A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non-acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra.


Introduction
Cluster algebras were introduced by Fomin and Zelevinsky in [7]. A (skew-symmetrizable) cluster algebra A is a subalgebra of a rational function field with a distinguished set of generators, called cluster variables, that are generated by an iterative procedure called mutation. By construction cluster variables are rational functions, but it is shown in loc. cit. that they are Laurent polynomials with integer coefficients. Moreover, these coefficients are known to be non-negative for skew-symmetric cluster algebras [11].
Each cluster algebra A also determines an upper cluster algebra U, where A ⊆ U [4]. It is believed, especially in the context of algebraic geometry, that U is better behaved than A (for instance, see [5,9]). Matherne and Muller [12] gave a general algorithm to compute generators of U. Plamondon [14,15] obtained a (not-necessarily positive) formula for certain elements of skew-symmetric upper cluster algebras using quiver representations. However a directly computable and manifestly positive formula for (non-trivial) elements in U is not available yet.
In this paper we develop an elementary formula for upper cluster algebra elements in terms of Dyck paths. These elements have positive coefficients. For an equioriented quiver of type A, these elements form a canonical basis [3]. For an acyclic cluster algebra, we conjecture that these elements form a basis. Using this formula, we show that A = U for all non-acyclic skew-symmetric cluster algebras of rank 3.
The paper is organized as follows. In Section 2 we review definitions of cluster algebras and upper cluster algebras. Section 3 is devoted to the construction of a family of elements in U, and Section 4 to the proof of A = U for non-acyclic rank 3 skew-symmetric cluster algebras.

Cluster Algebras and Upper Cluster Algebras
A square integer matrix B = (b ij ) is said to be sign-skew-symmetric if either b ij = b ji = 0, or else b ji and b ij are of opposite sign; in particular, b ii = 0 for all i. Furthermore, B is said to be skew-symmetric if b ij = −b ji for all i, j. If there exists a diagonal matrix D with positive integer entries d i such that DB is skew-symmetric then B is said to be skew-symmetrizable.
Let (P, ⊕, ·) be a semifield, that is an abelian multiplicative group with a binary operation of auxiliary addition ⊕ which is commutative associative, and distributive with respect to the multiplication in P. The group P is torsion free, so we will use its group ring ZP as the ground ring for our cluster algebra. As an ambient field for our cluster algebra, we take a field F which is isomorphic to the field of rational fractions in m variables with coefficients in QP.
One choice for P that we will use in this paper is the tropical semifield. Let Trop(u 1 , . . . , u m ) be a multiplicative abelian group freely generated by the u j . We define ⊕ in Trop(u 1 , . . . , u m ) by When we choose P = Trop(u 1 , . . . , u m ) we say that the cluster algebra is of geometric type. It is also important to note that in this case ZP is the ring of Laurent polynomials in the variables u i .
A seed Σ = (x, y, B) is a triple where x = {x 1 , . . . , x n } is an n-tuple of elements of F that form a free generating set over QP, y = {y 1 , . . . , y n } is an n-tuple of elements in P and B is an n × n integer sign-skew-symmetric matrix. We call x a cluster, y a coefficient tuple, B the exchange matrix, and the elements of a cluster cluster variables. The rank of the seed is the number of columns of the exchange matrix, or equivalently the number of the cluster variables in a cluster.
For any integer a, let [a] + := max(0, a). Given a seed (x, y, B) and a specified index 1 ≤ k ≤ n, we define mutation of (x, y, B) at k, denoted µ k (x, y, B), to be a new seed If B ′ is also sign-skew-symmetric, we say that the mutation is well-defined. Note that welldefined mutation is an involution, that is mutating (x ′ , y ′ , B ′ ) at k will return our original seed (x, y, B).
Two seeds Σ 1 and Σ 2 are said to be mutation-equivalent or in the same mutation class if Σ 2 can be obtained by a sequence of well-defined mutations from Σ 1 . This is obviously an equivalence relation. A seed Σ is said to be totally mutable if every sequence of mutations from Σ consists of well-defined ones.
Definition 2.1. Given a totally mutable seed (x, y, B), the cluster algebra A(x, y, B) is the subring of F generated by where the union runs over all seeds (x ′ , y ′ , B ′ ) that are mutation-equivalent to (x, y, B). The seed (x, y, B) is called the initial seed of A(x, y, B).
Any seed in the same mutation class will generate the same cluster algebra up to isomorphism.
For any n × n sign-skew-symmetric matrix B, we associate a directed graph Q B with vertices 1, . . . , n and arrows from vertex i to vertex j for b ij > 0. We call B acyclic if there are no oriented cycles in Q B . We say the cluster algebra A(x, y, B) is non-acyclic if and only if no matrix in its mutation class is acyclic.
Definition 2.2. Given a cluster algebra A, the upper cluster algebra U is defined as where x runs over all clusters of A.
In certain cases it is sufficient to consider only the clusters of the initial seed and the seeds that are a single mutation away from it, rather than all the seeds in the entire mutation class. Definition 2.3. A seed (x, y, B) is coprime if the columns of B are pairwise linearly independent. A cluster algebra is totally coprime if every seed is coprime.
For a cluster x, U x is defined as the intersection in ZP(x 1 , . . . , x n ) of the n + 1 Laurent rings corresponding to x and its one-step mutations.
Theorem 2.4. [4,13] We have A ⊆ U ⊆ U x . Moreover, (ii) If A is totally coprime, then U = U x for any seed (x, y, B). In particular, this holds when A is of geometric type and the matrixB (defined in §3) has full rank.

Construction of some elements in the upper cluster algebra
In this section, we consider a cluster algebra A of geometric type. Let n be the rank of A, and m be an integer such that m ≥ n. Assume the coefficient semifield is i . Equivalently, the exchange relation is given by Since y is determined byB, we also denote Let (a 1 , a 2 ) be a pair of nonnegative integers. Let c = min(a 1 , a 2 ). The maximal Dyck path of type a 1 × a 2 , denoted by D = D a 1 ×a 2 , is a lattice path from (0, 0) to (a 1 , a 2 ) that is as close as possible to the diagonal joining (0, 0) and (a 1 , a 2 ), but never goes above it. A corner is a subpath consisting of a horizontal edge followed by a vertical edge.  Definition 3.4. Let a = (a 1 , a 2 , ..., a n ) ∈ Z n , a n+1 = · · · = a m = 0.
We label D (i,j) with the corner-first index (Definition ), whose horizontal edges are denoted u are locally compatible with respect to D (i,j) for all (i, j) ∈ E; (ii) Definex[a] to be the Laurent polynomial where the sum runs over all globally compatible collections (abbreviated GCCs).
Next, we give an equivalent definition ofx[a] based on the following simple observation: with (i, k) ∈ E determines the rest, and they are determined by a sequence if and only if s i,r = 1.
We define the complement of s i to bē It is easy to check that Definition 3.4 is equivalent to the following.
Definition 3.5. Let a = (a 1 , a 2 , ..., a n ) ∈ Z n , a n+1 = · · · = a m = 0. (ii) Definex[a] to be the Laurent polynomial Definition 3.6. Let a = (a 1 , a 2 , ..., a n ) ∈ Z n . For n + 1 ≤ i ≤ m, let ℓ i be the lowest degree of x i in all the monomials in ( n i=1 x a i i )x[a]. Definex red [a] to be the Laurent polynomial If we further assume b ≥ c, theñ (b) For a non-acyclic seed,x[a] is less interesting for certain choices of a: take n = m = 3, a = (1, 1, 1) andB Remark 3.8. For the cluster algebra with an acyclic seed (x, y, B), it is known [4, Theorem 1.16] that the set of monomials in x 1 , ..., x n , x ′ 1 , ..., x ′ n which contains no product of the form x j x ′ j is a basis. This is called a standard monomial basis. Let x[a] be defined in the same way asx[a] in Definition 3.4, except that we drop the local compatibility condition (i.e. drop (1) but keep (2)-(4)). Then x[a] are precisely the standard monomial basis elements.
Remark 3.9. The local compatibility condition in Definition 3.4(1) is weaker than compatibility given in [10]. In particular, not all cluster variables are of the formx[a] even for rank 2 cluster algebras.  For each a = (a 1 , a 2 , ..., a n ) ∈ Z n ,x[a] are in the upper cluster algebra; moreover if the initial seed is acyclic, is not divisible by x k for any 1 ≤ k ≤ n. As a consequence, the same holds forx red [a].
Proof. We first show that if the initial seed is acyclic and 1 ≤ k ≤ n, then ( n i=1 x a i i )x[a] is not divisible by x k , or equivalently, there exists s = (s 1 , . . . , s m ) ∈ S gcc such that is not divisible by x k . So we need to find s such that |s j | = 0 if (k, j) ∈ E, and |s i | = 0 if (i, k) ∈ E. Such an s can be constructed as follows: since the initial seed is acyclic, E determines a partial order generated by "i ≺ j if (i, j) ∈ E". Define To check that s is in S gcc , note that if s i ·s j = 0 for some (i, j) ∈ E, then there exists r ≤ [a j ] + that s i,r = 1, s j,r = 0. By our choice of s, we have k ≺ i, k ≺ j. This contradicts with i ≺ j.
Next, we show thatx[a] is in the upper cluster algebra.
Thenx[a] is the coefficient of z 0 in the following polynomial in Meanwhile, denote f k (a) = (f k (a 1 ), . . . , f k (a n )) ∈ {0, 1} an . Thenx[f k (a)] is the coefficient So we conclude that Sox[a] is in the upper cluster algebra by Lemma 3.11 (ii).
can be written as is in the upper cluster algebra.
Proof. (i) By Definition 3.5, (ii) We introduce n extra variables x m+1 , . . . , x m+n . Let P ′ = Trop(x n+1 , . . . , x m+n ). Then the group ring ZP ′ is the ring of Laurent polynomials in the variables x n+1 , . . . , x m+n . LetB ′ = B I n be the (m + n) × n matrix that encodes a new cluster algebra A ′ . Assume a n+1 = · · · = a m+n = 0 and define E ′ , S ′ ,x ′ [a] for the coefficient semifield P ′ similarly as the definition of E, S,x[a] for the coefficient semifield P. Sõ We will show thatx ′ [a] is in the upper bound n ] (where x = {x 1 , . . . , x n } and for 1 ≤ k ≤ n, the adjacent cluster x k is defined by is in the upper cluster algebra U ′ , thanks to the fact that U ′ x = U ′ whenB ′ is of full rank [4, Corollary 1.7 and Proposition 1.8]. Then we substitute x m+1 = · · · = x m+n = 1 and conclude thatx[a] is in the upper cluster algebra U.
k ] when a k = 0. So we may assume a k = 1. Let N ⊂ S ′ contains those s such that P s is not divisible by x k ; equivalently, Then it suffices to show that s∈N P s is divisible by A, where A is the binomial in the exchange relation It is easy to check that ϕ is a well-defined bijection. Thus
The following statement follows from [6, Theorem 1.2, Lemma 2.1] for a coefficient free cluster algebra, and it is easy to see that it holds more generally for a cluster algebra of geometric type.  Let (x, y, z) ∈ R 3 . We define a partial ordering "≤" on R 3 by (x, y, z) ≤ (x ′ , y ′ , z ′ ) if and only if x ≤ x ′ , y ≤ y ′ , and z ≤ z ′ .
Proof. It follows easily from Theorem 4.1. To see (iii): if it is false, then there are 1 ≤ j < j ′ ≤ l such that But this cannot hold because of (ii) and Theorem 4.1.

4.2.
Grading on A. We now adapt the grading introduced in [8, Definition 3.1] to the geometric type.  x, y, B) is a seed of rank n, and (ii) G = [g 1 , . . . , g n ] T ∈ Z n is an integer column vector such that BG = 0.
Set deg G (x i ) = g i and deg(x −1 i ) = −g i for i ≤ n, and set deg(y j ) = 0 for all j. Extend the grading additively to Laurent monomials hence to the cluster algebra A(x, y, B). In [8] it is proved that under this grading every exchange relation is homogeneous and thus the grading is compatible with mutation. In the rest of the paper we assume that B is a 3 × 3 skew-symmetric non-acyclic matrix.
The following two propositions come from the work in [1] to show that rank three nonacyclic cluster algebras have no maximal green sequences.  In other words, if τ (B) = (a ′ , b ′ , c ′ ) then deg( any  seed (x, y, B). Notice that if we look at the quiver associated to our exchange matrix we see that the grading we chose gives that the degree of the cluster variable associated to vertex i is the number of arrows on the edge not incident to i. Furthermore, there is a canonical grading for a non-acyclic rank three cluster algebras since regardless of your choice of initial seed the grading imposed on A is the same.

4.3.
Construction of an element in U \A. Here we shall prove the following main theorem of the paper. where the last inequality follows from the computation in the previous case. This completes the proof of the claim that all cluster variables, except possibly x 1 , x 2 , x 3 , z 3 , have degree strictly larger than ac − b − a.
Therefore it is sufficient to check that Y cannot be written as a linear combination of products of the cluster variables x 1 , x 2 , x 3 , and z 3 since all other cluster variable have larger degree than Y . This is clear as the numerator of Y is irreducible and none of these cluster variables have a factor of x 1 in the denominator.