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Positivity and tameness in rank 2 cluster algebras

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.


Introduction and main results
This note continues the study of the positivity structure for coefficient-free rank 2 cluster algebras initiated in [7,6]. These algebras can be quickly defined as follows: they form a 2-parametric family depending on a pair of positive integers (b, c), and the cluster algebra A(b, c) is the subring of the ambient field F = Q(x 1 , x 2 ) generated by the cluster variables x m for all m ∈ Z, where the (two-sided) sequence of cluster variables is given recursively by the relations (1.1) x m−1 x m+1 = x b m + 1 for m odd; x c m + 1 for m even.
Recall also that the sets {x m , x m+1 } for m ∈ Z are called clusters, and the ambient field F is naturally identified with Q(x m , x m+1 ) for any m ∈ Z. Despite such an elementary definition, we find the structure theory of these algebras rather deep, and some of the natural questions surprisingly difficult. Here is a fundamental result, which is a special case of the Laurent phenomenon discovered and proved in [3,4,1]: every cluster variable is not just a rational function in the two elements of any given cluster but a Laurent polynomials with integer coefficients. The following stronger result is a special case of the results in [1]: where Z[x ±1 m , x ±1 m+1 ] denotes the ring of Laurent polynomials with integer coefficients in x m and x m+1 .
The main focus in our study of the algebras A(b, c) is positivity. Recall that a non-zero element x ∈ A(b, c) is positive at a cluster {x m , x m+1 } if all the coefficients in the expansion of x as a Laurent polynomial in x m and x m+1 are positive. We say that x ∈ A(b, c) is positive if it is positive at all the clusters.
Following [7], we introduce the following important definition.
Recall that in [6] for each (b, c) there was introduced a family {x[a 1 , a 2 ] : (a 1 , a 2 ) ∈ Z 2 } of greedy elements in A(b, c), and it was proved among other things that they are indecomposable positive, and that they form a Z-basis in A(b, c). Taking this into account, it is easy to see that the following conditions on (b, c) are equivalent: (1) Indecomposable positive elements do not form a Z-basis for A(b, c).
(2) Indecomposable positive elements in A(b, c) are linearly dependent.
(3) There exists a non-greedy indecomposable positive element in A(b, c).
(4) There exists a positive element p ∈ A(b, c) whose expansion in the basis of greedy elements has at least one negative coefficient. For instance, to deduce (3) from (4) take any expansion of p into the sum of indecomposable positive elements, and note that it is different from the expansion of p in the basis of greedy elements. Hence at least one of indecomposable positive components of p must be non-greedy.
We see that to prove Theorem 1.2 it suffices to show that in the wild case there always exists an element p satisfying (4). We exhibit such an element explicitly as follows. Theorem 1.3. Suppose that bc > 4, i.e., A(b, c) is wild. Define an element p ∈ A(b, c) as follows: Then p is positive, hence satisfies condition (4) above.
To show that the element p given by (1.3) is positive, we use the group of automorphisms W of A(b, c) introduced in [6]. By the definition, W is generated by the involutions σ ℓ for ℓ ∈ Z, where σ ℓ acts on cluster variables by a permutation σ ℓ (x m ) = x 2ℓ−m . It is easy to see that W is a dihedral group generated by σ 1 and σ 2 (this group is finite if A is of finite type, and infinite otherwise). As shown in [6, Proposition 1.8], the set of greedy elements is W -invariant, and the automorphisms σ 1 and σ 2 act on greedy elements as follows: for all (a 1 , a 2 ) ∈ Z 2 , where we use the standard notation [a] + = max(a, 0). Clearly, W acts transitively on the set of all clusters of A(b, c). Thus the positivity of p is equivalent to the property that σ(p) is positive at the initial cluster {x 1 , x 2 } for every σ ∈ W .
We identify Z 2 with the root lattice associated with the (generalized) Cartan matrix using an unorthodox convention that the simple roots α 1 and α 2 are identified with (0, 1) and (1, 0) respectively. The Weyl group W (A) is a group of linear transformations of Z 2 generated by two simple reflections s 1 and s 2 whose action in Z 2 is given by Comparing this with (1.4) we see that s 1 agrees with σ 1 , and s 2 agrees with σ 2 on Z 2 ≥0 . It is well-known (see e.g., [5]) that the difference between tame and wild cases manifests itself in the appearance and behavior of imaginary roots. According to [5], the set Φ im + of positive imaginary roots can be defined as follows: Q(a 1 , a 2 ) = ca 2 1 − bca 1 a 2 + ba 2 2 . It is known (and easy to check) that: • in the finite type bc < 4 the form Q is positive definite, and so Φ im + = ∅; • in the affine type bc = 4, we have Q(a 1 , a 2 ) ≥ 0 on Z 2 , and Φ im + = {(a 1 , a 2 ) ∈ Z >0 : Q(a 1 , a 2 )) = 0} is the set of integer positive multiples of the minimal imaginary root; • in the wild case bc > 4, the form Q does not vanish on Z 2 , and we have In this note our main interest is in the wild case. In this case we can and will identify the group W of automorphisms of A(b, c) generated by σ 1 and σ 2 with the Weyl group W (A) by identifying σ 1 with s 1 , and σ 2 with s 2 . The above facts imply that Φ im + is W (A)-invariant, and we have the following useful property. Proposition 1.4. In the wild case bc > 4, if σ ∈ W and w ∈ W (A) are identified with each other, and (a 1 , Returning to Theorem 1.3, it is easy to check that all the elements x[a 1 , a 2 ] appearing in the right hand side of (1.3) correspond to positive imaginary roots (see Lemma 4.2). Thus to prove Theorems 1.2 and 1.3 it suffices to establish the following key lemma. Lemma 1.5. In the setup of Theorem 1.3, for every w ∈ W , the element obtained from p by replacing each element x[a 1 , a 2 ] in the right hand side of (1.3) with x[w(a 1 , a 2 )] is positive at the initial cluster {x 1 , x 2 }.
The proof of Lemma 1.5 is carried out in Section 4. All the tools in our proof are already developed in [6]. The two most important ingredients are as follows: • A combinatorial definition of greedy elements in terms of compatible pairs (to be recalled later). The combinatorics of compatible pairs plays a crucial part in our argument. • A precise description of the set of Laurent monomials that appear in the expansion of x[a 1 , a 2 ] in terms of the initial cluster {x 1 , x 2 }, for an arbitrary positive imaginary root (a 1 , a 2 ). Specifically, we show that the upper bound for this set given in [6, Proposition 4.1, Case 6] is exact.

Compatibility and greedy elements in rank 2 cluster algebras
In this section we recall some definitions and results from [6]. Let (a 1 , a 2 ) be a pair of nonnegative integers. A Dyck path of type a 1 × a 2 is a lattice path from (0, 0) to (a 1 , a 2 ) that never goes above the main diagonal joining (0, 0) and (a 1 , a 2 ). Among the Dyck paths of a given type a 1 × a 2 , there is a (unique) maximal one denoted by D = D a 1 ×a 2 . It is defined by the property that any lattice point strictly above D is also strictly above the main diagonal.
Let D = D a 1 ×a 2 . Let D 1 = {u 1 , . . . , u a 1 } be the set of horizontal edges of D indexed from left to right, and D 2 = {v 1 , . . . , v a 2 } the set of vertical edges of D indexed from bottom to top. Given any points A and B on D, let AB be the subpath starting from A, and going in the Northeast direction until it reaches B (if we reach (a 1 , a 2 ) first, we continue from (0, 0)). By convention, if A = B, then AA is the subpath that starts from A, then passes (a 1 , a 2 ) and ends at A. If we represent a subpath of D by its set of edges, then for A = (i, j) and B = (i ′ , j ′ ), we have We denote by (AB) 1 the set of horizontal edges in AB, and by (AB) 2 the set of vertical edges in AB. Also let AB • denote the set of lattice points on the subpath AB excluding the endpoints A and B (here (0, 0) and (a 1 , a 2 ) are regarded as the same point).
[6] For S 1 ⊆ D 1 , S 2 ⊆ D 2 , we say that the pair (S 1 , S 2 ) is compatible if for every u ∈ S 1 and v ∈ S 2 , denoting by E the left endpoint of u and F the upper endpoint of v, there exists a lattice point A ∈ EF • such that One of the main results of [6] is the following combinatorial expression for greedy elements.
where the sum is over all compatible pairs For the purposes of this paper we can view (2.2) as a definition of greedy elements.

extremal pairs and their compatibility
In this section we define extremal pairs and study their compatibility condition. As a consequence, we give a precise description of the set of Laurent monomials that appear in the expansion of x[a 1 , a 2 ] in terms of the initial cluster {x 1 , x 2 }, for an arbitrary positive imaginary root (a 1 , a 2 ).
>0 is a positive imaginary root if and only if a 1 p + a 2 q ≤ a 1 a 2 for every (p, q) ∈ P , in other words, P is contained in the triangle with vertices (0, 0), (a 2 , 0) and (0, a 1 ).
Proof. The latter condition is equivalent to the condition that the vertex (a 1 /b, a 2 /c) lies inside or on the boundary of the triangle, which is equivalent to the inequality a 1 · a 1 /b + a 2 · a 2 /c ≤ a 1 a 2 . This is in turn equivalent to Q(a 1 , a 2 ) ≤ 0, i.e., to the condition that (a 1 , a 2 ) is a positive imaginary root. (1) every u ∈ S 1 precedes every v ∈ S 2 ; (2) a 1 p + a 2 q < a 1 a 2 + a 1 + a 2 .
Proof. The condition (1) is equivalent to the condition that the edge u q is not higher than the lower endpoint of v a 2 −p+1 , that is, ⌊(q − 1)a 2 /a 1 ⌋ ≤ a 2 − p. This inequality is equivalent to (q − 1)a 2 /a 1 − 1 < a 2 − p, thus is equivalent to (2).
Recall that in [6, Proposition 4.1 (6)] we showed that P S[a 1 , a 2 ] is contained in P . Our goal in the rest of this section is to prove the following strengthening of this result.
Proposition 3.4. Assume that (a 1 , a 2 ) is a positive imaginary root. If (p, q) is a lattice point in P and both p and q are positive, then the extremal pair (S 1 , S 2 ) of size (q; p) is compatible.
is the set of all lattice points in the region P .
Proof. Because P S[a 1 , a 2 ] is contained in P , we only need to show that for every lattice point (p, q) in P , the extremal pair (S 1 , S 2 ) of size (q; p) is compatible. If p = 0 or q = 0, the conclusion is immediate. Other cases follow from Proposition 3.4.
The following lemma plays a key role in our proof of Proposition 3.4. Lemma 3.6. Suppose (a 1 , a 2 ) is a positive imaginary root, and (p, q) a pair of positive integers satisfying a 1 p + a 2 q < a 1 a 2 + a 1 + a 2 . Let (S 1 , S 2 ) be the extremal pair of size (q; p) in D. Then (S 1 , S 2 ) is compatible if every horizontal edge u ∈ S 1 with the left endpoint E, and every vertical edge v ∈ S 2 with the top endpoint F satisfy at least one of the following inequalities: Equivalently, the extremal pair (S 1 , S 2 ) is compatible if for every 1 ≤ p ′ ≤ p, 1 ≤ q ′ ≤ q, at least one of the following inequalities holds: Proof Without loss of generality, we assume (3.1), that is, f (EF ) < 0. As A moves through the lattice points along D from E to the lower endpoint of v, f (AF ) either decreases by b, stays constant, or increase by 1 at each step; moreover, it starts with the negative value f (EF ) and end at the positive value b. Thus there exists A ∈ EF • such that f (AF ) = 0. Since the same argument works for every u and v, we conclude that (S 1 , S 2 ) is compatible.
Proof of Proposition 3.4. It is then easy to check that (p, q) lies in P if and only if at least one of the following two conditions hold: Without loss of generality we assume (3.5) holds. Thanks to Lemma 3.6, to show the compatibility of (S 1 , S 2 ), it suffices to prove Note that H 1 is the half plane below the line passing through (1, a 1 − bp + 1) with slope m 1 = (ba 2 − a 1 )/a 2 > 0, and H 2 is a half plane above a line with slope m 2 = a 1 /(ca 1 − a 2 ) > 0. Also note that m 1 > m 2 because Q(a 1 , a 2 ) < 0. Thus to show that R ⊂ H 1 ∪ H 2 , it suffices to show that (1, a 1 − bp + 1) ∈ H 2 (as illustrated in Figure 1). But this is exactly the statement of (3.5).

Proofs of main resuls
In this section we prove Theorems 1.2 and 1.3. As discussed in Section 1, it is enough to prove Lemma 1.5, which will be our goal.
Due to obvious symmetry, we can and will assume in the rest of this Section that   For (a 1 , a 2 ) ∈ Z 2 , we define a sequence (a(k)) k≥−1 with initial data (a 1 , a 2 ) as follows: let a(−1) = a 2 , a(0) = a 1 , and for k > 0 recursively define We define sequences (α(k)), (β(k)), (γ(k)) with initial data given in the table: The next step is to show that all components of the element p in (1.3) correspond to positive imaginary roots. Thus we show the following.  c) is a sum of two non-positive terms. Furthermore, we see that the equality b + c − bc = 0 is achieved only when b = c = 2, which is not a wild case. So in the wild case we have a strict equality b+c−bc < 0, and therefore Q(1, 1) < 0, Q(b+1, bc−c+1) < 0.
As a consequence, for p defined in (1.3), Proof. Straightforward induction on k.
Proof of Lemma 1.5. We only show that w(1; r)(p) is positive at {x 1 , x 2 } (recall that p is defined in (1.3)), since the treatment of w(2; r)(p) is completely similar and will be left to the reader.
First, note that O obviously lies strictly inside the region bounded by Next, we claim that M also lies strictly inside the same region. Indeed, it suffices to show that L ′ M ′ M is anti-clockwise oriented, or equivalently, the following determinant is positive, which is obviously the case. Finally, the area of the triangle KLK ′ is Thus KLK ′ is anti-clockwise oriented. Moreover, since the area is less than 1/2, Pick's theorem asserts that there is no lattice points P other than K and K ′ that lie inside or on the boundary of the triangle KLK ′ (otherwise the triangle KP K ′ , which is contained in the triangle KLK ′ , would have area at least 1/2). Therefore every lattice point other than K in the region bounded by OKLMO must be in the region bounded by Lemma 4.9. Let k, p, q ≥ 0, p < α(k − 1). If the extremal pair (S 1 , S 2 ) of size (q; p) is compatible in D α(k)×α(k−1) , then the extremal pair (S ′ 1 , S ′ 2 ) of size (q + α(k); p + α(k + 1)) is compatible in D β(k)×β(k−1) .
Proof. It follows immediately from Proposition 3.4 and Lemma 4.8.
For any horizontal or vertical edge u, we denote E u = the left/lower endpoint of u, F u = the right/upper endpoint of u.
We introduce some notation (see Figure 3) T be the intersection of the line CH ′ with the diagonal O ′ P ′ with O ′ = (0, 0) and P ′ = (β(k), β(k − 1)). Let BG ′ H ′ C be the path obtained by taking the union of the edge BG ′ , the subpath G ′ H ′ , and the edge H ′ C. Moreover, we denote B by B k (and we use the subscript k in a similar fashion for other letters) if we need to specify the dependence on k.  (1) The points B, C are above O ′ P ′ , and B ′ has coordinates (α(k), α(k− 1) − 1) and is below O ′ P ′ , for all k ≥ 0; G ′ is below O ′ P ′ for all k ≥ 1; H ′ is below O ′ P ′ for k ≥ 1 in the case min(b, c) > 1, and for k ≥ 2 in the case c = 1.
(2) The path BG ′ H ′ C has the same shape as D α(k)×α(k−1) for k ≥ 1 in the case min(b, c) > 1, and for k ≥ 2 in the case c ≥ 1.
We will prove by contradiction by assuming that (S ′ 1 , S ′ 2 ) is not compatible. Then by Lemma 4.11, there exist u ′ ∈ S ′ 1 and v ′ ∈ S ′ 2 such that f S ′ 2 (AF v ′ ) > 0 and There are four cases to be considered.
(In other words, (S 1 ,S 2 ) is obtained from (S 1 , S 2 ) by moving forward all horizontal edges after u i in S 1 so that they become immediately after u i , moving backward all horizontal edges after u i in S 2 to the northeast corner of the Dyck path, and adding all horizontal edges in front of u i to S 1 and removing all vertical edges in front of u i in S 2 .) We claim that (S 1 ,S 2 ) is compatible. To see this, let We shall show that (W 1 , V 2 ) is compatible, i.e., every u j ∈ W 1 and v ℓ ∈ V 2 are separated by (W 1 , V 2 ). First, consider the case j ≤ i. Since (S 1 , S 2 ) is compatible, Next, consider the case j > i. Since (W 1 , V 2 ) and (S 1 , S 2 ) coincides on the subpath E u j F v ℓ , a lattice point A that satisfies (2.1) for (S 1 , S 2 ) also satisfies (2.1) for (W 1 , V 2 ). Since |W 1 | = |S 1 |, |V 2 | =S 2 , Proposition 3.4 implies that (S 1 ,S 2 ) is compatible.
Thus we have a compatible extremal pair (S 1 ,S 2 ) and an incompatible extremal pair (S ′ 1 ,S ′ 2 ) which contradict Lemma 4.9. Case 3: v ′ is on B ′ C ′ but u ′ is not. The proof of this case is similar to Case 2.
Case 4: neither u ′ nor v ′ is on B ′ C ′ . We let (S ′ 1 ,S 2 ) be the extremal pair of size (|S ′ 1 |; |S ′ 2 |) (which is obtained from (S ′ 1 , S ′ 2 ) by moving all horizontal edges in B ′ C ′ ∩ S ′ 1 forward so that they are immediately after B ′ , and move all vertical edges in B ′ C ′ ∩ S ′ 2 backward so that they are immediately in front of C ′ ). Then (S ′ 1 ,S ′ 2 ) is not compatible because for any Similarly define (S 1 ,S 2 ) to be the extremal pair of size (|S 1 |; |S 2 |). Then by Proposition 3.4, (S 1 ,S 2 ) is compatible. This again contradicts Lemma 4.9.
Since in all the four cases we get contradictions, (S ′ 1 , S ′ 2 ) must be compatible. This completes the proof of Lemma 4.7.