 = m−1 m−1 m−2 L (e1−e3)(m−1−e3)(e1−1)ye⊕1 x x x e e2−e3 m−1−e1 e3 1 2 3 6 GIOVANNI CERULLI IRELLI

t where ε(e) = (e2 +e3, m−1−e1 +e3, 2m−e2 −e1 −2) . P e1−1m−e2 m−1−e3yexε(e) +xmx2m−1 e e3 e1−e2 e2−e3 2 3 x2m+2 = L e1−1m−e2 m−1−e3ye ⊕ 1 xmxm−1xm−1 e e3 e1−e2 e2−e3 1 2 3 t where ε(e) = (e2 +e3, m−e1 +e3, 2m−1−e1 −e2) . For every n ≥ 1: h i ynynynx2nxn +xnx2n+P e1−e3 n−e3e1−1+n−e3−1e1−1 yexε(e) 1 2 3 1 2 2 3 e e1−e2 n−e1 e3 n−e1 e3−1 un = n n n x1 x2 x3 t where ε(e) = (e2 +e3, n−e1 +e3, 2n−e1 −e2) . In the cluster {x , w, x } let y := y (y ⊕ 1), y := 1 and y := y2y3 1 3 1 1 2 2 y2 3 y2⊕1 three elements of P (the coeﬃcients of the seed with cluster {x1, w, x3}). Then for every m ≥ 0:

P m−1−e e −1 e e 2e e −e 2m−2e −2 2m−2 ( 3)( 1 )y 1 y 3 x 3 w 1 3 x 1 +x e1,e3 m−1−e1 e3 1 3 1 3 3 x2m+1 = e e m−1 m−2 L (m−1−e3)(e1−1)y 1 y 3 ⊕1 x x e1,e3 m−1−e1 e3 1 3 1 3

P m−1−e +e e −1 1 e 2e3+1−e2 e −e 2m−2e1−1+e2 2m−1 ( 3 2)( 1 )( )y x w 1 3 x +x (y2x3+x1) e m−1−e1+e2 e3 e2 1 3 3 x2m+2 = L m−1−e +e e −1 1 e m m−1 ( 3 2)( 1 )( )y ⊕y2⊕1 x wx e m−1−e1+e2 e3 e2 1 3 For every n ≥ 1:

n n 2n 2n Xh i e e 2e e −e 2n−2e y y x +x + (n−e3)(e1−1)+(n−e3−1)(e1−1) y 1 y 3 x 3 w 1 3 x 1 1 3 1 3 n−e1 e3 n−e1 e3−1 1 3 1 3 e1,e3 un = n n x1 x3

In the cluster {x1, w, x3} the expansion of z is given by 2 2 2 y1y2y3x1 + y1y2y3x1x3 + y1y2w + y2x3 + x1x3 z = 2 y1y2y3 ⊕ y1y2y3 ⊕ y1y2 ⊕ y2 ⊕ 1 x1wx3 Theorem 1.3 can be deduced from the theory of cluster categories by computing explicitly the cluster character. We do not use this approach here, even if we will look at it in a forthcoming paper. The strategy of our proof uses the following observation which extends the g–vector parametrization of cluster monomials (see [7, section 7]) to all the elements of B. We recall that given a polynomial F ∈ Z≥0 [z1, ··· , zn] with positive coefficients in n variables and a semifield P = (P, ·, ⊕), its evaluation F |P(y1, ··· , yn) at (y1, ··· , yn) ∈ P × · · · × P is the element of P obtained by replacing the addition in Z[z1, ··· , zn] with the auxiliary addition ⊕ of P in the expression F (y1, ··· , yn) . For example the evaluation of F (z1, z2) := z1 +1 at (y1, y2) ∈ T rop(y1, y2) × T rop(y1, y2) is 1. C Proposition 1.1. Let us first assume P = T rop(y1;C, y2;C, y3;C). Let Σ = C {H , C, {y1;C, y2;C,y3;C}} be a seed of FP and let AP be a cluster algebra of (1) C type A2 with principal coefficients at Σ . Then for every element b of B , C there exist a (unique) polynomial Fb in three variables with constant term C 3 1 and a (unique) integer vector gb ∈ Z such that the expansion of b in the cluster C = {s1, s2, s3} is given by: C C C C C h1 h2 h3 g (7) b = Fb (y1;Cs , y2;Cs , y3;Cs )s b , (1) CLUSTER ALGEBRAS OF TYPE A2 7

C C where hi is the i–th column vector of the exchange matrix H . Let now P be any semifield. Then if b is a cluster monomial then the expansion of b in the seed ΣC is given by: C hC hC hC F (y1;Cs 1 , y2;Cs 2 , y3;Cs 3 ) C b gb (8) b = C s . Fb |P(y1;C, y2;C, y3;C) C If b = un, n ≥ 1, then its expansion in the seed Σ is still given by (7). C C We call Fb and gb respectively the F –polynomial and the g–vector of b in the cluster C. The previous proposition is the key step for proving theorem 1.3. Explicit expression of F –polynomials and g–vectors in every cluster are given in section 4.1. We can define more intrinsically F –polynomials and g–vectors as follows: C in view of (7), the F –polynomial Fb of b in the cluster C := {s1, s2, s3} is the polynomial obtained in the following way: consider the tropical semifield C P = T rop(y1;C, y2;C, y3;C) generated by the coefficients of Σ and expand b ∈ AP in the cluster C. In this expression replace s1, s2 and s3 with 1. The g–vector of b can be defined as follows: following [7] we notice that hC the elementy î := yis i , i = 1, 2, 3, has degree zero with respect to the 3 principal Z –grading of AP (when P = T rop(y1;C, y2;C, y3;C)) given by (9) deg(xi) = ei, deg(yi) = −bi, i = 1, 2, 3 3 (ei is the i–th basis vector of Z ). Therefore, by formula (7), every element C b of B is homogeneous with respect to such grading and the g–vector gb is its degree. C The g–vector map in the cluster C is by definition the map b 7→ gb which associates to an element of B its g–vector in the cluster C. In contrast with the denominator vector map, the g–vector map provides a parametrization of B in every cluster of AP as shown by the following proposition. C Proposition 1.2. Given a cluster C of AP , the map b 7→ gb which sends C an element b of B to its g–vector gb in the cluster C, is a bijection between 3 B and Z The relation between denominator vectors and g–vector in the cluster {x1, x2, x3} is given by the following proposition.

Proposition 1.3. Let b be an element of B not divisible by x1, x2 or x3. The g–vector gb of b and its denominator vector in the cluster {x1, x2, x3} d(b) are related by

−1 0 0 ! (10) gb = 1 −1 0 d(b) 1 1 −1

If b is a cluster monomial in the initial cluster {x1, x2, x3} then gb = −d(b). The expression (10) can be seen as a generalization of a similar formula found in [7] for rank three cluster algebras of bipartite type (see [3]) and it is a special case of the formula found in [8]. We conclude this section with another remark about proposition 1.1. We notice that since F –polynomials have constant term 1, the expansion (7) 8 GIOVANNI CERULLI IRELLI in the principal coefficient setting follows from the expansion (8). On the other hand the elements of a canonically positive basis are defined up to a factor in P. Then one do not loose too much by considering the expansion (7) also in the situation of a general tropical semifield. For every cluster variable s and every cluster C = {s1, s2, s3} we define the principal cluster variable S in C to be the expansion (7). Then clearly the set of principal k k cluster variables and of {unW , unZ } still form a canonically positive basis of AP. Actually the elements un’s in definition 1.1 are defined in terms of the principal cluster variables Z and W in the cluster {x1, x2, x3}. This can be a naive explanation of why they are always “principal”, i.e. expansion (8) does not involve any denominator in P. The following proposition says that the definition of the un’s can be given in terms of principal cluster variables in every seed. By the symmetries of the exchange relations (see remark 2.1) it is sufficient to consider only the seed

0 −1 2 ! cyc Σ := { 1 0 −1 , {x1, w, x3}, {y1, y2, y3}} −2 1 0 obtained from the seed (1) by a mutation in direction two (see lemma 2.1).

Proposition 1.4. Let P be a tropical semifield. Let us consider the principal {x1,w,x3} cyc cluster variable Z := Fz |P(y1, y2, y3)z in the seed Σ associated to the cluster variable z. Then for every n ≥ 1 the family of rational functions {un} defined in definition 1.1 satisfies the initial conditions: 2 2 (11) u0 = 1, u1 = Zw − y1y2y3 − 1, u2 = u1 − 2y1y3, together with the recurrence relations

(12) un+1 = u1un − y1y3un−1. The paper is organized as follows: in section 2 we give an overview of the results that we need about cluster algebras. In this section we also ﬁnd the algebraic structure of AP. In section 3 we prove theorem 1.2. In section 4 we prove theorem 1.2 and proposition 1.1. Finally in section 5 we prove theorem 1.1.

(1) 2. Cluster algebras of type A2 2.1. Background on cluster algebras. Let P be a semifield as defined in section 1. Let FP = QP(x1, ··· , xn) be the field of rational functions in n independent variables x1, ··· , xn.A seed in FP is a triple Σ = {H, C, y} where H is an n × n integer matrix which is skew–symmetrizable, i.e. there exists a diagonal matrix D = diag(d1, ··· , dn) with di > 0 for all i such that DB is skew–symmetric; C = {s1, ··· , sn} is an n–tuple of elements of FP which is a free generating set for FP so that FP ' QP(s1, ··· , sn); and finally y = {y1, ··· , yn} is an n-tuple of elements of P. The matrix H is called the exchange matrix of Σ, the set C is called the cluster of Σ and its elements are called cluster variables of Σ and the set y is called the coefficients tuple of the seed Σ.

We ﬁx an integer k ∈ [1, n]. Given a seed Σ of FP it is deﬁned another seed Σk := {Hk, Ck, yk} by the following mutation rules (see [7]): (1) CLUSTER ALGEBRAS OF TYPE A2 9

0 (1) the exchange matrix Hk = (hij) is obtained from H = (hij) by 0 −hij if i = k or j = k (13) hij = hij + sg(hik)[hikhkj]+ otherwise

where [c]+ := max(c, 0) for every integer c; 0 0 (2) the new coeﬃcients tuple yk = {y1, ··· , yn} is given by: ( 1 if j = k 0 yk (14) yj := [h ] kj + −hkj yjyk (yk ⊕ 1) otherwise. 0 (3) the new cluster Ck is given by Ck = C\{sk} ∪ {sk} where

Q [hik]+ Q [−hik]+ 0 yk i si + i si (15) sk := ; (yk ⊕ 1)sk

It is not hard to verify that Ck is again a seed of FP. We say that the seed Σk is obtained from the seed Σ by a mutation in direction k. Every seed can be mutated in all the directions. Given a seed Σ we consider the set χ(Σ) of all the cluster variables of all the seeds obtained by a sequence of mutations. The rank n cluster algebra with initial seed Σ with coefficients in P is by definition the ZP–subalgebra of FP generated by χ(Σ); we denote it by AP(Σ). The cluster algebra AP(Σ) is said to have principal coefficients at Σ and it is denoted by A•(Σ) if P = T rop(y1, ··· yn). If the semifield P = T rop(c1, ··· , cr) is a tropical semifield, the elements of P are monomials in the yj’s. Therefore the coefficient yj of a seed Σ = Qr hn+r,j {H, C, {y1, ··· , yn}} is a monomial of the form yj = i=1 ci for some integers hn+1,j, ··· , hn+r,j. It is convenient to “complete” the exchange n×n matrix H to a rectangular (n + r) × n matrix H˜ whose (i, j)–th entry is hij. The seed Σ can be hence seen as a couple {H,˜ C} and the mutation of the coefficients tuple (14) translates into the mutation (13) of the rectangular matrix H˜ . We sometimes use this formalism. In this paper we often consider unlabeled seeds, i.e. we consider two seeds equivalent if one can be obtained from the other by a permutation of the index set {1, ··· , n}. We use the cyclic representation of a permutation σ so that σ = (i1, i2, ··· , it) denotes the permutation σ such that σ(ik) = ik+1 if k = 1, ··· , t − 1 and σ(it) = i1; all the other indices are fixed by σ. Example 2.1. The seed

0 1 1 ! Σ := {H = −1 0 1 , x = {x1, x2, x3}, y = {y1, y2, y3}} −1 −1 0 is equivalent to the seed

0 −1 −1 ! 0 0 0 0 Σ := {H = 1 0 1 , x = {x3, x1, x2}, y = {y3, y1, y2}} 1 −1 0 by the permutation (132). We remark that even if every mutation of a seed in a ﬁxed direction is involutive, this is not true for unlabeled seeds. 10 GIOVANNI CERULLI IRELLI

2.2. Algebraic structure of AP. Let P = (P, ·, ⊕) be a semiﬁeld as deﬁned above. Let AP be the cluster algebra with initial seed   0 1 1 Σ := {H =  −1 0 1  , x = {x1, x2, x3}, y = {y1, y2, y3}}. −1 −1 0

The following lemma gives the algebraic structure of AP.

Lemma 2.1. The seeds of the cluster algebra AP with initial seed Σ = Σ1 are the following:

(16) Σm := {Hm, {xm, xm+1, xm+2}, {y1;m, y2;m, y3;m}}, cyc cyc cyc cyc cyc (17)Σ2m−1 := {Hm , {x2m−1, w, x2m+1}, {y1;2m−1, y2;2m−1, y3;2m−1}}, cyc cyc cyc cyc cyc (18) Σ2m := {Hm , {x2m, z, x2m+2}, {y1;2m, y2;2m, y3;2m}} for every m ∈ Z; they are mutually related by the following diagram of mutations:

µ1 Σw / Σw (19) 2m−1 o µ 2m+1 O 3 O µ2 µ2 µ2 µ2 µ1 µ1 µ1 / / / Σ2m−1 o Σ2m o Σ2m+1 o Σ2m+2 µ3 O µ3 µ3 O µ2 µ2 µ2 µ2 µ1 Σz o / Σz 2m µ3 2m+2 where arrows from left to right (resp. from right to left) are mutations in direction 1 (resp. 3) and vertical arrows (in both directions) are mutations in direction 2. The seed Σm is not equivalent to the Σn if m 6= n, in particular the exchange graph of AP is given by ﬁgure 1 and every cluster C determines C cyclic a unique seed Σ . The exchange matrices Hm and Hm are the following:     0 1 1 0 −1 2 Cyc (20) Hm =  −1 0 1  ,Hm =  1 0 −1  −1 −1 0 −2 1 0 for every m ∈ Z. The exchange relations for the coeﬃcient tuples are the following:

(21) y1;1 = y1; y2;1 = y2; y3;1 = y3

y = y2;my1;m ; y = y3;my1;m ; y = 1 (22) 1;m+1 y1;m⊕1 2;m+1 y1;m⊕1 3;m+1 y1;m

y = 1 ; y = y (y ⊕ 1); y = y (y ⊕ 1) (23) 1;m−1 y3;m 2;m−1 1;m 3;m 3;m−1 2;m 3;m

ycyc = y1;m ; ycyc = 1 ; ycyc = y3;my2;m (24) 1;m y2;m⊕1 2;m y2;m 3;m y2;m⊕1

cyc cyc cyc cyc cyc 2 cyc y2;my1;m cyc 1 (25) y1;m+2 = y3;m(y1;m ⊕ 1) ; y2;m+2 = cyc ; y3;m+2 = cyc y1;m⊕1 y1;m

cyc cyc y1;my2;m 1 cyc cyc (26) y1;m = cyc ; y2;m = cyc ; y3;m = y3;m(y2;m ⊕ 1) y2;m⊕1 y2;m

cyc cyc cyc 1 cyc y2;my3;m cyc cyc cyc 2 (27) y1;m−2 = cyc ; y2;m−2 = cyc ; y3;m−2 = y1;m(y3;m ⊕ 1) y3;m y3;m⊕1 (1) CLUSTER ALGEBRAS OF TYPE A2 11 The exchange relations are the following: xm+1xm+2 + y1;m y3;m+1xm+1xm+2 + 1 (28) xmxm+3 = = y1;m ⊕ 1 y3;m+1 ⊕ 1 cyc y2;2m−1x2m−1 + x2m+1 x2m−1 + y2;2m−1x2m+1 (29) wx2m = = cyc y2;2m−1 ⊕ 1 y2;2m−1 ⊕ 1 cyc y2;2mx2m + x2m+2 x2m + y2;2mx2m+2 (30) zx2m+1 = = cyc y2;2m ⊕ 1 y2;2m ⊕ 1 2 cyc cyc 2 x2m + y1;2m−2z y3;2mx2m + z (31) x2m−2x2m+2 = cyc = cyc y1;2m−2 ⊕ 1 y3;2m ⊕ 1 2 cyc cyc 2 x2m+1 + y1;2m−1w y3;2m+1x2m+1 + w (32) x2m−1x2m+3 = cyc = cyc y1;2m−1 ⊕ 1 y3;2m+1 ⊕ 1

Proof of Lemma 2.1. We consider the seeds of AP up to a symulataneous reordering of the index set (in the terminology of [7] these are called unlabeled seeds). We need to prove the diagram (19) for every m ∈ Z. Clearly Σ1 coincides with the initial seed Σ. For m ≥ 2 let Σm (resp. Σ−m+1) be the seed of AP obtained from Σ1 by applying m − 1 times the following operation: first we mutate in direction 1 (resp. 3) and then we reorder the index set of the obtained seed by the permutation (132) (resp. (123)) of the index set (as in example 2.1). Suppose that Σm (resp.Σ−m+1) has the form (16), i.e. its exchange matrix is Hm (resp. H−m+1), its cluster is {xm, xm+1, xm+2} and its coefficient tuple is {y1;m, y2;m, y3;m}. Then it is straightforward to check by induction on m that Hm is given by (20) and the exchange relations passing from Σm to Σm+1 are given by (22) (resp. (23)) for the coefficient tuple and by (28) for the cluster variables. Now it is straightforward to verify that, for every m ∈ Z,Σm is obtained from Σm+1 by mutating in direction 3 and then by reordering with the permutation (123). The central line of the diagram is hence proved. cyc Let Σm be the seed obtained from Σm by the mutation in direction 2. cyc cyc cyc cyc cyc Suppose that Σm has the form {Hm , {xm, cm, xm+2}, {y1;m, y2;m, y3;m}}. Then it is straightforward to verify the following: that by (13) the exchange cyc matrix Hm is given by (20); that by (14) the coefficient tuple satisfy (24) and by the exchange relation (15) the cluster variable cm is given by: y2;mxm + xm+2 (33) cm = . (y2;m ⊕ 1)xm+1

By using (22), (23) and (28), it is straightforward to verify that cm = cm+2 for every m ∈ Z. We define w := c1 and z := c2 and we hence find that cyc cyc Σ2m−1 has the form (17) and Σ2m has the form (18). Now since two cluster variables can belong to at most two clusters, we conclude that the mutation cyc cyc cyc in direction 1 (resp. 3) of Σm is Σm+2 (resp. Σm−2). We have hence proved the diagram (19) and the fact that every cluster determines a unique seed. cyc It remains to prove that the seeds {Σm} (resp.{Σm }) are not equivalent for every m ∈ Z. It is sufficient to prove that xm is different from x1, x2 and 12 GIOVANNI CERULLI IRELLI x3 for every m ≥ 4. In view of the exchange relation (28), the denominator vector of xm in the cluster {x1, x2, x3} satisfies the initial conditions d(xi) = −ei, for i = 1, 2, 3 and

(34) d(xm+3) + d(xm) = d(xm+1) + d(xm+2). The solution of this recursion are give in lemma 2.2 below and it is hence not periodic.

Remark 2.1. The expansion of a cluster variable xm+n (resp. x2m+n) in the cluster {xm, c, xm+2} for c = w or c = xm+1, (resp. {x2m, z, x2m+2}) is obtained by the expansion of x1+n (resp. x2m+1+n) in the cluster {x1, c, x3} (resp. {x2m+1, w, x2m+3}) by replacing x1 with xm, c with x2 when c 6= w, x3 with xm+2 and yi with yi;m (resp. x2m+1 with x2m, w with z, x2m+3 with cyc cyc x2m+2 and yi;2m+1 with yi;2m) for i = 1, 2, 3 and n, m ∈ Z. Moreover the expansion of x−m is obtained from the expansion of xm+2 by replacing x1 −1 −1 −1 with x3, x3 with x1 and y1 with y3 , y2 with y2 and y3 with y1 . The same symmetries hence naturally hold for F –polynomials and g–vectors. 2.2.1. Denominator vectors. In this subsection we compute denominator vectors of the elements of B in every cluster of AP. By the symmetries in the exchange relations it is suﬃcient to consider only the two clusters {x1, x2, x3} and {x1, w, x3}.

Lemma 2.2. Denominator vectors in the cluster {x1, x2, x3} of cluster variables diﬀerent from x1, x2 and x3 are the following: for m ≥ 1  m−1  " m # (35) d(x2m+1) =  m−1  d(x2m+2) = m−1 m−2 m−1

" m−1 # " m−1 # (36) d(x−2m+1) = m d(x−2m+2) = m−1 m m " 0 # " 1 # (37) d(w) = 1 d(z) = 0 0 1

For every n ≥ 1 the denominator vector (in {x1, x2, x3}) of un is given by " n # (38) d(un) = n . n In particular the denominator vector of all the cluster variables and of all the elements {unw, unz| n ≥ 1} are all the positive real roots of the root (1) system of type A2 . Proof. By induction on m, one veriﬁes that such vectors satisfy the recurrence relation (34) together with the initial condition d(xi) = −ei for 3 i = 1, 2, 3 (being ei the i–th basis vector of Z ). Then (35) and (36) follow. The equalities in (37) follow by (2). It remains to prove (38). By its deﬁnition (4), and the fact that the denominator vector map is additive, the denominator vector of u1 is the t sum of d(w) and of d(z), i.e. d(u1) = δ = (1, 1, 1) . Then by (5), d(un) = nd(u1) = nδ. (1) CLUSTER ALGEBRAS OF TYPE A2 13 The proof follows now by knowing the structure of a root system of type (1) A2 , recalled in section 1. Lemma 2.3. Let dw(b) be the denominator vector of b ∈ B in the cluster {x1, w, x3}. The following formulas hold: for m ≥ 1     m−1 m−1 w w (39) d (x2m+1) =  0  d (x2m+2) =  1  m−2 m−2   " m−1 # m−2 w (40) d(x−2m+1) = 0 d (x−2m+2) =  1  m m−1

" 0 # " 1 # w w (41) d (x2) = 1 d (z) = 1 0 1

For every n ≥ 1 the denominator vector (in {x1, w, x3}) of un is given by

" n # (42) d(un) = 0 . n

We conclude this section by pointing out an important property of denominator vectors.

n Deﬁnition 2.1. [7, Deﬁnition 6.12] A collection of vectors in Z (or in n R ) are sign–coherent (to each other) if, for every i ∈ {1, ··· , n}, the i–th coordinates of all of these vectors are either all non–negative or all non– positive.

Corollary 2.1. Denominator vectors of cluster variables belonging to the same cluster are sign–coherent.

2.3. Explicit expression of the coefficient tuples. In this section we solve the recurrence relations (22)–(27) for the coefficient tuples of the seeds of AP, for every choice of the semifield P. Our solution of such recurrence is given in terms of denominator vectors and of F –polynomials in the cluster {x1, x2, x3} of the cluster variables of AP. Recall that given a cluster variable s, its F –polynomial Fs in the cluster {x1, x2, x3} is obtained from the expansion of s in such cluster by specializing x1, x2 and x3 to 1. We assume proposition 1.1, even if its proof will be given in section 4.1. Then Fs is a polynomial with positive coefficients. In particular we can consider its evaluation Fs|P at P.

Proposition 2.1. The family {y1;m : m ∈ Z} is given by

yd(xm+3) (43) y1;m = . Fm+1|P(y)Fm+2|P(y)

The family {y2;m : m ∈ Z} is the sequence of elements of P given by: (44) y = y , y = (y ⊕ 1)y , y = (y y ⊕ y ⊕ 1) 1 . 2;1 2 2;0 3 1 2;−1 2 3 2 y3 14 GIOVANNI CERULLI IRELLI

For every m ≥ 1

F2m|P(y) (45) y2;2m = y1y3 F2m+2|P(y) F−2m|P(y) 1 (46) y2;−2m = F−2m+2|P(y) y2 F2m+1|P(y) (47) y2;2m+1 = y2 F2m+3|P(y) F−2m−1|P(y) 1 (48) y2;−2m−1 = F−2m+1|P(y) y1y3 The family {y3;m| m ∈ Z} is given by y3;m+1 = 1/y1;m. (cm) The families {yi;m | m ∈ Z} for i = 1, 2, 3 are given by ycyc = y1;m ; ycyc = 1/y ; ycyc = 1/ycyc (49) 1;m y2;m⊕1 2;m 2;m 3;m+2 1;m As a corollary of the preceding proposition we get relations between principal cluster variables: we recall that given a cluster variable s the corresponding principal cluster variable in the cluster {x1, x2, x3} is the element S := Fs|P(y1, y2, y3)s.

Corollary 2.2. The principal cluster variables in the cluster {x1, x2, x3} of AP satisfy the following relations:

(50) X0X3 =y3X1X2 +1; X−1X2 =y2X0X1 +1; X−2X1 =y1X−1X0 +1.

d(xm+3) (51) XmXm+3 = Xm+1Xm+2 + y for m ≥ 1 and m ≤ −3   y2X2m−1 + X2m+1 if m ≥ 1 (52) WX2m = X−1 + y3X1 if m = 0  X2m−1 + y1y3X2m+1 if m ≤ −1

  y1y3X2m + X2m+2 if m ≥ 1 (53) ZX2m+1 = y1X0 + X2 if m = 0  X2m + y2X2m+2 if m ≤ −1

2 d(x2m+1) (54) X2m−2X2m+2 = X2m + y Z 2 d(x2m+2) (55) X2m−1X2m+3 = X2m+1 + y W

In particular, if P = Trop(y1, y2, y3), these are precisely the exchange relations of the cluster algebra AP = A•(Σ) with principal coeﬃcients at the seed Σ. Proof of proposition 2.1. Formuas (44)–(49) follows directly by (43). It is suﬃcient to proveBy (25) it follows that the family {y1;m : m ∈ Z} is the sequence of elements of P uniquely determined by the initial data: y1;m = 1/ym+3 if m = 0, −1, −2, y1;1 = y1, y1;−3 = y3 together with the recurrence relations y1;m+2y1;m+1 (56) y1;m+3 = y1;m(y1;m+2 ⊕ 1)(y1;m+1 ⊕ 1) (1) CLUSTER ALGEBRAS OF TYPE A2 15

We ﬁrst suppose P = T rop(y1, y2, y3), it follows by induction on m that the unique solution of recurrence (56) in this special case is given by:

d(xm+3) (57) y1;m = y . In particular formulas (50) and (51) hold. Let us suppose now that P is a semiﬁeld. Recall that the F –polynomial Fm of the cluster variable xm in the cluster {x1, x2, x3} is obtained from the Laurent expansion of xm in {x1, x2, x3} by specializing x1 = x2 = x3 = 1. In view of (51), the family {Fm : m ∈ Z} is hence recursively deﬁned by the initial data: F1 = F2 = F3 = 1

F0(y) = y3 + 1 F−1(y) = y2y3 + y2 + 1 (58) 2 F−2(y) = y1y2y3 + 2y1y2y3 + y1y3 + y1y2 + y1 + 1 together with the recurrence relations for m ≥ 1 and m ≤ −3:

d(xm+3) (59) Fm(y)Fm+3(y) = Fm+1(y)Fm+2(y) + y

We use the notation F (y) := F (y1, y2, y3). Then an easy induction gives the desired (43). The other families of coeﬃcients are given by {y1:m} and {y2;m}. Mainly: In particular if P = T rop(y1, y2, y3) then cyc cyc cyc (60) y1;m = y1;m; y2;m = 1/y2;m; y3;m+2 = 1/y1;m the exchange relations (31) and (32) become

3. Proof of theorem 1.2 In this section we prove that the denominator vector map d : A → Q in the initial cluster {x1, x2, x3} restricts to a bijection between B and the root (1) lattice Q of type A2 . a b c Clearly the denominator vector of a cluster monomial s1s2s3 is ad(s1) + bd(s2)+cd(s3). We also know that d(u1) = d(w)+d(z) and d(un) = nd(u1). Hence the cone CIm = Z≥0d(w) + Z≥0d(z) generated by d(w) and d(z) is in k k bijection with the set {d(unw ), d(unz )| n, k ≥ 0}. To complete the proof of theorem 1.2 it is enough to show the following:

For every cluster {s1, s2, s3}, the vectors d(s1), d(s2) and d(s3)(61) form a Z–basis of Q. For every cluster {s1, s2, s3}, the vectors d(s1), d(s2) and d(s3)(62) are the only positive real roots contained in the additive

semigroup C{s1,s2,s3} := Z≥0d(s1) + Z≥0d(s2) + Z≥0d(s3).

(63) The union ∪C{s1,s2,s3} is equal to Q \CIm. This is done in the subsequent sections.

Proof of (61). By the explicit formulas of denominator vectors of cluster variables given in lemma 2.2 one checks directly that the absolute value of the determinant |det(d(s1), d(s2), d(s3))| = 1 for every cluster {s1, s2, s3}. 16 GIOVANNI CERULLI IRELLI

Proof of (62) and (63). We consider the basis of simple roots α1, α2, α3 of Q and the corresponding coordinate system (e1, e2, e3). By using lemma 2.2, we notice that there are four aﬃne lines in Q which contain the denominator vectors of all the cluster variables diﬀerent from x2. They contain respectively “negative odd”, “positive odd”, “negative even” and “positive even” cluster variables: they are

P ∩ Q+ T ∩ Q+ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦

◦ ◦ ◦ ◦ ◦ d(x13) ◦ ◦ ◦ ◦ ◦ y< yy yy ◦ ◦ ◦ ◦ d(yxy ) ◦ ◦ ◦ ◦ ◦ d(x ) yy 11 10 yyw; w; yyww ww yyww ww ◦ ◦ ◦ dyw(wx ) ◦ ◦ ◦ ◦ ◦ dw(wx ) ◦ ywyw 9 ww 8 ywywt: wwt: ywywtt wwtt ywywtt wwtt ◦ ◦ ywywdtt(x ) ◦ ◦ ◦ ◦ ◦ wwdtt(x ) ◦ ◦ ywywtt 7 wwtt 6 O ywywtto7 O wwtto7 v2 ywttoo w2 wttoo ywtywotoo wtwotoo tywywoto wtwoto ◦ o / d(x5) ◦ ◦ ◦ ◦ ◦ o / d(x4) ◦ ◦ ◦ v1 w1

Figure 3. The cones P ∩ Q+ and T ∩ Q+ : they intersect themselves into the dotted line of equation e1 = e2 = e3 .

− e2 = e3 + e1 = e2 òdd := ; òdd := ; e1 = e2 − 1 e3 = e2 − 1 − e1 = e2 + e2 = e3 èven := ; èven := . e3 = e2 + 1 e1 = e2 + 1

We define the two-dimensional subspaces P and T of QR containing respec- + − − + tively both òdd and èven and both òdd and èven of equation: P := {e1 = e2} and T := {e2 = e3}. In corollary 2.1 it is shown that given a cluster C = {s1, s2, s3} the corresponding denominator vectors {d(s1), d(s2), d(s3)} are sign–coherent, i.e. the i–th coordinates of all of them are either all non–negative or all non–positive, for i = 1, 2, 3. It implies that if the initial cluster variable xi lies in the cluster C, then the i–th coordinates of the other two elements of C are zero. By lemma 2.2 also the opposite holds: if the i–th coordinate of the denominator vector of a cluster variable is zero, then it lies in the same cluster as xi. Figure 4 shows denominator vectors of the cluster variables having at least one coordinate equal to zero. Let us analyze this

figure: the cones C{s1,s2,s3} involved here satisfies property (62), i.e. they do not overlap themselves. Moreover their union is the whole lattice except the interior of the positive octant Q+ = Z≥0d(x4) + Z≥0d(w) + Z≥0d(x0). We now concentrate our attention on the other cones. One can use figure 2 to visualize the situation. Let CP be the (open) cone inside P ∩ Q+ defined by CP := {0 < e3 < e1} ∪ {0}. By (35), d(x2n+1) ∈ CP for every n ≥ 2. The t t vectors v1 := (1, 1, 0) = d(x5) and v2 = (0, 0, 1) = d(x0) form a Z–basis (1) CLUSTER ALGEBRAS OF TYPE A2 17 of P such that CP is given by Z≥0v1 + Z≥0(v1 + v2) (see figure 3). In this basis d(x2n+1) = an1v1 + an2v2 where an1 = n − 1 and an2 = n − 2. The sequence a /a is strictly increasing. It has limit lim an2 = 1. We n2 n1 n→∞ an1 conclude that the set of cones {{Z≥0d(x2n+1) + Z≥0d(x2n+3)} : n ≥ 2} is a fan in P whose union is CP . Hence we have [ C{x2n+1,w,x2n+3} = Z≥0d(w) + CP n≥2

and these cones have no common interior points. Similarly let CT be the

______n n l l l l n l l n x0 l ______xl−l1 n n l l l n l O l l w; n l l ww n z l ______l ww _> ww >> w > ww >> ww x 1 ______p >> ww ll5 m m p p > wwlll m x p lwll m 2po • m m/ w p pp EEE m p ppp E m p p x wp " x m 4 ______5 ______m m k k m k k m m x3 k m k k m m k m ______k k Figure 4. Denominator vectors of cluster variables having at least one coordinate equal to zero. We wrote xm for d(xm). The clusters involving here form a fan whose union is Q \ Q+

(open) cone inside T ∪ Q+ defined by CT = {0 < e2 < e1} ∪ {0}. By t (35), d(x2n) ∈ CT for every n ≥ 2. The vectors w1 = (1, 0, 0) = d(x4) and t w2 = (0, 1, 1) form a Z–basis of T such that CT = Z≥0w1 +Z≥0(w1 +w2). In this basis d(x2n) = bn1w1 +bn2w2 with bn1 = n−1 and bn2 = n−2. (Figure 3 shows the cones P ∩Q+ and T ∩Q+ in the chosen basis {v1, v2} and {w1, w2} respectively). The strictly increasing sequence {bn2/bn1} has limit 1 for n → ∞. We conclude that the set of cones {{Z≥0d(x2n)+Z≥0d(x2n+2)} : n ≥ 2} is a fan in T whose union is the closure of CT . Hence we have [ C{x2n,z,x2n+2} = Z≥0d(z) + CT . n≥2 S It follows from the previous arguments that CP +CT = m≥4 C{xm,xm+1,xm+2} and that the interiors of two different cones involved here are disjoint. By now in figure 2 we have obtained all the points in the triangle between z, w and x4. We are going to obtain the others by reflecting through the line between z and w. We consider the orthogonal reflection rIm with respect to the imaginary cone CIm. It acts on vectors by exchanging the first coordinate with the third one. In particular it fixes CIm. By remark 2.1, rIm sends d(xm) to d(x4−m) for m ≥ 3. We have just obtained CIm +Z≥0d(x4) as union of mutually disjoint cones of the form C{s1,s2,s3} with si = xm, w or z for m ≥ 4. By applying rIm 18 GIOVANNI CERULLI IRELLI we obtain CIm + Z≥0d(x0) as mutually disjoint union of cones of the form

C{s1,s2,s3} with si = xm, w or z with m ≤ 0. Since Q+ = CIm + Z≥0d(x4) + Z≥0d(x0) we are done.

4. Proof of theorem 1.3 As already mentioned in the introduction, our proof of theorem 1.3 is based on proposition 1.1 by finding explicit expression for F –polynomials and g–vectors of every element of B. Once again it is sufficient to consider only the two clusters {x1, x2, x3} and {x1, w, x3} (see remark 2.1). The F – polynomials in the cluster {x1, x2, x3} and {x1, w, x3} are given respectively in lemma 4.3 (resp, lemma 4.4) and lemma 4.6 (resp. lemma 4.7). It remains to prove proposition 1.1. 4.1. Proof of proposition 1.1. We first prove that the elements of B have the form (8) in the cluster {x1, x2, x3}. Let hence P = T rop(y1, y2, y3) and let AP = A•(Σ) be the cluster algebra with principal coefficients at the initial seed Σ = {H, {x1, x2, x3}, {y1, y2, y3}} given in (1). The exchange relations for this algebra are (51), (52), (53), (54), (55) given in section 2.3. We need to prove that given an element b of B there exist a polynomial Fb in three variables with non–negative integer coefficients and an integer 3 vector gb ∈ Z such that the expansion of b in the cluster {x1, x2, x3} is g b = Fb(ˆy1, yˆ2, yˆ3)x b where (64) yˆ := y1 = y xh1 yˆ := y2x1 y xh2 yˆ := y x x = y xh3 1 x2x3 1 2 x3 2 3 3 1 2 3

Following [7, section 7] we deﬁne M to be the set of all the elements of FP that can be written in this form. Clearly M is closed under multiplication.

We hence need to prove that all the cluster variables of AP and all the elements {un : n ≥ 1} of definition 1.1 belong to M. In view of remark 2.1, it is sufficient to prove the statement only for the cluster variables w, z and xm with m ≥ 1 and for un, n ≥ 1. We first look at w, z and at the family {un : n ≥ 1}. By their definition (2) both w and z belong to M and their F –polynomials and g–vectors are respectively

" 0 # Fw(y) = y2 + 1, gw = −1 (65) 1 " −1 # Fz(y) = y1y3 + y1 + 1 gz = 1 0

By deﬁnition u1 = ZW −y1y3−y2. We notice that by inverting the equalities in (64) we get: x3 (66) u1 = [Fz(ˆy1, yˆ2, yˆ3)Fw(ˆy1, yˆ2, yˆ3) − yˆ1yˆ3 − yˆ2] x1 and hence u1 belongs to M and its F –polynomial and its g–vector are respectively

" −1 #

(67) Fu1 (y1, y2, y3) = y1y2y3 + y1y2 + y1 + 1 gu1 = 0 . 1 (1) CLUSTER ALGEBRAS OF TYPE A2 19 Similarly, by using their deﬁnition and induction on n, we get:

2 x3 2 u2 = [Fu (ˆy1, yˆ2, yˆ3) − 2ˆy1yˆ2yˆ3]( ) (68) 1 x1 u = [F F (ˆy , yˆ , yˆ ) − yˆ yˆ yˆ F (ˆy , yˆ , yˆ )]( x3 )n+1 n+1 u1 un 1 2 3 1 2 3 un−1 1 2 3 x1

We hence have that un belongs to M and its F –polynomial and g–vector is given by the following lemma. Lemma 4.1. For n ≥ 1 X h i (69) F (y) = ynδ + e1−e3 e1−1n−e3 + e1−1n−e3−1 ye + 1. un e2−e3 e3 n−e1 e3−1 n−e1 e and " −n #

(70) gun = 0 n

Proof. We need to show that the family {Fun : n ≥ 1} of polynomials deﬁned by (69) satisﬁes the initial conditions:

Fu1 (y1, y2, y3) = y1y2y3 + y1y2 + y1 + 1 2 Fu2 (y1, y2, y3) = Fu1 (y1, y2, y3) − 2y1y2y3 together with the recurrence relation:

Fun+1 (y1, y2, y3) = Fun Fu1 (y1, y2, y3) − y1y2y3Fun−1 (y1, y2, y3). This is done by direct check. We notice that we have not made any assumption about the semifield of coefficients and hence the expansion of un in the cluster {x1, x2, x3} has the form (7) for every choice of the coefficient semifield. Moreover its F – polynomial Fun has constant term 1. We now consider cluster variables xm, m ≥ 1. We proceed by induction on m ≥ 1. For m = 1, 2 or 3, xm belongs to M: its F –polynomial is 1 and 3 its g–vector is em, the m–th basis vector of Z . We then assume m + 3 ≥ 4 and we prove the statement for xm+3. By the exchange relation (51) and by inductive hypothesis we have the following relation:

gm gm+1+gm+2 d(xm+3) (71) Fm(yˆ)x xm+3 = Fm+1(yˆ)Fm+2(yˆ)x + y where yˆ = {yˆ1, yˆ2, yˆ3} and (72) yˆ := y1 = y xh1 yˆ := y2x1 y xh2 yˆ := y x x = y xh3 1 x2x3 1 2 x3 2 3 3 1 2 3 being hi the i–th column vector of the exchange matrix H deﬁned in (1). In particular, by inverting these expressions, (71) becomes:

t gm gm+1+gm+2 d(xm+3) H d(xm+3) (73) Fm(yˆ)x xm+3 = Fm+1(yˆ)Fm+2(yˆ)x +yˆ x where Ht is the transpose of the matrix H. We have the following elementary lemma. 3 Lemma 4.2. Let gm : m ≥ 0 ⊂ Z be a family of integer vectors deﬁned by the initial conditions gi = ei, i = 1, 2, 3, together with the recurrence relations: gm+3 = gm+2 + gm+1 − gm. 20 GIOVANNI CERULLI IRELLI

3 Let {dm} ⊂ Z be a family of integer vectors deﬁned by the initial conditions di = −ei, i = 1, 2, 3, together with the recurrence relations:

dm+3 = dm+2 + dm+1 − dm. t t Then H dm+3 = gm+1 + gm+2, where H is the transpose of the exchange matrix H given in (1). By lemma 4.2, (73) becomes

gm d(xm+3) gm+1+gm+2 (74) Fm(yˆ)x xm+3 = (Fm+1(yˆ)Fm+2(yˆ) + yˆ )x .

Every cluster variable xm, m ≥ 1 can be hence written in the form xm = gm Fm(ˆy1, yˆ2, yˆ3)x where the family {Fm(y1, y2, y3): m ≥ 1} ⊂ Q(y1, y2, y3) is a family of rational functions satisfying the initial condition F1 = F2 = F3 = 1 together with the recurrence relation:

d(xm+3) (75) FmFm+3 = Fm+1Fm+2 + y 3 and {gm : m ≥ 1} ⊂ Z is a family of integer vectors satisfying the initial conditions gi = ei, for i = 1, 2, 3, together with the recurrence relations: for m ≥ 1

(76) gm+3 = gm+1 + gm+2 − gm

The following lemma shows that for every m, the rational function Fm is a polynomial with positive coeﬃcients and constant term 1. Lemma 4.3. The unique solution of the recurrence (75) is given by: for m ≥ 0 X e1 − e3 m − 1 − e3 e1 − 1 (77) F (y) = ye + 1. 2m+1 e − e m − 1 − e e e 2 3 1 3 For m ≥ 1 X e1 − 1 m − e2 m − 1 − e3 (78) F (y) = ye + 1. 2m+2 e e − e e − e e 3 1 2 2 3 Lemma 4.4. The unique solution of the recurrence (76) is given by: For every m ≥ 0,:

"1−m# "−m# (79) g2m+1 = 0 g2m+2 = 1 m m

"−m# " −m # (80) g−(2m+1) = −1 g−2m = 0 m m−1

And (7) is proved for the cluster C = {x1, x2, x3}. Let now P be any semiﬁeld and let us prove that every cluster variable xm with m ≥ 1, has the form (8) in the cluster {x1, x2, x3}. This follows by induction on m by using the exchange relation (28) and the relation (75) between F –polynomials. It remains to prove that the elements of B have the form (7) in every cluster of AP . It is suﬃcient to show that they have this form in the seed 0 −1 2 ! cyc cyc (81) Σ := {H = 1 0 −1 , {x1, w, x3}, {y1, y2, y3}}. −2 1 0 (1) CLUSTER ALGEBRAS OF TYPE A2 21

Let hence P be any semiﬁeld. We introduce the elements ˆy1, ˆy2 and ˆy3 of FP in analogy with (64) as follows:

2 y1w y1x3 y3x1 (82) ˆy1 := 2 ˆy2 := ˆy3 := x3 x1 (y2⊕1)w Moreover, by definition of the coefficient mutation (14) in direction 2, the coefficients y1, y2 and y3 of the seed Σ in the semifield P, are given by: (83) y = y1y2 y = 1 y = y (y ⊕ 1) 1 y2⊕1 2 y2 3 3 2

The following lemma shows that the elements {ˆyi} are obtained from {yî} by the mutation (14) in direction 2 (in the terminology of [7] this means that the families {yî;C} form a Y –pattern): Lemma 4.5. yˆ = ˆy1ˆy2 yˆ = 1 yˆ = ˆy (ˆy + 1) 1 ˆy2+1 2 ˆy2 3 3 2 Proof. By definition we have

yˆ := y1 yˆ = y2x1 y yˆ := y x x x = x1+y2x3 1 x2x3 2 x3 2 3 3 1 2 2 w and hence the proof follows by direct check.

Let b be an element of B. Let Fb and gb(g1, g2, g3) be respectively the F –polynomial and the g–vector of b in {x1, x2, x3}. The expansion of b in the seed Σcyc is hence given by:

Fb(ˆy1, yˆ2, yˆ3) g1 g2 g3 b = x1 x2 x3 = Fb|P(y1, y2, y3) ˆy1ˆy2 1 Fb( , , ˆy3(ˆy2 + 1)) ˆy2+1 ˆy2 g2 g1 −g2 g3 = y y (x1 + y2x3) x1 w x3 = F | ( 1 2 , 1 , y (y ⊕ 1))(y ⊕ 1)g2 b P y2⊕1 y2 3 2 2

ˆy1ˆy2 1 g2 Fb( , , ˆy3(ˆy2 + 1))(ˆy2 + 1) ˆy2+1 ˆy2 g1−g2 −g2 g3 = y y x1 w x3 = F | ( 1 2 , 1 , y (y ⊕ 1))(y ⊕ 1)g2 b P y2⊕1 y2 3 2 2 t Definition 4.1. For every b ∈ B let gb = (g1, g2, g3) be the g–vector of b in w {x1, x2, x3}. We define the rational function Fb (y1, y2, y3) ∈ Q(y1, y2, y3) as follows: y y 1 w 1 2 g2 Fb (y1, y2, y3) := Fb( , , y3(y2 + 1))(y2 + 1) . y2 + 1 y2 w t 3 We define the vector g = (g1;w, g2;w, g3;w) ∈ Z as follows:

(84) g1;w := g1 − g2 g2;w = −g2 g3;w = g3 In view of deﬁnition 4.1 every element b of B has the form:

w gw b = Fb (y1, y2, y3)(x1, w, x3) b . w The following lemma shows that Fb is a polynomial with positive integer coeﬃcients and constant term 1. 22 GIOVANNI CERULLI IRELLI

Lemma 4.6. For every m ≥ 0:

w X e e (85) F (y) = (m−1−e3)(e1−1)y 1 y 3 +1; 2m+1 m−1−e1 e3 1 3 e1,e3

w X e −1 m−1−e +e 1 e (86) F (y) = ( 1 )( 3 2)( )y +y2+1; 2m+2 e3 m−1−e1+e2 e2 e w X e e (87) F (y) = (m−e3)(e1+1)y 1 y 3 +ymym+1; −(2m+1) m−e1 e3 1 3 1 3 e1,e3

w X e +1−e m−e 1 e m m+1 (88) F (y) = ( 1 2)( 3)( )y +y y (y2+1). −2m e3−e2 m−e1 e2 1 3 e

w 2 (89) Fz (y) = y1y2y3 + y1y2y3 + y1y2 + y2 + 1. For every n ≥ 1:

w n n X h i e e (90) F (y) = y y + (n−e3)(e1−1)+(n−e3−1)(e1−1) y 1 y 3 + 1. un 1 3 n−e1 e3 n−e1 e3−1 1 3 e1,e3 Lemma 4.7. The unique solution of (84) is given by: for every m ≥ 0

" 1−m # " 1−m # w w (91) g2m+1 = 0 g2m+2 = −1 m m " −m # " −m # w w (92) g−(2m+1) = 1 g−2m = 0 m−1 m−1 For every n ≥ 1

" 0 # " 0 # " −n # w w −1 (93) gw = 1 gz = gun = 0 0 0 n 4.2. Proof of proposition 1.3. It follows from the explicit formulas for the g–vectors given in lemma 4.4 and from the explicit formulas for the denominator vectors given in (35), that formula (10) holds for cluster variables and for the un’s. By corollary 2.1 denominator vectors of cluster variables belonging to the same cluster are sign–coherent. The matrix of (10) is hence linear in every cone generated by such vectors. Then the claim follows for cluster monomials. By lemma 2.3 denominator vectors of the un ’s, w and z lie in the positive octant Q+ in which EQ is linear. The claim is hence true k k for unw and unz , n, k ≥ 0.

5. Proof of theorem 1.1 Let P be a tropical semiﬁeld. In this section we prove that the set B of k k cluster monomials and of the elements {unw , unz |n ≥ 1, k ≥ 0} of the cluster algebra AP has the following properties: • B is a linearly independent set overZP (section 5.1); • the elements of B are positive (section 5.2);

• B spans AP over ZP (section 5.3); • the elements of B are positive indecomposable (section 5.4). and hence B is a canonically positive basis of AP. (1) CLUSTER ALGEBRAS OF TYPE A2 23

5.1. Linear independence of B. Let P be an arbitrary semifield and let AP be the cluster algebra with initial seed Σ given by (1). In view of proposition 1.1 the expansion of an element b of B in the seed Σ has the form: F (y xh1 , y xh2 , y xh3 )xgb b = b 1 2 3 Fb|P(y1, y2, y3) where Fb and gb are respectively the polynomial and the vector given in section 4.1, hi is the i–th column vector of the exchange matrix H of the seed P e Σ. Moreover the polynomial Fb has the form: Fb(y1, y2, y3) = 1+ e χe(b)y where the sum is over a finite set of non–negative integer vectors E(b) = 3 {e ∈ Z≥0 \{0}|χe(b) 6= 0} and the coefficients {χe(b): e ∈ E(b)} are positive integer numbers. The expansion of b has hence the form: g e g +P3 e h x b + P χ (b)y x b i=1 i i e=(e1,e2,e3) e b = L e 1 ⊕ e χe(b)y 3 3 We introduce in Z the following binary relation: given s, t ∈ Z we say that s ≤H t if and only if there exist non–negative integers e1, e2, e3 such that P3 t = s+ i=1 eihi. The vectors h1, h2 and h3 form a pointed cone, i.e. a non– negative linear combination of them is zero if and only if all the coefficients 3 are zero. The relation ≤H is hence a partial order on Z . The map b 7→ gb 3 between B and Z is injective (actually bijective) by proposition 1.2 and hence the partial order ≤H induces a partial order on B given by: 0 (94) b ≤ b ⇐⇒ gb ≤H gb0 . 0 In particular every finite subset B of B has a minimal object b0. Then the g 0 monomial x b0 is not a summand of every other element of B . We conclude that B is a linearly independent set over ZP. 5.2. Positivity of the elements of B. We now show that the elements of the set B defined in theorem 1.1 are positive, i.e. their laurent expansion in every cluster of AP has coefficients in Z≥0P. In view of remark 2.1 it is sufficient to show that they have such property only in the two clusters {x1, x2, x3} and {x1, w, x3}. By theorem 1.3 the cluster variables w, z , un and xm with m, n ≥ 1 have such property. By remark 2.1 it follows that all the cluster variables are positive and since a product of positive elements is positive all the cluster monomials are positive.

5.3. The set B spans AP over ZP. In this section we show the set B defined in theorem 1.1 spans AP over ZP. The strategy of the proof is the following: since B contains cluster variables, the monomials in its elements span AP over ZP. It is then sufficient to express every such monomial as a ZP–linear combination of elements of B. We introduce the following multi–degree on monomials in the elements of B: the generic monomial M a1 as b1 bt c d has the form M = un1 ··· uns xm1 ··· xmt w z where 0 < n1 < ··· < ns, m1 < ··· < mt and the exponents are positive integers. We define the 3 multi-degree µ(M) = (µ1(M), µ2(M), µ3(M)) ∈ Z≥0 by setting  Ps Pt  µ1(M) := i=1 ai + j=1 bj + c + d (95) µ2(M) := mt − m1  µ3(M) := b1 + bt 24 GIOVANNI CERULLI IRELLI

3 The lexicographic order of Z makes it a total ordering on the set of monomials in the elements of B. In section 5.3.2 we show that every such monomial can be expressed as a linear combination of monomials of (lexicographically) smaller multi-degree. In section 5.3.1 we ﬁnd the minimal monomials which do not belong to B and express them as linear combinations of elements of B. We refer to such expressions as straightening relations.

5.3.1. Straightening relations. In this section we express the monomials in the elements of B which do not belong to B and are minimal with respect to the multi–degree (95) as ZP–linear combinations of elements of B. We notice that the multi–degree (95) do not depend on coeﬃcients, and hence a cluster variable s and the relative principal cluster variable S := Fs|P(y1, y2, y3)s in the initial seed Σ have the same multi–degree. It is convenient to consider principal cluster variables in the initial seed Σ since the exchange relations are simpler and are given by (50), (51), (52), (53), (54) and (55). Such minimal monomials are the following:

(96) unup; unXm; XmXm+2+n; X2mW ; X2m+1Z; ZW for every n, p ≥ 1 and m ∈ Z. Indeed every monomial M in (96) satisﬁes µ1(M) = 2 and hence they are minimal (it follows from the deﬁnition that µ1(M) = 1 if and only if M is either a cluster variable or one of the un’s). Moreover they are the only monomials not belonging to B with this property. The straightening relations for the monomials X2mW , X2m+1Z, ZW are given respectively by (52), (53) and (4). Propositions 5.1 and 5.2 give the remaining ones. Proposition 5.1. For every n, p ≥ 1: pδ un+p + y un−p if n > p (97) unup = nδ u2n + 2y if n = p where δ := (1, 1, 1)t.

Proof. We use the deﬁnition of the un’s given in (5). For simplicity we δ assume now that u0 := 2, so that the relation u1un = un+1 + y un−1 holds for every n ≥ 1 (instead of holding only for n ≥ 2). Moreover, with this convention, we have to prove that for every p : 1 ≤ p ≤ n we have: pδ (98) unup = un+p + y un−p

If n = p = 1 then (98) is the definition (4) of u2; we assume n ≥ 2 and we proceed by induction on p ≥ 1: if p = 1, then (98) is just the definition (5) of un+1. We then assume 2 ≤ p + 1 ≤ n and we get: δ unup+1 = un[u1up − y up−1] = pδ δ (p−1)δ = u1[un+p + y un−p] − y [un+p−1 + y un−p+1] = δ pδ δ = un+1+p + y un+p−1 + y [un+1−p + y un−p−1] δ (p−1)δ −y [un+p−1 + y un−p+1] = (p+1)δ = un+p+1 + y un−(p+1) (1) CLUSTER ALGEBRAS OF TYPE A2 25 In order to give the remaining straightening relations we need to introduce the following coefficients.

Definition 5.1. For every m ∈ Z we define yd(xm+3) = y if m ≥ 1 (99) ξ := 1;m m d(xm) y = y1;m−3 if m ≤ 0 and also ( min ( 1 if m ≥ 1 − nδ + (100) ζ (m) = ξm ⊕y if m ≥ 1 ; ζ (m) = min n n nδ 1 if m ≤ 0 ξm ⊕y if m ≤ 1 where nδ = (n, n, n)t. For every integer k ≥ 0 we define (101) k d k e b k c d k e b k c d k e b k c δ γ (k) = y 2 y 2 y 2 ; γ (k) = y 2 y 2 y 2 ; γ (k) = y 2 if k is even, 1 1 2 3 2 1 2 3 3 0 if k is odd. and we define for i = 1, 2, 3 the corresponding elements of AP : X k Γ (n) = (b c + 1) · γ (k) · u . i 2 i n−k k≥0

We also deﬁne for every m ∈ Z and m1 ≥ 0: ( min − ξm ⊕ ξm+m if m ≤ 0 < m + m1 ηm;m := 1 1 1 otherwise and ( 1 if m ≤ 0 < m + m + 1 ηm;m1 := min ξm ⊕ ξm+m1 otherwise where we use the following notation: min e d min(e1,d1) min(e2,d2) min(e3,d3) y ⊕ y := y1 y2 y3 Proposition 5.2. With the notations of deﬁnition 5.1 the following equalities hold. (i) : For every m ∈ Z and n ≥ 1: − + (102) unXm = ζn (m)Xm−2n + ζn (m)Xm+2n (ii) : For every m ∈ Z even and n ≥ 0: − + (103) XmXm+2n+3 = ηm;2n+3Xm+n+1Xm+n+2 + ηm;2n+3Γ1(n) (iii) : For every m ∈ Z odd and n ≥ 0: − + (104) XmXm+2n+3 = ηm;2n+3Xm+n+1Xm+n+2 + ηm;2n+3Γ2(n) (iv) : For every m ∈ Z even and n ≥ 2: − + (105) XmXm+2n = η X n X n + η Γ3(n − 2)Z m;2n m+2b 2 c m+2d 2 e m;2n (v) : For every m ∈ Z odd and n ≥ 2: − + (106) XmXm+2n = η X n X n + η Γ3(n − 2)W m;2n m+2b 2 c m+2d 2 e m;2n 26 GIOVANNI CERULLI IRELLI

Proof. We prove part (i) by induction on n ≥ 1. By using relations (52) and (53) it is easy to verify that for every m ∈ Z we have:  δ  y Xm−2 + Xm+2 if m ≥ 2   y1X−1 + X3 if m = 1 u1Xm = X−2 + y3X2 if m = 0  X−3 + y2y3X1 if m = −1  δ  Xm−2 + y Xm+2 if m ≤ −2 and (102) follows for n = 1. We now proceed by induction on n ≥ 1. We use δ the convention that u0 = 2 so that the relation un+1 = u1un −y un−1 (given in deﬁnition 1.1) holds for every n ≥ 1. Moreover, with this convention, since ± ζ0 (m) = 1, (102) still holds for n = 0. We have δ un+1Xm = u1unXm − y un−1Xm = − + un[ζ1 (m)Xm−2 + ζ1 (m)Xm+2]+ δ − + −y [ζn−1(m)Xm−2n+2 + ζn−1(m)Xm+2n−2] =

− − + ζ1 (m)[ζn (m − 2)Xm−2−2n + ζn (m − 2)Xm−2+2n]+ + − + +ζ1 (m)[ζn (m + 2)Xm+2−2n + ζn (m + 2)Xm+2+2n]+ δ − δ + −y ζn−1(m)Xm+2−2n − y ζn−1(m)Xm−2+2n =

− − − + δ + Xm−2−2n[ζ1 (m)ζn (m − 2)]+Xm−2+2n[ζ1 (m)ζn (m − 2)−y ζn−1(m)]+ + − δ − + + +Xm+2−2n[ζ1 (m)ζn (m + 2) − y ζn−1(m)]+Xm+2+2n[ζ1 (m)ζn (m + 2)] The claim follows by Lemma 5.1 below.

Lemma 5.1. For every m ∈ Z and n ≥ 1 the following equalities hold: − − − (1) ζ1 (m)ζn (m − 2) = ζn+1(m); − + δ + (2) ζ1 (m)ζn (m − 2) − y ζn−1(m) = 0; + − δ − (3) ζ1 (m)ζn (m + 2) − y ζn−1(m) = 0; + + + (4) ζ1 (m)ζn (m + 2) = ζn+1(m). The proof of lemma 5.1 is by direct check. We prove part (ii) and (iii) together. It is convenient to prove that the following relation holds for every m ∈ Z, n ≥ 0 and i = im = 1 if m is even and 2 if m is odd: + − (107) ηm;2n+3Γi(n) = XmXm+2n+3 − ηm;2n+3Xm+n+1Xm+n+2 We proceed by induction on n ≥ 0. We ﬁrst prove (107) for n = 0. In this case Γ1(0) = Γ2(0) = 1. By the exchange relations (50) and (51) we know that for every m ∈ Z the following relation holds: ym+3Xm+1Xm+2 + 1 if m = 0, −1, −2; (108) XmXm+3 = Xm+1Xm+2 + y1;m otherwise. We need the following lemma. Lemma 5.2. With notations of deﬁnition 5.1 we have the following: (1) CLUSTER ALGEBRAS OF TYPE A2 27

(1) For every m ∈ Z and k ≥ 0: ( min ξm if m ≥ 1, (109) ξm ⊕ ξm+k = min ξm ⊕ ξm+k if m + k ≤ 0. min ξ if − 1 ≤ m ≤ 2, (110) ξ ⊕ yδ = m m yδ otherwise. If m ≥ 0 and n ≥ 1 we get   ξ−m if m < n − 1, min  ykδ if m = n − 1 = 2k, (111) ξ−m ⊕ ξn = kδ y2y if m = n − 1 = 2k + 1,   ξn if m > n − 1. (2) For every m ∈ Z and n ≥ 1 the following relation holds: + − − + (112) ζn (m) = (13)ζn (1 − m); ζn (m) = (13)ζn (1 − m)

where (13) is the automorphism of P that exchanges y1 with y3. (3) For every n ≥ 1 and i = 1, 2, 3 we have: δ (113) u1Γi(n) = Γi(n + 1) + y Γi(n − 1) − γi(n + 1) of lemma 5.2. (109) and (110) follow directly by the deﬁnition of ξm and ξm+k by using lemma 2.2: indeed one can see that d(xm) ≤ d(xm +k) (resp. d(xm) ≥ d(xm+k)) if m ≥ 1 (resp. m + k ≤ 0). (Here ≤ is understood term by term). . min min d(x ) d(x ) We now want to compute ξ−m ⊕ ξn := y −m ⊕ y n+3 . By re- 3 mark 2.1, d(x−m) = (13)d(xm+4), where (13) is the linear operator on Z that exchanges the ﬁrst entry with the third one. We now consider all the possible cases:

If m + 4 < n + 3 then d(xm+4) ≤ d(xn+3); since m + 4 and n + 3 are positive integers, d(xm+4) and d(x−m) have respectively the form 0 0 0 0 0 (d3 + 1, d2, d3) and (d3 + 1, d2, d3) for some d2, d3 , d2 , d3 ≥ 0; in 0 particular d(x−m) = (d3, d2, d3 + 1). Since by hypothesis d3 < d3 min 0 d(x−m and d2 < d2, we conclude d(x−m) ≤ d(xn+3) so that y ) ⊕ yd(xn+3) = yd(x−m). If m + 4 = n + 3 = 2k + 4 for some k ≥ 0, then by (35), d(xm+4) = min t t d(x ) (k + 1, k, k) so that (13)d(xm+4) = (k, k, k + 1) ; then y −m ⊕ yd(xn+3) = ykδ. If m + 4 = n + 3 = 2k + 5 for some k ≥ 0, then by (35), d(xm+4) = t t (k + 1, k + 1, k) so that (13)d(xm+4) = (k, k + 1, k + 1) ; then min d(x ) d(x ) kδ y −m ⊕ y n+3 = y2y . If m + 4 > n + 3 then d(xm+4) ≥ d(xn+3), then also (13)d(xm+4) ≥ d(xn+3). and (111) is proved. Formula (113) follows by (97). By part 1 of lemma 5.2, it is immediate to verify that − ym+3 if m = 0, −1, −2 + 1 if m = 0, −1, −2 ηm;3 = ; ηm;3 = 1 otherwise y1;m otherwise 28 GIOVANNI CERULLI IRELLI so that (107) specializes to (108) when n = 0, i.e. for every m ∈ Z the following relation holds − + (114) XmXm+3 = ηm;3Xm+1Xm+1 + ηm;3. We now assume n ≥ 1. In this case, by using the inductive hypothesis we have:

δ Γi(n + 1) = u1Γi(n) − y Γi(n − 1) + γi(n + 1) =

u1 − + · [XmXm+2n+3 − ηm;2n+3Xm+n+1Xm+n+2]+ ηm;2n+3 yδ − − + · [XmXm+2n+1 − ηm;2n+1Xm+nXm+n+1] + γi(n + 1) = ηm;2n+1

Xm − + + · [ζ1 (m + 2n + 3)Xm+2n+1 + ζ1 (m + 2n + 3)Xm+2n+5]+ ηm;2n+3 − ηm;2n+3 − + − + · [ζ1 (m+n+2)Xm+nXm+n+1 + ζ1 (m+n+2)Xm+n+1Xm+n+4]+ ηm;2n+3 yδ − − + · [XmXm+2n+1 − ηm;2n+1Xm+nXm+n+1] + γi(n + 1) = ηm;2n+1

ζ−(m+2n+3)η+ −yδη+ + 1 m;2n+1 m;2n+3 ζ1 (m+2n+3) = XmXm+2n+1[ + + ]+XmXm+2n+5[ + ]+ ηm;2n+1ηm;2n+3 ηm;2n+3 δ − + − − + y ηm;2n+1ηm;2n+3−ηm;2n+3ζ1 (m+n+2)ηm;2n+1 +Xm+nXm+n+1[ + + + ηm;2n+1ηm;2n+3 + − ζ1 (m+n+2)ηm;2n+3 − + Xm+n+1Xm+n+4 + γi(n + 1) = ηm;2n+3

ζ−(m+2n+3)η+ −yδη+ + 1 m;2n+1 m;2n+3 ζ1 (m+2n+3) = XmXm+2n+1[ + + + XmXm+2n+5[ + ]+ ηm;2n+1ηm;2n+3 ηm;2n+3 δ − + − − + y ηm;2n+1ηm;2n+3−ηm;2n+3ζ1 (m+n+2)ηm;2n+1 +Xm+nXm+n+1[ + + ]+ ηm;2n+1ηm;2n+3 + − ζ1 (m+n+2)ηm;2n+3 + + + [ηm+n+1;3Xm+n+2Xm+n+3 + ηm+n+1;3] + γi(n + 1) ηm;2n+3 Lemma 5.3 below shows that this polynomial is equal to

1 − + [XmXm+2n+5 − ηm;2n+5Xm+n+2Xm+n+3] ηm;2n+5 and we are done. We prove (iii) and (iv) together. In order to do that we introduce the variable c = c(m) depending on m ∈ Z in the following way: c is w if m is odd and c is z if m is even. With this convention, both (105) and (106) are equivalent to the following:

1 − (115) c(m)Γ3(n − 2) = [XmXm+2n − η X n X n ]. + m;2n m+2b 2 c m+2d 2 e ηm;2n In order to prove (115) we proceed by induction on n ≥ 2. We verify directly the formula for n = 2 and n = 3. We then assume n ≥ 4. By using (113) (1) CLUSTER ALGEBRAS OF TYPE A2 29 and the inductive hypothesis we get the following equality: − + δ + ζ1 (m + 2n − 2)ηm;2n−4 − y ηm;2n−2 c(m)Γ3(n − 2) = XmXm+2n+4[ + + ] ηm;2n−2ηm;2n−4 + ζ1 (m + 2n − 2) +XmXm+2n[ + ] ηm;2n−2 δ − + − + − n−1 y ηm;2n−4ηm;2n−2 −ηm;2n−2ηm;2n−4ζ1 (m+2d 2 e) +X n−2 X n−2 [ ] m+2b 2 c m+2d 2 e + + ηm;2n−2ηm;2n−4 − + n−1 ηm;2n−2ζ1 (m + 2d 2 e) − X n−1 X n−1 + c(m)γ3(n − 2) + m+2b 2 c m+2d 2 e ηm;2n−2 Lemma 5.3 concludes the proof.

Lemma 5.3. For every n ≥ 1, m1 ≥ 3 and m ∈ Z the following equalities hold in ZP: − + δ (1) ζ1 (m + m1 + 2)ηm;m1 − y ηm;m1+2 = 0; (2) ζ+(m + m ) = η+ /η+ ; 1 1 m;m1 m;m1+2 (3) yδη− η+ − η− η+ ζ−(m + d m1+2 e) = 0; m;m1 m;m1+2 m;m1+2 m;m1 1 2 + − − + − + (4) ζ1 (m + n + 2)ηm;2n+3ηm+n+1;3 = ηm;2n+3ηm;2n+5/ηm;2n+5; (5) For i = i(m) := 1 if m is even and i = i(m) := 2 if m is odd we have for every n ≥ 1: + + − + γi(n + 1)ηm;2n+3 − ζ1 (m + n + 2)ηm;2n+3ηm+n+1;3 = 0;

− + n−1 ηm;2n−2ζ1 (m+2d 2 e) (6) + Xm+2b n−1 cXm+2d n−1 e − cγ3(n − 2) = ηm;2n−2 2 2 − ηm;2n + Xm+2b n cXm+2d n e ηm;2n 2 2 The proof of lemma 5.3 follows by direct check.

5.3.2. Span property. In this section we prove that a monomial of the form a1 as b1 bt c d M = un1 ··· uns xm1 ··· xmt w z in the elements of B is a ZP–linear combination of elements of B. We proceed by induction on the multi–degree µ(M) (deﬁned in (95)). If µ1(M) = 1 then M is either a cluster variable Ps or one of the un’s. If i=1 ai ≥ 2 (resp. 1) then one can apply (97) (resp. (102)), expressing M as a linear combination of monomials with smaller b1 bt c d value of µ1. So we can assume that M = xm1 ··· xmt w z . If both c and d are positive, by using the fact that ZW = u1 + y1y3 + y2, one obtains again a sum of two monomials with smaller value of µ1. So we assume that d = 0 (resp. c = 0) and that we can apply the relation (52) (resp. (53)), i.e. some mi is odd (resp. even). We again obtain a sum of two monomials having smaller value of µ1 than the initial one. So we can assume that M has one of the following forms: M := (Q xbi )wc or M := (Q xbi )zd 1 mi even mi 2 mi odd mi b1 bt or M3 := xm1 ··· xmt with mt − m1 ≥ 3. We apply either (105) or (106) or (114) to the product xm1 xmt . By inspection, in the resulting expression for both M1 and M2, all the monomials except at most one that has smaller value of µ1 have the same value of µ1. By further inspection, for 0 every such monomial M , if min(b1, bt) = 1 (resp. min(b1, bt) ≥ 2) then 30 GIOVANNI CERULLI IRELLI

0 0 0 µ2(M ) < µ2(M) (resp. µ2(M ) = µ2(M) and µ3(M ) = µ3(M) − 2). Anal- ogously in the resulting expression for M3 , there is precisely one monomial 0 M with µ1(M ) = µ1(M), while the rest of the terms have smaller value of 0 µ1. Moreover if min(b1, bt) = 1 (resp. min(b1, bt) ≥ 2) then µ2(M ) < µ2(M) 0 0 (resp. µ2(M ) = µ2(M) and µ3(M ) = µ3(M) − 2).

5.4. The elements of B are positive indecomposable. In this section we prove the elements of the spanning set (section 5.3 ) of positive elements (section 5.2) B are positive indecomposable, i.e. they cannot be written as a sum of two positive elements. The proof is based on the following lemma.

Lemma 5.4. Let P be a semifield and let A be a cluster algebra of rank n with coefficients in P. Let B be a ZP–basis of A that satisfies the following hypothesis:

[Ind] for every element b of B there exist a cluster Cb = {c1, ··· , cn} and a (Laurent) monomial cγb such that cγb is a summand of the Laurent expansion of b in Cb but it is not a summand of the Laurent expansion in the same cluster Cb of every other element of B. Moreover the γ coefficient of c b is an element of P. Then if the elements of B are positive then they are positive indecomposable and B is a canonically positive basis of A . Proof. We need to show that the linear expansion in B of a positive element p of A has coefficients in Z≥0P. We express p as a ZP–linear combination π of elements of B with coefficients in ZP. Suppose that b ∈ B appears in π with non–zero coefficient ab. We expand p in the cluster Cb = {c1, ··· , cn} of hypothesis [ind]. Then the Laurent monomial cγb appears with coefficient γ abyb, where yb ∈ P is the coefficient of c b in the expansion of b in the cluster Cb. Since p is positive, abyb ∈ Z≥0P; since yb ∈ P, ab ∈ Z≥0P. By repeating the argument for all the elements of B appearing in π, we get that all the coefficients of π lie in Z≥0P. In section 5.4.3 we prove that B satisfies hypothesis [Ind] of lemma 5.4 and hence the elements of B are positive indecomposable. In order to do that we need to study Newton polytopes of the elements of B in every cluster of

AP. By the symmetries in the exchange relations, it is suﬃcient to consider only the two clusters {x1, x2, x3} and {x1, w, x3}.

5.4.1. Newton polygons in the cluster {x1, x2, x3}. In this section we study Newton polytopes in the initial cluster CIn = {x1, x2, x3} of the elements of the basis B of the cluster algebra AP . Recall that the Newton polytope ±1 ±1 ±1 NewtC(x) of a Laurent polynomial x ∈ Z[s1 , s2 , s3 ] with respect to the ordered set C = {s1, s2, s3} is the convex hull in QR = Rα1 ⊕ Rα2 ⊕ Rα3 g g g g of all lattice points g = (g1, g2, g3) such that the monomial s := s 1 s 2 s 3 appears with a non-zero coeﬃcient in x. We say that a vertex γ of NewtC(x) is monic if the corresponding monomial sγ appears in the expansion of x in C with a coeﬃcient in P. With some abuse of language we say that an element x of AP is monic in the cluster C if all the vertices of NewtC(x) are monic. (1) CLUSTER ALGEBRAS OF TYPE A2 31 Example 5.1. By (2) we immediately have

" 1 # " 0 # −1 −1 (116) Newt{x1,x2,x3}(w) = Conv{ , } 0 1   " 0 # −1 " −1 # 1 0 (117) Newt{x1,x2,x3}(z) = Conv{ ,   , 1 } −1 −1 0

3 where Conv means convex hull in QR = R . Moreover both w and z are monic in the initial cluster {x1, x2, x3} and their Newton polytopes are polygons contained respectively in the plane P2 := {(g1, g2, g3): g1 − g2 + g3 = 2} and P−2 := {(g1, g2, g3): g1 − g2 + g3 = −2}. By the symmetries in the exchange relations, the Newton polytope of the cluster variable x−m is obtained from the Newton polytope of the cluster 3 variable xm+4 by the automorphism (13) of Q = Z that exchanges the ﬁrst coordinate with the third one:

(118) Newt{x1,x2,x3}(x−m) = (13)Newt{x1,x2,x3}(xm+4). The following proposition gives the Newton polytopes in the initial cluster CIn = {x1, x2, x3} of the elements of B.

Proposition 5.3. The elements {xm : m ∈ Z} and {un : n ≥ 1} of AP are monic in the initial cluster {x1, x2, x3}. Moreover, for m ≥ 2 and n ≥ 1, the following explicit formulas hold: 2 1 − m 3 2 1 − m 3 2 0 3 2 m − 2 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 Newt (x ) = Conv{6 7,6 7,6 7,6 7} {x1,x2,x3} 2m+1 6 0 7 6 1 − m 7 6 1 − m 7 6 −1 7 4 m 5 4 1 5 4 2 − m 5 4 2 − m 5 (119) 2 1 − m 3 2 1 − m 3 2 −1 3 2 m − 3 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 Newt (x ) = Conv{6 7,6 7,6 7,6 7} {x1,x2,x3} 2m 6 1 7 6 2 − m 7 6 2 − m 7 6 0 7 4 m − 1 5 4 0 5 4 2 − m 5 4 2 − m 5

2 −n 3 2 −n 3 2 0 3 2 n 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 (120) Newt (u ) = Conv{6 7,6 7,6 7,6 7} {x1,x2,x3} n 6 0 7 6 −n 7 6 −n 7 6 0 7 4 n 5 4 0 5 4 −n 5 4 −n 5

Proof. By proposition 1.1, up to a multiple in P which does not modify the Newton polytope, every element b of B has the form

X e gb+He b = χe(b)y x e∈E(b) e where χe(b) is the coeﬃcient of y in the polynomial Fb given in lemma 4.1, E(b) = E (b) := Conv{e ∈ 3|χ (b) 6= 0} is the convex hull in Q of {x1,x2,x3} Z e R the support of e 7→ χe(b), gb is the g–vector of b given by lemma 4.4 and H is the initial exchange matrix given in (1). The aﬃne map Nb : e 7→ gb + He sends convex sets to convex sets and generators to generators. In particular if E(b) = Conv{e1, ··· , en}, then Newt(b) = Conv{Nb(e1), ··· ,Nb(en)}. The proof is based on the following lemma. 32 GIOVANNI CERULLI IRELLI

Lemma 5.5. With the previous notations: 2 0 3 2 m − 1 3 2 m − 1 3 2 m − 1 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 E(x ) = Conv{6 7,6 7,6 7,6 7}; 2m+1 6 0 7 6 0 7 6 m − 1 7 6 m − 1 7 4 0 5 4 0 5 4 0 5 4 m − 2 5

2 0 3 2 m − 1 3 2 m − 2 3 2 m − 1 3 2 m − 1 3 2 m − 2 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 E(x ) = Conv{6 7,6 7,6 7,6 7,6 7,6 7}; 2m 6 0 7 6 0 7 6 m − 2 7 6 m − 2 7 6 m − 2 7 6 m − 2 7 4 0 5 4 0 5 4 0 5 4 0 5 4 m − 2 5 4 m − 3 5 2 0 3 2 n 3 2 n 3 2 n 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 E(u ) = Conv{6 7,6 7,6 7,6 7}. n 6 0 7 6 0 7 6 n 7 6 n 7 4 0 5 4 0 5 4 0 5 4 n 5

e Proof. By definition χe(b) is the coefficient of y of the polynomial Fb given in lemma 4.1. The proof hence follows by direct check. In order to finish the proof of proposition 5.3 we need to show that Nb(E(b)) coincides with the expression of Newt(b) given there. This follows from lemma 5.5 by applying the affine transformation Nb to every generator of Conv(E(b)): if b = x2m+1 or b = un this is immediate; if b = x2m we apply Nx2m to Conv(E(x2m)) and we get 2 1 − m 3 2 1 − m 3 2 −1 3 2 −1 3 2 m − 3 3 2 m − 4 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 Newt(x )=Conv{6 7,6 7,6 7,6 7,6 7,6 7}. 2m 6 1 7 6 2 − m 7 6 3 − m 7 6 2 − m 7 6 0 7 6 0 7 4 m − 1 5 4 0 5 4 3 − m 5 4 2 − m 5 4 2 − m 5 4 3 − m 5 To conclude (119) we show that the third and the last generators are convex combinations of the others: indeed let v1, ··· , v6 be the generators enu- 1 2m−5 1 merated from left to right. Then v3 = 2m−3 v1 + 2m−3 v4 + 2m−3 v5 and 1 v6 = (m−1)(2m−4) [(m − 2)v1 + v2 + (m − 1)(2m − 5)v5]. Remark 5.1. The Newton polytopes of the elements of B are actually polygons. Indeed let Pi := {(e1, e2, e3) ∈ QR|e1 − e2 + e3 = i}. Then, for every m ∈ Z, NewtCIn (x2m+1) ⊂ P1, NewtCIn (un) ⊂ P0 and NewtCIn (x2m) ⊂ P−1. This is equivalent to the fact that the algebra AP is graded. Indeed if we choose the gradation deg(x1) = deg(x3) = 1, deg(y) = 0 for every y ∈ P and deg(x2) = −1 then, from the exchange relations it follows that deg(w) = 2, deg(x2m+1) = 1, deg(un) = 0, deg(x2m) = −1, deg(z) = −2 for every m ∈ Z. Proposition 5.4. (1) If b is a cluster monomial divisible by at least one cluster variable different from x1, x2 and x3, then there exists a non- zero linear form ϕb : QR → R, (e1, e2, e3) 7→ αbe1 + βbe2 + γbe3 with non–negative coefficients, αb, βb, γb ≥ 0 such that Newt{x1,x2,x3}(b) ⊂

{φb < 0}. In particular Newt{x1,x2,x3}(b) does not intersect the positive cone Q+ = Z≥0α1 + Z≥0α2 + Z≥0α3. k (2) For k ≥ 0 and n ≥ 1: Newt{x1,x2,x3}(unw ) ⊂ {g1 + 2g2 + g3 ≤ 0} k (3) For k > 0 and n ≥ 1 Newt{x1,x2,x3}(unz ) ⊂ {2g1 + g2 + 2g3 ≤ 0} a b c Moreover the cluster monomial x1x2x3, for every non-negative integers a, b and c, appears with zero coeﬃcient in the Laurent expansion of every other element of B in the initial cluster {x1, x2, x3}. (1) CLUSTER ALGEBRAS OF TYPE A2 33

p q r Proof. Since Newt(s1s2s3) = pNewt(s1) + qNewt(s2) + rNewt(s3), it is suf- ﬁcient to ﬁnd a linear form which has negative values on the vertices of Newt(s1), Newt(s2) and Newt(s3). For every b ∈ B, table 1 shows a linear form φb with the desired property. From (116), (117) and (120), (2) and (3) follow by direct check. k The only monomial that could appear in the Laurent expansion of unw k or in unz is 1. We now check that this is not the case. For k > 0 follows by k k observing that deg(unw ) = 2k and deg(unz ) = −2k (see remark 5.1), and hence 1 cannot appear in their Laurent expansion. For k = 0 we observe that by (102) unx3 = x3−2n + x3+2n. So 1 appears in the Laurent expansion of un if and only if x3 appears in the Laurent expansion of x3−2n and of x2n+3. By part (1) this cannot be the case since their Newton polygons do not intersect the positive octant Q+(in particular they do not contain the t point (0, 0, 1) ).

b ϕb p q r 2 1 x2m+1x2m+2x2m+3 [m(m − 1), m(m − 1), (m − 2m + 2 )] m ≥ 2 p r x3x5 [1, 1, 0] r > 0 p q r x3x4x5 [1, 1, 0] q > 0 p q r 1 3 2 1 x2mx2m+1x2m+2 [m(m − 1), (m − 2 )(m − 2 ), (m − 2m + 2 )] m ≥ 4 p q r x2x3x4 [1, 0, 0] p q r x4x5x6 [6, 3, 2] p q r x6x7x8 [9, 5, 5] p q r x2m+1w x2m+3 [m, 2m, (m − 2)] m ≥ 2 p q r x1w x3 [0, 1, 0] q > 0 p q r x3w x5 [0, 1, 0] q > 0 p q r m 3 x2mz x2m+2 [m(m − 1), 4 (m − 1), m(m − 2 )] m ≥ 2 p q r x2z x4 [1, 0, 0] r > 0 p q x2z [1, 0, 0] q > 0 e1 e2 e3 Table 1. Every laurent monomial s = x1 x2 x3 appearing with non–zero coefficient in the laurent expansion of the element b ∈ B satisfies ϕb(e1, e2, e3) < 0. In the second column it is written the (row) vector that defines ϕb. In the first column b is assumed to be different from a monomial in x1, x2 and x3; it hence satisfies conditions given in the third column on the right.

5.4.2. Newton polygons in the cluster {x1, w, x3}. In this section we ﬁnd explicit formulas for the Newton polytopes of the elements of B in the cluster {x1, w, x3}. Proposition 5.5. For m ≥ 2 and n ≥ 1 the following explicit formulas hold: 2 1 − m 3 2 m − 3 3 2 1 − m 3 6 7 6 7 6 7 6 7 6 7 6 7 Newt (x ) = Conv{6 7,6 7,6 7}; {x1,w,x3} 2m+1 6 0 7 6 1 7 6 m − 1 7 4 m 5 4 2 − m 5 4 2 − m 5 34 GIOVANNI CERULLI IRELLI

2 1 − m 3 2 m − 2 3 2 −m 3 2 −m 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 (121) Newt (x ) = Conv{6 7,6 7,6 7,6 7}; {x1,w,x3} 2m+2 6 −1 7 6 0 7 6 −1 7 6 m − 1 7 4 m 5 4 1 − m 5 4 m + 1 5 4 1 − m 5 2 n 3 2 −n 3 2 −n 3 2 1 − m 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 Newt (u ) = Conv{6 7,6 7,6 7,6 7}; {x1,w,x3} n 6 0 7 6 0 7 6 n 7 6 m − 2 7 4 −n 5 4 n 5 4 −n 5 4 2 − m 5

Moreover un is monic in the cluster {x1, w, x3}. 2 1 3 2 −1 3 2 −1 3 6 7 6 7 6 7 6 7 6 7 6 7 Newt (z) = Conv{6 7,6 7,6 7}; {x1,w,x3} 6 −1 7 6 0 7 6 −1 7 4 −1 5 4 −1 5 4 1 5 Proof. By proposition 1.1, every element b of B has the form X w e gw+HCyce b = χe (b)y x b e∈E(b) w e w where χe (b) is the coeﬃcient of y in the polynomial Fb given in lemma 4.6, Ew(b) = E (b) := Conv{e ∈ 3|χw(b) 6= 0} is the convex hull in Q {x1,w,x3} Z e R w w of the support of e 7→ χe (b), gb is the g–vector in the cluster {x1, w, x3} of b given by lemma 4.7 and HCyc is the exchange matrix given in (81). w w Cyc The aﬃne map Nb : e 7→ gb + H e sends convex sets to convex sets w and generators to generators. In particular if E (b) = Conv{e1, ··· , en}, w w then Newt{x1,w,x3}(b) = Conv{Nb (e1), ··· ,Nb (en)}. The proof is hence based on the following lemma. Lemma 5.6. With the previous notations: 2 0 3 2 m − 1 3 2 m − 1 3 6 7 6 7 6 7 6 7 6 7 6 7 E(x ) = Conv{6 7,6 7,6 7}; 2m+1 6 0 7 6 0 7 6 0 7 4 0 5 4 m − 2 5 4 0 5 For m ≥ 2: 2 0 3 2 m − 1 3 2 m − 1 3 2 0 3 2 m 3 2 m 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 E(x ) = Conv{6 7,6 7,6 7,6 7,6 7,6 7}; 2m+2 6 0 7 6 0 7 6 0 7 6 1 7 6 1 7 6 1 7 4 0 5 4 m − 2 5 4 0 5 4 0 5 4 0 5 4 m − 1 5 2 0 3 2 n 3 2 n 3 6 7 6 7 6 7 6 7 6 7 6 7 E(u ) = Conv{6 7,6 7,6 7}. n 6 0 7 6 0 7 6 0 7 4 0 5 4 0 5 4 n 5

Proof. It follows by a case by case inspection, by using lemma 4.6. In order to ﬁnish the proof of proposition 5.5 we need to show that w w Nb (E (b)) coincides with the expression of Newt{x1,w,x3}(b) given there. This follows from lemma 5.5 by applying the aﬃne transformation Nb to every generator of Conv(E(b)): if b = x2m+1 or b = un this is immediate; if b = x2m+2 we apply Nx2m to Conv(E(x2m)) and we get 2 1 − m 3 2 m − 3 3 2 1 − m 3 2 −m 3 2 −m 3 2 m − 2 3 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 Newt(x )=Conv{6 7,6 7,6 7,6 7,6 7,6 7}. 2m 6 −1 7 6 0 7 6 m − 2 7 6 −1 7 6 m − 1 7 6 0 7 4 m 5 4 3 − m 5 4 2 − m 5 4 m + 1 5 4 1 − m 5 4 1 − m 5 To conclude (121) we show that the second and the third generators are convex combinations of the others: indeed let v1, ··· , v6 be the generators (1) CLUSTER ALGEBRAS OF TYPE A2 35

1 1 2m−3 enumerated from left to right. Then v2 = 2m v4 + 2m(m−1) v5 + 2m−2 v6 and 1 2m−3 1 v3 = 2m−1 v1 + 2m−1 v5 + 2m−1 v6. It remains to prove that every element of B is monic. But this follows directly from proposition 5.5 and from the explicit formulas for these elements by checking that every monomial corresponding to a vertex of the newton polygon has coefficient in P. The next result is the analogous of the proposition 5.4 for the cluster {x1, w, x3}. Proposition 5.6. If b is an element of B divisible by at least one cluster variable different from x1, w and x3, then there exists a non-zero linear form ϕb : QR → R, (e1, e2, e3) 7→ αbe1 + βbe2 + γbe3 with non–negative coefficients, αb, βb, γb ≥ 0 such that Newt{x1,w,x3}(b) ⊂ {φb < 0}. In particular

Newt{x1,w,x3}(b) does not intersect the positive cone Q+ = Z≥0α1 +Z≥0α2 + Z≥0α3. a b c In particular the cluster monomial x1w x3, for every non-negative integers a, b and c, appears with zero coeﬃcient in the Laurent expansion of every other element of B in the cluster {x1, w, x3}.

Proof. Table 2 shows a desired linear form φb, for every choice of the element a b c b of B. It remains only to prove that the monomial x1w x3 does not appear k k in the Laurent expansion of unw and 1 and unz in {x1, w, x3}. It is clear k k that unw could have only w with this property. But this happens if and only if 1 appears in the Laurent expansion (in {x1, w, x3}) of un . But this cannot be the case because by (102) unx3 = x3−2n + x3+2n. So 1 appears in the Laurent expansion of un if and only if x3 appears in the Laurent expansion of either x3−2n or x2n+3. This cannot be the case because their Newton polygons do not intersect the positive octant Q+ (in particular they do not contain the point (0, 0, 1)t). k The Laurent expansion of unz cannot contain any such monomial, since its Newton polygon does not intersect the positive octant.

5.4.3. Indecomposability. By lemma 5.7 below, the set B satisﬁes hypothesis of lemma 5.4. This concludes the proof of theorem 1.1. Lemma 5.7. The set B satisﬁes hypothesis (5.4) of lemma 5.4, i.e. for every element b of B there exists a cluster C = Cb and a monic vertex γb of 0 0 NewtC(b) such that γb does not lie in NewtC(b ) if b 6= b is another element of B. Proof. If b is a cluster monomial in the elements of a cluster C, by proposi- k k tions 5.4 and 5.6 we can choose Cb = C. If b is an element of {unw , unz | n ≥ 1, k ≥ 0} we claim that the following couples have all the desired properties:

if b = un then C = {x1, x2, x3} and γ = (−n, 0, n); k if b = unw then C = {x1, w, x3} and γ = (−n, k, n); k if b = unz then C = {x2, z, x4} and γ = (−n, k, n). γ Indeed let us consider ﬁrst the case b = un. By proposition 5.3, x occurs in the Laurent expansion of un with coeﬃcient 1 and by the explicit formulas γ γ x does not occur in up for p 6= n. Moreover x do not appear in the 36 GIOVANNI CERULLI IRELLI

w b ϕb p q r 1 2 1 x2m+1x2m+2x2m+3 [m(m − 1), 4 (m − 1), (m − 2m + 2 )] m ≥ 2 p q r x1x2x3 [0, 1, 0] p q x3x4 [1, 1, 0] p q r x3x4x5 [1, 0, 0] r > 0 p q r 2 1 x2mx2m+1x2m+2 [m(m − 1), (m − 1), (m − 2m + 2 )] m ≥ 2 p q r x2x3x4 [0, 1, 0] p q r 2 1 x2m+1w x2m+3 [m(m − 1), 0, (m − 2m + 2 )] m ≥ 2 p q r x3w x5 [1, 0, 0] p q r x2mz x2m+2 [1, 2, 1] m ≥ 1 e1 e2 e3 Table 2. Every laurent monomial s = x1 w x3 appearing with non–zero coefficient in the laurent expansion of the element b ∈ B satisfies ϕb(e1, e2, e3) < 0. In the second column it is written the (row) vector that defines ϕb. In the first column b is assumed to be different from a monomial in x1, w and x3; it hence satisfies conditions given in the third column on the right.

k k γ Laurent expansion of upw and upz if k > 0, because deg(x ) = 0 whereas k k deg(unw ) = 2k and deg(unz ) = −2k (see remark 5.1). The other two cases follow by the same arguments.

Acknowledgements This paper is part of my phd thesis [3] at the “Universit`adegli studi di Padova”. I am grateful to the doctoral school of mathematics of the university of Padua and in particular to its director Professor Bruno Chiarellotto, for the opportunity he gave me to participate at several conferences and periods abroad. I want to thank the group of algebra of the university of Padova, in particular Alberto Tonolo and Silvana Bazzoni, for the possibil- ity I had to give several talks on this subject and for their useful comments. I spent three semesters of my PhD program at Northeastern University of Boston under the supervision of Professor Andrei Zelevinsky. I want to thank him for the patience he had in introducing me to this subject and for his encouraging behavior. I also want to thank the department of mathematics of the Northeastern for its kind hospitality; in particular Sachin, Shih–Wei and Daniel for several conversations about cluster algebras.

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Universita` degli studi di Padova, Dipartimento di Matematica Pura ed Ap- plicata. Via Trieste 63, 35121, Padova (ITALY) E-mail address: [email protected]