CANONICALLY POSITIVE BASIS OF CLUSTER ALGEBRAS OF TYPE A(1) 2

ALGEBRAS OF TYPE A⁽¹⁾₂

GIOVANNI CERULLI IRELLI

Abstract. We study cluster algebras of type A⁽¹⁾2 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

2. Cluster algebras of type A⁽¹⁾₂ 8

2.1. Background on cluster algebras 8

2.2. Algebraic structure of A_P 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 A_P 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 particular class of such algebras called of type A⁽¹⁾₂ 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(y_j : j ∈ J ) is an abelian multiplicative group freely generated by the elements {yj : j ∈ J } (for some set of indices J ) endowed

1

with the auxiliary addition ⊕ given by:

Y

j

y_j^aj ⊕Y

j

y^b_j^j :=Y

j

y_j^min(aj,bj).

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 semi- field 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 F_P 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.

Here we study rank three cluster algebras of type A⁽¹⁾₂ 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

(1) Σ := {H =





0 1 1

−1 0 1

−1 −1 0



, x = {x₁, x₂, x₃}, y = {y₁, y₂, y₃}}.

The algebra A_Pis 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 A_P 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;0 = _y¹

3, y_1;1 = y₁, y_1;2= _y^y1y2

1⊕1

together with the recursive relations:

y1;my1;m+3= y_1;m+2y_1;m+1 (y1;m+1⊕ 1)(y_1;m+2⊕ 1)

for m ≥ 1 and m ≤ −4 (a solution of this recurrence is given in (43)). The cluster variables of A_P are the elements w, z and xm (m ∈ Z) defined as follows:

(2) w := _(y^y2x1+x3

2⊕1)x₂, z := ^y_(y^{1y3x1x2+y1+x2x3}

1y3⊕y₁⊕1)x₁x3

while xm is defined recursively by the exchange relation:

(3) x_mx_m+3 = x_m+1x_m+2+ y_1;m y1;m⊕ 1 . By definition A_P := ZP[w, z, xm : m ∈ Z] ⊂ F_P.

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 F_P, so that F_P ' QP(s1, s2, s3). Such a set C is called a cluster of A_P. The clusters of A_P are

the sets {x_m, x_m+1, x_m+2}, {x_2m+1, w, x_2m+3} and the set {x_2m, z, x_2m+2} for every m ∈ Z. The following figure 1 shows (a piece of) the “exchange graph” of A_P. 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 A_P

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 x_m’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 A_P appears also in [5] as the exchange graph of a coefficient–free cluster algebra of type A⁽¹⁾₂ (i.e. P = 1) and it is called the “brick wall”.

We have already observed that F_P ' QP(s1, s₂, s₃) for every cluster {s₁, s2, s3} of A_P and hence every element of A_P 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 A_Pis 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 A_P are the monomials x^a_mx^b_m+1x^c_m+2, x^a_2m+1w^bx^c_2m+3 and x^a_2mz^bx^c_2m+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.

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

Definition 1.1. Let W := (y₂⊕ 1)w and Z := (y₁y₃⊕ y₁⊕ 1)z. We define elements {un| n ≥ 0} of A_P by the initial conditions:

(4) u0 = 1, u1 = ZW − y1y3− y₂, u2 = u²₁− 2y₁y2y3

together with the recurrence relation

(5) u_n+1= u₁u_n− y₁y₂y₃u_n−1, n ≥ 2.

The following is the main theorem of the paper.

Theorem 1.1. The set

(6) B := {cluster monomials} ∪ {unw^k, unz^k| n > 1, k > 0}

is a canonically positive basis of A_P. It is unique up to rescaling by elements of P.

1.1. Main properties of the elements of B. The elements of the canon- ically positive basis B have some distinguished properties. We start with the connection with the root system of type A⁽¹⁾₂ . By the already mentioned Laurent phenomenon [5, Theorem 3.1] every element b of A_P is a Laurent polynomial in {x₁, x₂, x₃} of the form ^Nb(x1,x2,x3)

x^d1₁ x^d2₂ x^d3)₃ for some primitive, i.e. not divisible by any x_i, polynomial N_b ∈ ZP[x1, x₂, x₃] in x₁, x₂ and x₃, and some integers d1, d2, d3. We consider the root lattice Q generated by an affine root system of type A⁽¹⁾₂ . The choice of a simple system {α₁, α₂, α₃}, with coordinates {e1, e2, e3}, identifies Q to Z³. We usually write elements of Q as column vectors with integer coefficients and we denote them by a bold type letter. The map b 7→ d(b) = (d1, d2, d3)^tis hence a map between A_Pand 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 δ := (1, 1, 1)^t, 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 bijection positive real roots of the root system of type A⁽¹⁾₂ correspond to the set of cluster variables together with the set {u_nw, u_nz| 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 Q_R 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

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 of unw^k and unz^k, 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 decomposition of representations of the acyclic quiver of type A⁽¹⁾₂ . This is because denominator vectors of cluster variables are all the positive real

“Schur roots”, but we do not use this fact here.

•

8• 88 88 88 88 88 88 88 88 88 88 88 8

88 88 88 88 88 88 88 88 88 88 88 88

d(x0)

•

d(x−2)

•

d(x−4)

•

d(x−6)

•

••

•

d(x−2m)

•_d(x

5)

•d(x7)

•d(x9)

••

••

d(x2m+1)

•

(0, 0, 1)

•d(x−1)

•^ll

d(x−3)

•^ll

d(x−5)

•^ll

••

••

d(x−(2m+1))

d(x4)

•^llllllll

llllllllllllllllll

d(x6) ^d(x8•^)d(x10•^)d(x•••^2m••⁾ lllllllllllllllllllllllllllll

•

•

















































(0, 1, 0)

• •d(w)

RRRRRR

RRRRRRRRRRRRRRRRRRRRRRRR

RRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRR

RRRRRRRRRRRRRRRRRRRRRRRRR (1, 0, 0)

d(z)

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

Our next result provides explicit formulas for the elements of B in every cluster of A_P. 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

x_2m+1= P

e e1−e₃ e2−e₃

_m−1−e3

m−1−e1

_e1−1

e3
y^ex^ε(e)+ x^m−1₂ x^2m−2₃



L

e(^e1−e3_e2−e3)(^m−1−e3_m−1−e1)(^e1−1_e3 )^ye⊕1



x^m−1₁ x^m−1₂ x^m−2₃

where ε(e) = (e₂+e₃, m−1−e₁+e₃, 2m−e₂−e₁−2)^t. x2m+2=

P

e e1−1

e3

_m−e2

e1−e2

_m−1−e3

e2−e3
y^ex^ε(e)+x^m₂ x^2m−1₃

L

e e1−1

e3

_m−e2

e1−e2

_m−1−e3

e2−e3
y^e⊕ 1

x^m₁ x^m−1₂ x^m−1₃ where ε(e) = (e₂+e₃, m−e₁+e₃, 2m−1−e₁−e₂)^t. For every n ≥ 1:

un=

yⁿ₁yⁿ₂y₃ⁿx²ⁿ₁ xⁿ₂+xⁿ₂x²ⁿ₃ +P

e e1−e3

e1−e2

h

n−e3

n−e1

_e1−1

e3
+ ^n−e_n−e³⁻¹

1

_e1−1

e3−1

i

y^ex^ε(e) xⁿ₁xⁿ₂xⁿ₃

where ε(e) = (e2+e3, n−e1+e3, 2n−e1−e₂)^t.

In the cluster {x₁, w, x₃} let y₁ := y₁(y₂ ⊕ 1), y₂ := _y¹

2 and y₃ := _y^y2y3

2⊕1

three elements of P (the coefficients of the seed with cluster {x1, w, x₃}).

Then for every m ≥ 0:

x2m+1= P

e1,e3(^m−1−e3_m−1−e1)(^e1−1_e3 )^ye11 y^e3₃ x^2e3₁ w^e1−e3x^2m−2e1−2₃ +x^2m−2₃

L

e1,e3(^m−1−e3_m−1−e1)(^e1−1_e3 )^ye11 y^e3₃ ⊕1



x^m−1₁ x^m−2₃ x2m+2=

P

e(^{m−1−e3+e2}_{m−1−e1+e2})(^e1−1_e3 )(_e2¹)^{yex2e3+1−e2}1 w^e1−e3x2m−2e1−1+e2

3 +x^2m−1₃ (y2x3+x1)

L

e(^{m−1−e3+e2}_{m−1−e1+e2})(^e1−1_e3 )(_e2¹)^ye⊕y2⊕1



x^m₁ wx^m−1₃ For every n ≥ 1:

u_n=

yⁿ₁y₃ⁿx²ⁿ₁ +x²ⁿ₃ +X

e1,e3

h

(^n−e3_n−e1)(^e1−1_e3 )⁺(^n−e3−1_n−e1 )(^e1−1_e3−1) i

y^e₁¹y^e₃³x^2e₁ ³w^e1−e3x^2n−2e₃ ¹ xⁿ₁xⁿ₃

In the cluster {x₁, w, x₃} the expansion of z is given by

z = y1y2y3x²₁+ y1y²₂y3x1x3+ y1y2w + y2x²₃+ x1x3

y1y2y3⊕ y₁y²₂y3⊕ 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, · · · , y_n) at (y1, · · · , yn) ∈ P × · · · × P is the element of P obtained by replacing the addition in Z[z1, · · · , z_n] 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.

Proposition 1.1. Let us first assume P = T rop(y1;C, y_2;C, y_3;C). Let Σ^C = {H^C, C, {y1;C, y2;C,y3;C}} be a seed of F_P and let A_P be a cluster algebra of type A⁽¹⁾₂ with principal coefficients at Σ^C. Then for every element b of B , there exist a (unique) polynomial F_b^C in three variables with constant term 1 and a (unique) integer vector g^C_b ∈ Z³ such that the expansion of b in the cluster C = {s1, s2, s3} is given by:

(7) b = F_b^C(y1;Cs^hC1, y2;Cs^hC2, y3;Cs^hC3)s^gbC,

where h^C_i is the i–th column vector of the exchange matrix H^C.

Let now P be any semifield. Then if b is a cluster monomial then the expansion of b in the seed Σ^C is given by:

(8) b = F_b^C(y_1;Cs^hC1, y_2;Cs^hC2, y_3;Cs^hC3) F_b^C|_P(y_1;C, y_2;C, y_3;C) s^gCb.

If b = u_n, n ≥ 1, then its expansion in the seed Σ^C is still given by (7).

We call F_b^C and g_b^C 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:

in view of (7), the F –polynomial F_b^C of b in the cluster C := {s₁, s₂, s₃} is the polynomial obtained in the following way: consider the tropical semifield P = T rop(y1;C, y_2;C, y_3;C) generated by the coefficients of Σ^C and expand b ∈ A_P 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 the element ˆyi := yis^hCi, i = 1, 2, 3, has degree zero with respect to the principal Z³–grading of A_P (when P = T rop(y1;C, y_2;C, y_3;C)) given by (9) deg(xi) = ei, deg(yi) = −bi, i = 1, 2, 3

(ei is the i–th basis vector of Z³ ). Therefore, by formula (7), every element b of B is homogeneous with respect to such grading and the g–vector g^C_b is its degree.

The g–vector map in the cluster C is by definition the map b 7→ g^C_b 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 A_P as shown by the following proposition.

Proposition 1.2. Given a cluster C of A_P , the map b 7→ g^C_b which sends an element b of B to its g–vector g_b^C in the cluster C, is a bijection between B and Z³

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 g_b of b and its denominator vector in the cluster {x1, x2, x3} d(b) are related by

(10) g_b =

−1 0 0 1 −1 0 1 1 −1

! d(b)

If b is a cluster monomial in the initial cluster {x1, x2, x3} then g_b = −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)

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 cluster variables and of {u_nW^k, u_nZ^k} still form a canonically positive basis of A_P. 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 u_n’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

Σ^cyc:= {

0 −1 2 1 0 −1

−2 1 0

!

, {x₁, w, x₃}, {y₁, y₂, y₃}}

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 cluster variable Z := F_z^{x1,w,x3}|_P(y₁, y₂, y₃)z in the seed Σ^cyc associated to the cluster variable z. Then for every n ≥ 1 the family of rational functions {u_n} defined in definition 1.1 satisfies the initial conditions:

(11) u0 = 1, u1 = Zw − y1y²₂y3− 1, u₂ = u²₁− 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 A_P. 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.

2. Cluster algebras of type A⁽¹⁾₂

2.1. Background on cluster algebras. Let P be a semifield as defined in section 1. Let F_P = QP(x1, · · · , x_n) be the field of rational functions in n independent variables x₁, · · · , x_n. A seed in F_P 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, · · · , d_n) with d_i > 0 for all i such that DB is skew–symmetric; C = {s1, · · · , sn} is an n–tuple of elements of F_P which is a free generating set for F_P so that F_P ' QP(s1, · · · , s_n); 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 F_P it is defined another seed Σk:= {Hk, Ck, yk} by the following mutation rules (see [7]):

(1) the exchange matrix H_k= (h⁰_ij) is obtained from H = (h_ij) by (13) h⁰_ij =

 −h_ij if i = k or j = k

h_ij+ sg(h_ik)[h_ikh_kj]₊ otherwise where [c]+ := max(c, 0) for every integer c;

(2) the new coefficients tuple y_k = {y₁⁰, · · · , y_n⁰} is given by:

(14) y⁰_j :=

( ₁

yk if j = k

y_jy_k^[hkj]+(y_k⊕ 1)^−hkj otherwise.

(3) the new cluster C_k is given by C_k= C \ {s_k} ∪ {s⁰_k} where

(15) s⁰_k:= ykQ

is^[h_i ^ik]+ +Q

is^[−h_i ^ik]+ (y_k⊕ 1)s_k ;

It is not hard to verify that Ck is again a seed of F_P. We say that the seed Σ_k 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 F_P generated by χ(Σ); we denote it by A_P(Σ).

The cluster algebra A_P(Σ) is said to have principal coefficients at Σ and it is denoted by A•(Σ) if P = T rop(y1, · · · y_n).

If the semifield P = T rop(c1, · · · , c_r) is a tropical semifield, the elements of P are monomials in the yj’s. Therefore the coefficient yj of a seed Σ = {H, C, {y₁, · · · , yn}} is a monomial of the form y_j = Qr

i=1c^h_i^n+r,j for some integers h_n+1,j, · · · , h_n+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 h_ij. 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 Σ := {H =

0 1 1

−1 0 1

−1 −1 0

!

, x = {x1, x2, x3}, y = {y₁, y2, y3}}

is equivalent to the seed Σ⁰ := {H⁰=

0 −1 −1 1 0 1 1 −1 0

!

, x⁰ = {x₃, x₁, x₂}, y⁰ = {y₃, y₁, y₂}}

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.

2.2. Algebraic structure of A_P. Let P = (P, ·, ⊕) be a semifield as defined above. Let A_P be the cluster algebra with initial seed

Σ := {H =





0 1 1

−1 0 1

−1 −1 0



, x = {x₁, x₂, x₃}, y = {y₁, y₂, y₃}}.

The following lemma gives the algebraic structure of A_P.

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

Σ_m := {H_m, {x_m, x_m+1, x_m+2}, {y_1;m, y_2;m, y_3;m}}, (16)

Σ^cyc_2m−1 := {H_m^cyc, {x_2m−1, w, x_2m+1}, {y_1;2m−1^cyc , y^cyc_2;2m−1, y_3;2m−1^cyc }}, (17)

Σ^cyc_2m := {H_m^cyc, {x2m, z, x2m+2}, {y^cyc_1;2m, y_2;2m^cyc , y_3;2m^cyc }}

(18)

for every m ∈ Z; they are mutually related by the following diagram of mutations:

(19) Σ^w_2m−1

µ1 //

µ2



Σ^w_2m+1

µ2



µ3

oo

Σ_2m−1

µ1 //

µ2

OO

Σ2m µ1//

µ2



µ3

oo Σ_2m+1

µ1 //

µ3

oo

µ2

OO

Σ_2m+2

µ3

oo

µ2



Σ^z_2m

µ1 //

µ2

OO

Σ^z_2m+2

µ2

OO

µ3

oo

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 Σ_m is not equivalent to the Σ_n if m 6= n, in particular the exchange graph of A_P is given by figure 1 and every cluster C determines a unique seed Σ^C. The exchange matrices Hm and Hm^cyclic are the following:

(20) H_m=





0 1 1

−1 0 1

−1 −1 0



, Hm^Cyc =





0 −1 2 1 0 −1

−2 1 0





for every m ∈ Z. The exchange relations for the coefficient tuples are the following:

(21) y_1;1 = y₁; y_2;1= y₂; y_3;1= y₃ (22) y1;m+1 = ^y_y^2;my1;m

1;m⊕1 ; y2;m+1 = ^y_y^3;my1;m

1;m⊕1 ; y3;m+1 = _y¹

1;m

(23) y1;m−1 = _y¹

3;m; y2;m−1 = y1;m(y3;m⊕ 1); y_3;m−1 = y2;m(y3;m⊕ 1) (24) y_1;m^cyc = _y^y1;m

2;m⊕1; y_2;m^cyc = _y¹

2;m; y^cyc_3;m= ^y_y^3;my2;m

2;m⊕1

(25) y_1;m+2^cyc = y^cyc_3;m(y_1;m^cyc ⊕ 1)²; y_2;m+2^cyc = ^y

cyc 2;my^cyc_1;m

y^cyc_1;m⊕1 ; y_3;m+2^cyc = _ycyc¹ 1;m

(26) y1;m = ^y

cyc 1;my^cyc_2;m

y^cyc_2;m⊕1 ; y2;m= _ycyc¹ 2;m

; y3;m = y_3;m^cyc(y_2;m^cyc ⊕ 1) (27) y_1;m−2^cyc = _y^cyc1

3;m; y_2;m−2^cyc = ^y

cyc 2;my^cyc_3;m

y^cyc_3;m⊕1 ; y_3;m−2^cyc = y_1;m^cyc(y^cyc_3;m⊕ 1)²

The exchange relations are the following:

x_mx_m+3 = x_m+1x_m+2+ y_1;m

y1;m⊕ 1 = y_3;m+1x_m+1x_m+2+ 1 y3;m+1⊕ 1 (28)

wx_2m = y_2;2m−1x_2m−1+ x_2m+1

y_2;2m−1⊕ 1 = x2m−1+ y_2;2m−1^cyc x2m+1

y_2;2m−1^cyc ⊕ 1 (29)

zx_2m+1 = y_2;2mx_2m+ x_2m+2

y2;2m⊕ 1 = x_2m+ y_2;2m^cyc x_2m+2 y_2;2m^cyc ⊕ 1 (30)

x_2m−2x_2m+2 = x²_2m+ y_1;2m−2^cyc z

y_1;2m−2^cyc ⊕ 1 = y_3;2m^cyc x²_2m+ z y_3;2m^cyc ⊕ 1 (31)

x2m−1x2m+3 = x²_2m+1+ y_1;2m−1^cyc w

y_1;2m−1^cyc ⊕ 1 = y_3;2m+1^cyc x²_2m+1+ w y_3;2m+1^cyc ⊕ 1 (32)

Proof of Lemma 2.1. We consider the seeds of A_Pup to a symulataneous re- ordering 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 Σ_m (resp. Σ−m+1) be the seed of A_P 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 Σ_m (resp.Σ−m+1) has the form (16), i.e. its exchange matrix is H_m (resp. H−m+1), its cluster is {x_m, x_m+1, x_m+2} and its coefficient tuple is {y1;m, y2;m, y3;m}. Then it is straightforward to check by induction on m that H_m 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 Σ_m+1 by mutating in direction 3 and then by reordering with the permutation (123).

The central line of the diagram is hence proved.

Let Σ^cyc_m be the seed obtained from Σ_m by the mutation in direction 2.

Suppose that Σ^cycm has the form {Hm^cyc, {xm, cm, xm+2}, {y_1;m^cyc, y^cyc_2;m, y^cyc_3;m}}.

Then it is straightforward to verify the following: that by (13) the exchange matrix Hm^cyc is given by (20); that by (14) the coefficient tuple satisfy (24) and by the exchange relation (15) the cluster variable c_m is given by:

(33) c_m= y_2;mx_m+ x_m+2

(y2;m⊕ 1)x_m+1.

By using (22), (23) and (28), it is straightforward to verify that c_m = c_m+2 for every m ∈ Z. We define w := c1 and z := c₂ and we hence find that Σ^cyc_2m−1 has the form (17) and Σ^cyc_2m has the form (18). Now since two cluster variables can belong to at most two clusters, we conclude that the mutation in direction 1 (resp. 3) of Σ^cycm is Σ^cyc_m+2 (resp. Σ^cyc_m−2).

We have hence proved the diagram (19) and the fact that every cluster determines a unique seed.

It remains to prove that the seeds {Σ_m} (resp.{Σ^cyc_m }) are not equivalent for every m ∈ Z. It is sufficient to prove that xm is different from x1, x2 and

x₃ for every m ≥ 4. In view of the exchange relation (28), the denominator vector of xmin the cluster {x1, x2, x3} satisfies the initial conditions d(x_i) =

−e_i, for i = 1, 2, 3 and

(34) d(x_m+3) + d(x_m) = d(x_m+1) + d(x_m+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 x_m+n (resp. x_2m+n) in the cluster {xm, c, xm+2} for c = w or c = x_m+1, (resp. {x2m, z, x2m+2}) is obtained by the expansion of x_1+n (resp. x_2m+1+n) in the cluster {x₁, c, x₃} (resp. {x_2m+1, w, x_2m+3}) by replacing x₁ with x_m, 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 x_2m+2 and y^cyc_i;2m+1 with y_i;2m^cyc ) for i = 1, 2, 3 and n, m ∈ Z. Moreover the expansion of x−m is obtained from the expansion of x_m+2 by replacing x₁ with x₃, x₃ with x₁ and y₁ with y₃⁻¹, y₂ with y⁻¹₂ and y₃ with y₁⁻¹. 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 A_P. 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 vari- ables different from x1, x2 and x3 are the following: for m ≥ 1

d(x2m+1) =





m−1 m−1 m−2



 d(x2m+2) =

" _m

m−1 m−1

# (35)

d(x−2m+1) =

"

m−1 m m

#

d(x−2m+2) =

"

m−1 m−1 m

# (36)

d(w) =

"

0 1 0

#

d(z) =

"

1 0 1

# (37)

For every n ≥ 1 the denominator vector (in {x1, x2, x3}) of u_n is given by

(38) d(un) =

"

n n n

# .

In particular the denominator vector of all the cluster variables and of all the elements {u_nw, u_nz| n ≥ 1} are all the positive real roots of the root system of type A⁽¹⁾₂ .

Proof. By induction on m, one verifies that such vectors satisfy the re- currence relation (34) together with the initial condition d(x_i) = −e_i for i = 1, 2, 3 (being ei the i–th basis vector of Z³). 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 sum of d(w) and of d(z), i.e. d(u₁) = δ = (1, 1, 1)^t. Then by (5), d(u_n) = nd(u1) = nδ.

The proof follows now by knowing the structure of a root system of type

A⁽¹⁾₂ , recalled in section 1. 

Lemma 2.3. Let d^w(b) be the denominator vector of b ∈ B in the cluster {x₁, w, x₃}. The following formulas hold: for m ≥ 1

d^w(x_2m+1) =





m−1 0 m−2



 d^w(x_2m+2) =





m−1 1 m−2

 (39) 

d(x−2m+1) =

"

m−1 0 m

#

d^w(x−2m+2) =





m−2 1 m−1

 (40) 

d^w(x₂) =

"

0 1 0

#

d^w(z) =

"

1 1 1

# (41)

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

(42) d(un) =

"_n

0 n

# .

We conclude this section by pointing out an important property of de- nominator vectors.

Definition 2.1. [7, Definition 6.12] A collection of vectors in Zⁿ (or in Rⁿ) 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 A_P, 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 clus- ter {x₁, x₂, x₃} of the cluster variables of A_P. 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 F_s|_P at P.

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

(43) y1;m = y^d(xm+3)

Fm+1|_P(y)Fm+2|_P(y).

The family {y_2;m: m ∈ Z} is the sequence of elements of P given by:

(44) y2;1 = y2, y2;0= (y3⊕ 1)y₁, y2;−1 = (y2y3⊕ y₂⊕ 1)_y¹

3.